Partial Differential Relations ||

Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge . Band 9

A Series of Modern Surveys in Mathematics

Editorial Board

E. Bombieri, Princeton S. Feferman, Stanford N. H. Kuiper, Bures-sur-Yvette P. Lax, New York R. Remmert (Managing Editor), Munster w. Schmid, Cambridge, Mass. J-P. Serre, Paris 1. Tits, Paris

~ildhaelCJronnov

Partial Differential Relations

Springer-Verlag Berlin Heidelberg GmbH

Mikhael Gromov

Institute des Hautes Etudes Scientifiques 35, route de Chartres F-91440 Bures-sur-Yvette France

Mathematics Subject Classification (1980): 53, 58

ISBN 978-3-642-05720-5

Library of Congress Cataloging in Publication Data Gromov, MikhaeI, 1943-Partial differential relations. (Ergebnisse der Mathematik und ihrer Grenzgebiete; 3. Folge, Bd. 9) Includes index. 1. Geometry, Differential. 2. Differential equations, Partial. 3. Immersions (Mathematics) 1. Title. II. Series: Ergebnisse der Mathematik und ihrer Grenzgebiete; 3. Folge, Bd. 9. QA641.G76 1986 515.3'53 86-13906 ISBN 978-3-642-05720-5 ISBN 978-3-662-02267-2 (eBook) DOl 10.1007/978-3-662-02267-2

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 ofthe German Copyright Law where copies are made for other than private use a lee is payable to "Verwertungsgesellschaft Wort", Munich.

© Springer-Verlag Berlin Heidelberg 1986 Originally published by Springer-Verlag Berlin Heidelberg New York in 1986 Softcover reprint of the hardcover 1st edition 1986

Typesetting: ASCO, Hong Kong

2141/3020-543210

Foreword

The classical theory of partial differential equations is rooted in physics, where equations (are assumed to) describe the laws of nature. Law abiding functions, which satisfy such an equation, are very rare in the space of all admissible functions (regardless of a particular topology in a function space).

Moreover, some additional (like initial or boundary) conditions often insure the uniqueness of solutions. The existence of these is usually established with some apriori estimates which locate a possible solution in a given function space.

We deal in this book with a completely different class of partial differential equations (and more general relations) which arise in differential geometry rather than in physics. Our equations are, for the most part, undetermined (or, at least, behave like those) and their solutions are rather dense in spaces of functions.

We solve and classify solutions of these equations by means of direct (and not so direct) geometric constructions.

Our exposition is elementary and the proofs of the basic results are selfcontained. However, there is a number of examples and exercises (of variable difficulty), where the treatment of a particular equation requires a certain knowledge of pertinent facts in the surrounding field.

The techniques we employ, though quite general, do not cover all geometrically interesting equations. The border of the unexplored territory is marked by a number of open questions throughout the book.

I am grateful to my friends and colleagues with whom I have discussed various aspects of the subject in the course of years. The book took final shape under unrelenting criticism by Nico Kuiper directed at earlier drafts. I thank Mme v. Houllet for typing the manuscript, Mari Anne Gazdick for rectifying my English and Mme J. Martin for the help with a multitude of last minute corrections.

Bures-sur-Yvette, May 1986 M.Gromov

Contents

Part 1. A Survey of Basic Problems and Results 1

1.1 Solvability and the Homotopy Principle. . . . 1

1.1.1 Jets, Relations, Holonomy . . . . . . . . . 1 1.1.2 The Cauchy-Riemann Relation, Oka's Principle and the Theorem of

Grauert . . . . . . . . . . . . . . . . . . . . . . .. 4 1.1.3 Differentiable Immersions and the h-Principle of Smale-Hirsch . 6 1.1.4 Osculating Spaces and Free Maps. . . . . . . . . . . . .. 8 1.1.5 Isometric Immersions of Riemannian Manifolds and the Theorems

of Nash and Kuiper. . . . . 10

1.2 Homotopy and Approximation 13

1.2.1 Classification of Solutions by Homotopy and the Parametric h-Principle. . . . . . . . . . . . . . . . . 13

1.2.2 Density of the h-Principle in the Fine Topologies 18 1.2.3 Functionally Closed Relations . . 22

1.3 Singularities and Non-singular Maps 26

1.3.1 1.3.2

1.4

1.4.1 1.4.2 1.4.3 1.4.4

Singularities as Differential Relations 26 Genericity, Transversality and Thorn's Equisingularity Theorem. 30

Localization and Extension of Solutions. . . . . . . . . .. 35

Local Solutions of Differential Relations . . . . . . . . .. 35 The h-Principle for Extensions; Flexibility and Micro-flexibility 39 Ordinary Differential Equations and "Zero-Dimensional" Relations 44 The h-Principle for the Cauchy Extension Problem . . . . . . 46

Part 2. Methods to Prove the h-Principle 48

2.1

2.1.1 2.1.2 2.1.3 2.1.4 2.1.5

Removal of Singularities . . . . . . .

Immersions and k-Mersions V ---+ IRq for q > k . Immersions and Submersions V ---+ W . . . . Folded Maps yn ---+ W q for q ~ n . . . . . . Singularities and the Curvature of Smooth Maps Holomorphic Immersions of Stein Manifolds . .

48

48 52 54 61 65

VIII Contents

2.2 Continuous Sheaves 74

2.2.1 Flexibility and the h-Principle for Continuous Sheaves . 75 2.2.2 Flexibility and Micro-flexibility of Equivariant Sheaves. 78 2.2.3 The Proof of the Main Flexibility Theorem . 80 2.2.4 Equivariant Microextensions . 84 2.2.5 Local Compressibility and the Proof of the Microextension

Theorem. 87 2.2.6 An Application: Inducing Euclidean Connections 93 2.2.7 Non-flexible Sheaves 98

2.3 Inversion of Differential Operators . 114

2.3.1 Linearization and the Linear Inversion. 114 2.3.2 Basic Properties of Infinitesimally Invertible Operators . 117 2.3.3 The Nash (Newton-Moser) Process 121 2.3.4 Deep Smoothing Operators 123 2.3.5 The Existence and Convergence of Nash's Process. 131 2.3.6 The Modified Nash Process and Special Inversions of the

Operator ~ 139 2.3.7 Infinite Dimensional Representations of the Group Diff(V) 145 2.3.8 Algebraic Solution of Differential Equations 148

2.4 Convex Integration 168

2.4.1 Integrals and Convex Hulls. . . . . . . . 168 2.4.2 Principal Extensions of Differential Relations 174 2.4.3 Ample Differential Relations 180 2.4.4 Fiber Connected Relations and Directed Immersions. 183 2.4.5 Directed Embeddings and the Relative h-Principle . 189 2.4.6 Convex Integration of Partial Differential Equations . 194 2.4.7 Underdetermined Evolution Equations. 195 2.4.8 Triangular Systems of P.D.E .. 198 2.4.9 Isometric C1-Immersions. 201 2.4.10 Isometric Maps with Singularities . · 207 2.4.11 Equidimensional Isometric Maps · 214 2.4.12 The Regularity Problem and Related Questions in the Convex

Integration. · 219

Part 3. Isometric CCXl-Immersions ...... . · 221

3.1 Isometric Immersions of Riemannian Manifolds · 221

3.1.1 Nash's Twist and Approximate Immersions; Isometric Embeddings into Rq . . . . . . . . . . . . . . . . . . . . . . 221

3.1.2 Isometric Immersions V" ---+ wq for q ~ (n + 2) (n + 5)/2 . . 224 3.1.3 Convex Cones in the Space of Metrics. . . 231 3.1.4 Inducing Forms of Degree d> 2 . . . . . 232 3.1.5 Immersions with a Prescribed Curvature. . 235

Contents IX

3.1.6 Extensions of Isometric Immersions . . . . . . . . . 240 3.1.7 Isometric Immersions V" -+ wq for q ~ (n + 2) (n + 3)/2 . 247 3.1.8 Isometric Cylinders V" x lR -+ wq for q ~ (n + 2) (n + 3)/2 250 3.1.9 Non-free Isometric Maps. . . . . . . . 254

3.2 Isometric Immersions in Low Codimension 259

3.2.1 Parabolic Immersions . . . . . . . . . 260 3.2.2 Hyperbolic Immersions . . . . . . . . 269 3.2.3 Geometric Obstructions to Isometric C2-Immersions V 2 -+ lR3 279 3.2.4 Isometric Coo-Immersions V 2 -+ lRq for 3 ~ q ~ 6 . . . . 289

3.3 Isometric Coo-Immersions of Pseudo-Riemannian Manifolds 306

3.3.1 Local Pseudo-Riemannian Immersions. . . . . . . . . 307 3.3.2 Global Immersions . . . . . . . . . . . . . . . . . 312 3.3.3 Immersions with a Prescribed Curvature and the C1-Approximation 316 3.3.4 Isotropic Maps and Non-unique Isometric Immersions. . 321 3.3.5 Isometric Coo-Immersions V" -+ wq for q ~ [n(n + 3)/2J + 2 324

3.4

3.4.1 3.4.2 3.4.3 3.4.4

Symplectic Isometric Immersions . . .

Immersions of Exterior Forms . . . . Symplectic Immersions and Embeddings Contact Manifolds and Their Immersions Basic Problems in the Symplectic Geometry

References. .

Author Index.

Subject Index

327

328 333 338 340

350

359

361

Part 1. A Survey of Basic Problems and Results

1.1 Solvability and the Homotopy Principle

1.1.1 Jets, Relations, Holonomy

Consider a COO-smooth fibration p: X --+ V and let Xl') be the space of r-jets (of germs) of smooth sections f: V --+ X. Denote by p': Xl') --+ V and p:: X(s) --+ Xl') for s > r ~ 0, where X(O) ~ X, the natural projections. The rlh-order jet (or r-jet) of a C-smooth section f: V --+ X is denoted by J;: V --+ Xl'). A section cp: V --+ Xl') is called holonomic if it is the r-jet of some C-section f: V --+ X. This f, if it exists, is unique. Namely f = p~ 0 cp, since p~ 0 J; = f for all sections f: V --+ X. Thus, sections f: V --+ X are identified with holonomic sections V --+ Xl') by fHJ;.

Recall basic properties of jets which, in fact, uniquely define the fibration Xl') --+ V and the jet operator f H J;.

(a) The space X(1) consists of the linear maps L: T,,(V) --+ T,,(X) for all x e X and v = p(x) e V, such that Dp 0 L = Id: T,,(v);:::> , where T,,(v) and T,,(x) are the respective tangent spaces. Here Dp: T(X) --+ T(V) denotes the differential of the map p and Id stands for the identity map. Thus, the fibration pt,: X(l) --+ X carries a natural structure of an affine bundle with fibers X~l) ~ IRnq, x e X, where n = dim V and q = dim X - dim V = dim Xv for the fiber Xv = p-l(V) C X, ve V.

The I-jet of a smooth section f: V --+ X sends each point ve V into the fiber X~l) c X(l) for x = f(v)eX, by J}(v) = DI : T,,(v) --+ T,,(X).

(b) The space X(,+1), for all r = 0, 1,2, ... , is naturally embedded into (X('»(l) as it consists ofthe I-jets of the smooth holonomic sections V --+ Xl'). The operator J,+l on sections V --+ X is the composition of J' with the I-jet operator on sections V --+ Xl').

(b/) The fibration p;+1: X(,+l) --+ Xl') carries a natural structure of an affine bundle over Xl') which makes it a subbundle in (X('»)<1) --+ Xl'), whose fiber X~+l) c

X(,+l) X e Xl') has dim X(,+l) = q ( (n + r)! ) for n = dim V and q = dim X . , , x (n _ 1)!(r + I)! ' v

(c) Take a trivial (split) subfibration Y = U x Wo c X over an open subset U c V, for an open subset Wo in the fiber Xvo C X over some Vo e U, and identify y(') --+ Y with the restriction X(,) I Y = (p~rl(y) --+ Y. If Ul' ... , un and Yl' ... , Yq are local coordinates in U and in Wo respectively, then sections U --+ y(') c Xl') are

2 1. A Survey of Basic Problems and Results

, , '( n(n + 1) gIVen by dr-tuples of functIOns U --+ IR for dr = dIm X~~ = q 1 + n + 2 +

, " + ' ,such that the holonomic sections Ji: U --+ y(r) are represented by (n + r - 1)') (n - 1)!r!

strings of partial derivatives,

r _ ( ofll 02fll orfll) Jf - f ll,-, "'" ,

au; OU;,OU;2 OUt" ,., oU;r

for J1 = 1,2, ... , q and i = 1,2, ... , n, where the functions fll = fll(u t ,···, un) represent sections f: U --+ Y c X.

Furthermore, the (trivial) fibration y(r) --+ U admits a (non-unique) holonomic splitting, y(r) = U X IRdr, such that the submanifold U x z c y(r) is the image of a holonomic section Ji: U --+ y(r) for some Coo -section f = fz: U --+ Y c X, and for all Z E IRdr. This may be achieved, for example, with the space Pr = Pr(lRn --+ IRq) ~ IRdr of maps f: U --+ Wo whose components f ll , J1 = 1, ... , q, are polynomials in U t , ... , Un

of degree ~ r, as the (tautological) map U x Pr --+ y(r) for (u,f) f--+ (u,Ji(u)) is a diffeomorphism. In fact, each Coo-section fo: U --+ X admits a split neighborhood y = U x Wo => fo(U) in X, such that fo(U) = U x wo, for Wo = (O,O, ... ,O)E Woo Thus every holonomic section Jio: U --+ x(r) admits a holonomically split neighborhood (for example y(r) = U x Pr) in x(r).

Definition. A differential relation (or condition) imposed on sections f: V --+ X is a subset fll c x(r), where r is called the order of fll. A C' -section f is said to satisfy (or to be a solution of) fll if the jet J'{: V --+ x(r) maps V into fJIl. Thus solutions of fll are (naturally identified with) holonomic sections V --+ fll c x(r).

Example. Let 'P: x(r) --+ IR be a continuous function. Then the relation imposed by the (zero) set fll = {X E x(r) I 'P(x) = O} is expressed by the (partial differential) equation 'P(JJ) = 0, that is

in local coordinates.

Solving a relation fJIl c x(r) can be done in two stages. We start with the topological problem of constructing a section V --+ fll. The topology can be hard, but even so this first stage looks analytically easy. For example, when fJIl is repre-

sented by the equation 'P(:~t , ... , :~J = 0, then sections V --+ fJIl correspond to

solutions of 'P( CPt, ... ,CPn) = ° with arbitrary functions cP; in place of the partial derivatives of/au;.

Our real problem appears at the second stage, when we try to pass from an arbitrary section V --+ fll to a holonomic one. The most optimistic expectation is expressed in the following

1.1 Solvability and the Homotopy Principle 3

Homotopy Principle. We say that fJl satisfies the h-principle and (or) that the h-principle holds for (obtaining) solutions of fJl if every continuous section V ---+ fJl is homotopic to a holonomic section V ---+ fJl by a continuous homotopy of sections V ---+ f7l.

If X is a trivial fibration, X = V x W ---+ V, then sections V ---+ X correspond to maps V ---+ Wand the h-principle for maps V ---+ W which satisfy f7l, by definition, refers to the h-principle for sections V ---+ X.

Remarks on CS-Solutions for s = r + k. Let us lift (or prolong) a given relation fJl c x(r) to a relation fJl1 c x(r+1) by mimicking the differentiation of an equation (like P(u;,f(u;), af/au;, ... ) = ° in the variables u;) as follows:

Let fJl' c (x(r»)(1) consist of the I-jets of germs of C1-sections V ---+ fJl and put fJl1 = f7l' n x(r+1) for the canonical embedding x(r+1) c (x(r»)(1). Then repeat this and define fJlk = (fJlk-1)1 C x(r+k) for all k = 1, 2, .... Now, the h-principle for CS-solutions of fJl for s = r + k, by definition, refers to the h-principle for (solutions of) fJlk.

Call fJl stable if fJlk+1 C x(r+k+1) is an affine subbundle in the bundle x(r+k+1) ---+

x(r+k) restricted to fJlk c x(r+k) for all k = 1, 2, .... Observe that the intersection fJl~+l = fJlk+1 n (p;tt+1 )-1 (x), x E f7lk typically is an affine subspace in the fiber Xr+k+1) = (p;tt+1r1(x), (since the top derivatives appear linearly as one differentiates a differential equation) but the dimension dim fJl~+1 need not be constant in x. The stability of fJl amounts to the constancy of dim fJl~+l and to the continuity of the subspace f7l~+1 c Xr+k+1) in x E fJlk.

If fJl is stable, then every section <p: V ---+ f7llifts to a section <p: V ---+ fJlk, for all k = 1, ... , since the projection fJlk ---+ fJl is a fibration with contractible fibers, and the lift is unique up to a homotopy of sections V ---+ fJlk.

The h-Principle for COO(CaO)-Solutions of q{. This h-principle claims, by definition, the stability of fJlk for some integer k ~ 0, as well as the possibility to homotope an arbitrary continuous section V ---+ fJlk to a holonomic Coo(Can)-section, where can stands for the real analyticity (which makes sense if the fibration X ---+ V is can).

Examples. (a) If fJl c x(r) is an open subset (relation), then it is obviously stable since fJlk = (p;+kr1(fJl) for all k = 1,2, .... In fact, the content of the h-principle for an open relation f7l is independent of the smoothness class cs:?r of solutions. Indeed, every C-solution fo: V ---+ X can be finely (see 1.2.2) C-approximated by COO-sections f: V ---+ X. These necessarily satisfy fJl when fJl c x(r) is open.

(b) If fJl is stable and if every C-solution of fJl is (known to be) COO-smooth, then, of course, there is no need to specify the smoothness of solutions for stating the h-principle.

In general, however, the stability of fJl insures no immediate passage from C to Cr +1.


1.1.2 The Cauchy-Riemann Relation, Oka's Principle and the Theorem of Grauert

Let the manifolds V and X be endowed with complex analytic structures and let p: X --+ Ybea complex analytic fibratli0n. The Cauchy-Riemann relation is the subset fJ/ c X(l) which consists of the complex linear maps T,,(V) --+ T,,(X) in X(l). The solutions of fJ/ are (well known to be) exactly the holomorphic (or complex analytic) sections V --+ X. The projection fJ/ --+ X is clearly an affine bundle (whose fibers fJ/x,

x E X, have dimlR fJ/x = tdimlR xii) = 2nq for 2n = dimR V and 2q = dimlR Xv, v E V). Hence every continuous section V --+ X lifts to a section V --+ fJ/ which is unique up to a homotopy. Therefore the h-principle for fJ/ reduces to the following.

Oka's Principle. Every continuous section V --+ X is homotopic to a holomorphic one.

Remarks. (a) One can easily see the Cauchy-Riemann relation to be stable. In fact, the projection p::f: fJ/k+1 --+ fJ/k is an affine bundle (with ck-dimensional fibers for

( (n+ k)!) . k Ck = 2q (n _ 1)!(k + I)! . Thus, passmg from fJ/ to fJ/ does not change the content

of the h-principle. (b) Oka's principle is older than the h-principle and it is rarely stated in a precise

form. Furthermore, one does not speak of Oka's principle unless the underlying manifold V is Stein. That is (according to one of several equivalent definitions) V is biholomorphic to a complex,analytic submanifold in eN for some N = N(V). [In fact, the greatest integer N:s; t(dime V + 1) will do, see 2.1.5.]

,Examples. (a) Every non-singular complex algebraic subvariety V c eN clearly is Stein.

(b) A Riemann surface V (i.e. dime V = 1) is Stein if and only if it is open (i.e. it contains no compact connected component). For instance, open subsets V eel are Stein (see Gunning and Rossi 1965).

(c) The tangent space T(Vo) of an arbitrary smooth manifold Vo admits a complex analytic structure which is Stein (Cartan 1957).

(d) Submanifolds in Stein manifolds:and Cartesian products of Stein manifolds, are obviously Stein.

(e) No compact manifolds of positive dimension is Stein by Liouville's theorem. The complement of a non-empty compact subset in en is non-Stein for n ~ 2 by the following

(A) Lefschetz Theorem. Every Stein manifold V admits a homotopy retraction onto a cell sUbcomplex K c V which has dim K = t dimlR V = dime V.

Proof Holomorphically embed V c; eN and observe that the Euclidean distance v ~ dist(zo, v) for a fixed generic (see 1.3.2) point Zo E eN, is a Morse function on V c eN with non-degenerate critical points. [This is, in fact, true for all smooth properly embedded submanifolds V c ~2N, see Milnor (1963).] Since the restriction


of dist(xo, v) to every complex submanifold V' c: V of dime V' = 1 assumes no local maximum on V~ by Liouville's theorem, the critical points of dist(xo, v) on V have Morse indices :::; dime V. Hence, the Morse subcomplex K c: V, which is a homotopy retract of V, satisfies dim K :::; dime V. Q.E.D. [See Narasimhan (1967) for a similar theorem for singular varieties.]

(A') Exercises. (a) Generalize (A) to complex submanifolds V c: Z where Z is a complete simply connected Kahler manifold of non-positive sectional curvature. [In fact, V and Z are known to be Stein under these assumptions, see Green-Wu (1979).]

(b) Take complex submanifolds V and V' in eN, such that the subset {(v, v') E

V x V'ldist(v, v'):::; const} is compact for all const ~o. Assume dime V + dime V' ~ N and show the intersection V n V' to be non-empty.

(b') Generalize (b) to submanifolds in a complete Kahler manifold Z of nonnegative sectional curvature for N = dime Z.

(c) Take an arbitrary (Stein or not) complex manifold V and let f: V -+ eN be a holomorphic map. Show for a fixed generic point Zo E en that the critical set of the function v~ [dist(zo,J(v))]2 on V lies in a disjoint union of "fibers" f- 1(z;) c: Vfor some points Zl' ... ' Z;, ... in eN. Let m = sUPzeudimef-1(z) and approximate the function [dist(zo,J(v))]2 by a function d' on V whose critical points are nondegenerate with the indices :::; 2m + dime V. Assume the map f is proper and contract V onto a subcomplex K c: V, such that dim K :::; 2m + dime V.

(d) Holomorphically map two complex manifolds into eN by f: V -+ eN and f': V -+ eN. Then, consider the function E(v, v') = [dist(f(v),J'(v'))]2 on V x V' and let Uo = (vo, v~) be a non-degenerate critical point of E. Show that index(uo) :::; dime V x V'. Assume E(uo) > 0 and prove that

index(uo) ~ dim V x V' - N + 1. e

(e) Let Z be a complete Kahler manifold of non-negative sectional curvature and let f: V -+ Z be a holomorphic map. Denote by 0/1 the space of triples u = (v, v', IX)

where v, v' E V and IX is a smooth path in Z betweenf(v)andf(v'). That is IX: [0,1] -+ Z is a smooth map for which IX(O) = f(v) and 1X(1) = f(v'). Define the (energy) functional E: 0/1 -+ IR+ by

E(u) = E(IX) = J: II ~: r dt,

for the Riemann-Kahler norm II II in T(z), and let Uo E 0/1 be a non-degenerate critical point of E, such that E(uo) > o. Show that

index(uo) ~ 2 dim V - dimZ + 1, e e

by combining (*) with the Morse index comparison theorem (see Milnor 1963). (e') Let the sectional curvature of Z be strictly positive. Assume V to be compact

and the map f to be finite-to-one. Show, by applying Morse theory to a small perturbation E' of E, that the homotopy groups ofthe pair (0/1, 0/10)' for 0/10 = E-1(0), satisfy 11:;(0/1,0/10) = 0 for i = 0, 1, ... , k = 2 dime V - dime Z. Identify 0/10 with the subset {(v, v') E V x Vlf(v) = f(v')}, and prove the following vanishing theorem of


Lefschetz-Barth-Larsen (see Barth 1975) for holomorphic embeddings V c:; Z = CpN,

'lti(Z, V) = 0, for 0::; i::; 2 dim V - N + 1. e

(B) The h-Principle of Grauert. Let G be a complex Lie group, take a complex analytic subgroup H c G and consider a complex analytic fibration X --+ V with the structure group G and with the (homogeneous) fiber G/H.

(B') Theorem (Grauert 1957). If V is Stein, then every continuous section V --+ X can be homo toped to a holomorphic section. In particular, every continuous map V --+ G/H is homotopic to a holomorphic one.

[See Cartan (1958) and Ramspot (1962) for the proof.]

Exercises. (a) Let M be the multiplicative group of non-vanishing holomorphic functions on V and let E c M consist of the functions ef for all holomorphic functionf: V --+ C. Show the factor group M/E to be isomorphic to the cohomology group Hl(V;Z) by applying (B') to maps V --+ C\{O}.

(b) Apply (B') to holomorphic maps V --+ CpN and construct a non-singular complex subvariety Vo c V of co dime Vo = 1 whose fundamental class is the Poincare dual of a given cohomology class ho E H2(V; Z).

(c) Construct with (A) and (B') k linearly independent holomorphic tangent vector fields on V, provided 2k ::; n + 1 for n = dime V.

(d) Let W be a Riemann surface and let V be an arbitrary (Stein or not) complex manifold. Assume the fundamental group 'lt1(W) to be non-Abelian and show hoi om orphic maps V --+ W to violate the h-principle, unless Hl (V; Z) = O.

(e) Let V be a compact (hence, non-Stein) Riemann surface. Show holomorphic maps V --+ cPq, q ;:::.: 1, to satisfy the h-principle if and only if 'lt1 (V) = 0 (i.e. V ~ Cpl).

(f) Consider a holomorphic map f of a compact complex manifold V to the complex torus Cq/Z 2q• Prove the induced homomorphism H 1(CQ/Z 2q ; IR) --+ Hi(v, IR) has even rank. [This contradicts the h-principle if q ;:::.: 1 and Hl(V; IR) # 0.]

We shall return to holomorphic maps in 2.1.5.

1.1.3 Differentiable Immersions and the h-Principle of Smale-Hirsch

A C1-map f: V --+ W is called an immersion ifrankf~ rankDf = dim V everywhere on V. For example, if dim W = dim V, then immersions V --+ Ware exactly locally diffeomorphic maps.

The pertinent jet space X(l), for X = V x W --+ V, consists of the linear maps T.,(V) --+ Tw(W) for all (v, W)EX. The immersion relation J c: X(1) is fibered over X by the projection X(1) --+ X and the fiber J x , x = (v, w) E X = V x W consists of the injective linear maps in X~l) = Hom(T.,(V) --+ Tw(W)). Now, sections V --+ J corre-

spond to fiberwise injective homomorphisms cp: T(V) -+ T(W), while holonomic sections are differentials DJ : T(V) -+ T(W) of immersions f: V -+ W

The subset J c X(l) is empty for dim W < dim V and thus trivially satisfies the h-principle. If dim W ~ dim V, then J is an open dense subset in X(l). This does not, however, insure the existence of immersions or (and) the validity of the h-principle. In fact, if V is a closed (i.e. compact without boundary) manifold and dim W = dim V then the h-principle may fail to be true. Moreover, if W is an open manifold (i.e. no connected component of W is a closed manifold), then no (equidimensional!) immersion V -+ Wexists at all (see 2.1.3) and the relation J violates the h-principle in so far as it admits a section V -+ f

The h-principle (in the parametric form, see 1.2.1) for immersions sn -+ [Rq,

q ~ n + 1, was discovered by Smale (1958, 1959) and the theory was completed by the following

(A) Theorem of Hirsch. Immersions V -+ W satisfy the h-principle in the following two cases:

(i) Extra dimension: dim W > dim V (Hirsch 1959). (ii) Critical dimension: dim W = dim V and the manifold V is open (Hirsch 1961).

See 2.1.1, 2.2.2, and 2.4.3 for three different proofs.

(B) Examples and Corollaries. A manifold V is called para lie liz able if its tangent bundle is trivial, T(V) = V x [Rn -+ V for n = dim V.

(B 1) If the manifolds V and Ware parallelizable, then in cases (i) and (ii), every continuous map V -+ W is homotopic to an immersion. In particular, every open parallelizable manifold V admits an immersion V -+ [Rn for n = dim V.

Proof. Since the bundles T(V) and T(W) are trivial, the fibration J -+ X = V x W is also trivial. Hence, every section V -+ X lifts to J and the h-principle applies.

(B~) Let G be a Lie group and let F c G be a discrete subgroup. The manifold V = GIF obviously is parallelizable (and it can be made open, if necessary, by deleting a point from each component of V or by multiplying V by [Rl). Yet, it seems unlikely that anybody can immerse these manifolds into [Rn+l (or into [Rn if they are open) bypassing Hirsch's theorem. (See p. 43 in Kirby-Siebenmann for a geometric immersion of the punctured n-torus into [Rn.)

(Bn Let V be an open manifold which admits a possibly non-complete Riemannian metric of constant negative curvature. Then there exists a finite parallelizable covering V -+ V (see Deligne-Sullivan 1975) which admits by (B 1) an immersion V -+ [Rn,

n = dim V = dim V.

(B 2) A manifold V is called stably parallelizable if V x [R is a parallelizable manifold. For example, every orientable hypersurface V C [Rn+1 is stably parallelizable since a small (tubular) neighborhood U C [Rn+1 of V is diffeomorphic to V x [R and since open subsets U in parallelizable manifolds (like [Rn+1) obviously, are parallelizable. Moreover, a manifold V (obviously) is stably parallelizable if and only if there is a

continuous map g: V --+ sn, such that the induced bundle g*(T(sn)) over V is isomorphic to T(V). (One may use the tangential Gauss map for hypersurfaces in IRn+l.)

(B~) Every n-dimensional stably parallelizable manifold V admits an immersion V --+ IRn+1.

Proof Construct a fiberwise homomorphism T(V) --+ T(sn) c T(lRn+l) and apply (i). Alternatively, apply (ii) to the (open!) manifold V x IR.

(B~) Exercises. (a) Assume V and W to be stably parallelizable and find with (i) an immersion in a given homotopy class of maps V --+ W for dim W > dim V.

(b) Derive (i) of Hirsch's theorem from (ii) which applies to the total space ofthe quotient bundle f*(T(W))/T(V) for the map f: V --+ W underlying a given fiberwise injective homomorphism T(V) --+ T(W).

(B3) (Whitney 1944). Every n-dimensional manifold V admits an immersion V --+ 1R2n-1.

Proof The obstruction to the existence of a section V --+ J can be easily identified in this case with the normal Stiefel-Whitney class Wn of V. This is zero by the Thom-Wu formulae for Wi.

Whitney's result is sharp for n = 2\ k = 1, 2, .... For example, the projective space IRpn, n = 2\ admits no immersion into 1R2n- 2, since the normal class Wn- 1 (lRpn) "# 0 for n = 2k. [See the surveys by Lannes (1982) and Cohen (1984) for the contemporary state of art.]

1.1.4 Osculating Spaces and Free Maps

Consider a C2-map f: V --+ IRq and fix local coordinates Ul' ... , Un in V near a given point v E V. Denote by 1?(V, v) c Tw(lRq) = IRq, W = f(v), the subspace spanned by

the vectors aaf (v) and aa2fa (v) in IRq for 1 ~ i,j ~ n. This subspace (obviously) is Ui Ui Uj

independent of the choice of U i and it is called the (second) osculating space of the map f [The first osculating space is D/Cf,,(V)) c Tw(lRq).] The dimension of 1?(V, v) can vary between zero and tn(n + 3). Call f free if dim 1/(V, v) = tn(n + 3) for all VE V.

The freedom relation ff c X(2),for X = V x IRq --+ V, is fibered over the immersion relation J c X(l) by the projection X(2) --+ X(l). The fiber over each point is (naturally isomorphic to) the Stiefel manifold Stm IRq, m = n(n + 1)/2, ofm-frames of independent vectors in IRq-n. If q < n(n + 3)/2 then ff is empty; otherwise it is an open dense subset in X(2) which is invariant under the natural actions of diffeomorphisms of V and of affine transformations of IRq. Moreover, the group Diff V x Aff IRq acts transitively on ff. In fact, ff is the only subset in X(2) with these properties.

Exercise. Classify open subsets in X(2) for X = V x IRn+1 --+ V, which lie over J c X(l) and which are invariant under DiffV x AfflRn+1.


There are relatively few natural examples of free maps. The simplest one is the map f: ~n _ ~q for q = !n(n + 3) given by the q monomials Xi and XiXj on ~n for 1 ::s; i ::s; j ::s; n. A more interesting example is provided by the Veronese map f: sn -~q+1, q = !n(n + 3), defined by the monomials xixj on ~n+1 :::l sn, 1 ::s; i ::s; j ::s; n + 1. The image f(sn) is diffeomorphic to ~pn lying in a hyperplane H ~ ~q C ~q+1, since Li~t xt = 1 on sn C ~n+1. Thus one obtains a free (Veronese) embedding of ~p" into ~q for q = !n(n + 3). [One does not know of a single closed manifold V besides S" and ~pn which can be freely mapped into ~q for q = !n(n + 3). Yet, see Wintgen (1978) for interesting specific examples.]

Exercise. Find a linear transformation of ~q :::l f(S") which makes the Veronese map f: S" - ~q isometric (compare 1.1.5) and which sends f(S") into a round sphere Sq-l c ~q.

(A) The h-Principle for Free Maps. If either q ~ !n(n + 3) + 1, (compare the extra dimension case of Hirsch's theorem) or q = !n(n + 3) and V is an open manifold (the critical dimension), then free maps V - ~q satisfy the h-principle. In particular, every stably parallelizable manifold admits a free map V - ~q for q = !n(n + 3) + 1, where n=dimV.

See 2.2.2 for the proof for open manifolds V and 2.2.1 and 2.4.3 for the extra dimension case.

Questions. Do free maps of closed manifolds V - ~q for q = !n(n + 3) and n ~ 2 satisfy the h-principle? Does every parallelizable manifold (e.g. the torus Tn, n ~ 2) admit a free map into this ~q?

Generalizations. The notion of freedom can be extended to higher order jets. For example, third order free maps SI _ ~3 are curves with nowhere vanishing curvature and torsion. Pohl conjectured the h-principle for the kth-order free maps V - ~q and the answer is positive for open manifolds V and in the extra dimensional case

( 1 n(n + 1) (n + k - 1)!) h h . . I . .. I q ~ + n + 2 + ... + k!(n _ 1)! . T e -pnnclp em one cnhca case,

namely concerning third order free maps of the circle into ~3, is also true by the work of Little (1971) (compare Hamenstadt (1986)].

Freedom makes sense also for maps into general Riemannian manifolds, where it is expressed by independence of covariant derivatives of various orders. Our state of knowledge here is approximately the same as for maps into ~q. The extradimensional second order case for n = 1 is due to Feldman (1968,1971). Closed free curves in S2 were classified by Little (1970). The techniques of 2.2.2 and 2.4.3 yield the h-principle for open and extra dimensional cases.

Exercises. (a) Study free maps S1 _ ~2 (that are closed immersed curves in ~2 with nowhere vanishing curvature) and decide which sections S1 - !IF are homotopic to holonomic sections (that are free maps S1 _ ~2).

(b) Show with (A) that every (orientable or not) surface admits a free map into ~6. (c) Show every n-dimensional manifold V to admit a free map V -+ ~q for

q = !n(n + 5) and find topological obstructions for q < !n(n + 5).

1.1.5 Isometric Immersions of Riemannian Manifolds and the Theorems of Nash and Kuiper

Let g be a Riemannian CO-metric on V, that is a continuous field of Euclidean structures gv in T,,(v), v E V, and let ~q always have its standard Euclidean metric. The isometric immersion relation Ja = Jag C X(l) for X = V x ~q -+ V consists of the isometric linear embeddings (T,,(V),gv) -+ Tw(~q) = ~q for all (v, W)E V X ~q.

Solutions of Ja are called isometric immersions V -+ ~q. These are indeed (differential) immersions, since the immersion condition J C X(l) (properly) contains Ja. In fact, Ja is a subfibration of J -+ X whose fiber Jax c J x , x = (v, w) E V X ~q = X, can be identified with the Stiefel manifold St~ ~q c Stn ~q of orthonormal n-frames in ~q for n = dim V, where Stn ~q ~ J x is the manifold of independent frames. Hence, Ja is a fiberwise homotopy retract of J and every section V -+ J can be homotoped to a (unique up to a homotopy) section V -+ Ja.

The condition Ja can be described geometrically by observing that a C1-map f: V -+ ~q is an isometric immersion if and only if it preserves the Riemannian length of the smooth curves in V. Alternatively, one can express the isometry property of D,: T(V) -+ T(~q), and off itself, in local coordinates Ul' ••• , Un in V by the following system of !n(n + 1) (that is the codimension of Ja c X(1» non-linear partial differential equations (P.D.E.) of the first order

(Ja) \:~;' :~) = gij' 1 ~ i ~j ~ n.

The unknown vector-function f has q components fl' ... , hand < , > denotes the scalar product

t of,. iJfk. k=l oUi oUj

The functions gij are the components of the metric tensor of V. Namely gij =

/ :l0 . ' :liJ .) . Analytically speaking, our problem is to solve (Ja) for a given positive \uu, uUJ 9 symmetric matrix of functions gij = gij(v) on V. This is achieved in 2.4.9 with the following

(A) Theorem. Isometric C1-immersions V -+ ~q satisfy the h-principle for all Riemannian manifolds V = (V, g) and for all q > dim V.

This fact is an immediate corollary of the h-principle of Hirsch for the extra dimension and of the following striking theorem discovered by Nash (1954) for q ;?: n + 2 and extended to q = n + 1 by Kuiper (1955).

(A') An arbitrary differentiable immersion fo: V --+ ~q admits a C1-continuous homotopy of immersions fr: V --+ ~q, t E [0,1], to an isometric immersion f1: V --+ ~q.

Remarks and Corollaries. (a) One obtains with the examples in (B) of 1.1.3 and with either (A) or (A') an impressive list of isometric C1-immersions V --+ ~q.

(b) The Theorems (A) and (A') remains true for isometric C1-immersions of V into a given convex open subset W c ~q. Thus, for example, the unit sphere sn C

~n+1 admits an isometric C1-immersion into an arbitrary small ball in ~n+1 for all n = 1,2, .... This result of Kuiper (1955) disproved a long standing conjecture which had claimed every isometric C1-immersion f: sn --+ ~n+1 for n ~ 2 to be congruent to the unit sphere. (If f is C2-smooth, then it is congruent to the unit sphere sn C ~n+1 by the classical rigidity theory).

Remark. An interesting (non-differential) functional equation whose solution is also sensible to the smoothness assumption is presented in Hilbert's 13th problem on superpositions of functions. A simple (counting parameters, compare 2.3.8) argument shows a generic Coo -function in k variables to be no· superposition of Coo-functions in k - 1 variables. Yet, every CO-function in k ~ 3 variables is a superposition of CO-functions in two variables (Kolmogorov 1956; Arnold 1957).

(c) The existence of an isometric C1-immersion of an arbitrarily small (yet, non-empty!) neighborhood U c V into ~n+1 is a non-trivial geometric phenomenon (discovered by Nash and Kuiper). This has no counterpart in the Smale-Hirsch theory where the emphasis is laid upon global immersions as the local (nonisometric) ones exist by the very definition of a smooth manifold. In fact, the local solution of an arbitrary open differential relation 91 c Xl') (e.g. J c X(l») is a trivial matter. But ifcodim91 = k > 0 [e.g. codimJo = tn(n + 1)], then the solutionsf of 91 satisfy a certain P.D.E. system ~(J;) = 0, i = 1, ... , k (where the functions ~ on Xl') have 91 for their zero set) and the local solvability of 91 becomes a non-trivial analytic problem. However, the relations J and Jo share some common geometric features (concerning their convex hulls in X(l») which allow us to treat them on an equal footing in 2.4.

(d) There is no meaningful h-principle for equidimensional isometric immersions. However, every n-dimensional Riemannian manifold admits a continuous (nonsmooth!) map V --+ ~n which preserves the Riemannian length of the smooth curves in V (see 2.4.11).

Exercise. Let W c ~q be an open subset bounded by a smooth closed hypersurface in ~q, and let Sl be the circle with the standard metric d02 on Sl. Prove the h-principle for isometric immersions Sl --+ W, provided W is simply connected and q ~ 2. Find counterexamples to this h-principle for 11:1 (W) 1= O.

(B) COO-Immersions (V, g) --+ ~q. Let the metric g on V be Coo-smooth and consider the lift (see 1.1.1) Jo' c X(,+l) of the relation Jo = JOg c X(1). If q < tn(n + 1) and if r ~ ro for some ro = ro(n), then Jo' is empty for generic (see 1.3.2) Coo-metrics g (see 2.3.8). Therefore, most COO-manifolds V admit no isometric COO-immersions into

IRq for q < !n(n + 1). It is unknown if a small neighborhood of each point VE V admits an isometric Coo-immersion into IRq, for q = !n(n + 1); however, if (V, g) is real analytic, such a local Can-immersion does exist by a theorem of Janet (1926) (see 3.1.6).

The relation Jo c X(l) is never stable (see 1.1.1) for n ~ 2 and q ~ n + 1. However, the lift Jor may (or may not) become stable for large r.

Exercises. (a) Prove the above mentioned non-stability of Jo by invoking Gauss theorema egregium (see 3.1.5).

(b) Prove the stability of Jo for n = 1 and for all q ~ 1. Then study q = nand n = 1,2, ....

(c) Let n = 2, q = 3 and show Jo; c X(3) to be stable for metrics g on V whose Gauss curvature is either everywhere positive or everywhere negative on V. Find a Coo-metric g on V for which Jo~ is non-stable for all r = 1,2, ....

(B l ) Free Isometric Immersions. Lift Jo to Jl c X(2), intersect it with the freedom relation (see 1.1.4), put ffJo = ff n Jo l c X(2) and observe that solutions of ffJo are those maps f: V ~ IRq which are both free and isometric. This ffJo is yet unstable; however, the lift (ffJo)1 c X(3) is stable for all Riemannian metrics g (see 3.1.6). Moreover, for every continuous section <Po: V ~ ff there exists a unique up to homotopy continuous section <PI: V ~ (ffJo)1 whose projection to ff is homotopic to <Po (see 3.1.6).

Question. Do free isometric Coo -immersions satisfy the h-principle for n ~ 2? (If n = 1 then the answer is, obviously, "no", for q = 2, and it is an easy "yes" for q ~ 3.)

The positive answer for q ~ !(n + 2)(n + 3) is given in the following

(B2) Theorem. Let (V,g) be a Ck-smooth Riemannian manifold for k = 5, 6, ... , 00, an. Then free isometric Ck-immersions V ~ IRq satisfy the h-principle for q ~ !(n + 2)(n + 3), where n = dim V.

This is proven in 3.1.7 by deforming a free immersion fo: V ~ IRq into a free isometric one [by using auxiliary isometric immersions V x 1R2 --.:. IRq which require q ~ !(n + 2)(n + 3)].

Remarks and Corollaries. (a) The relation ff c X(2) fibers over X = V x IRq with the fiber ~ Stm IRq, m = !n(n + 3). Hence, ff admits a section V ~ ff for q ~ !n(n + 5) and every two sections are homotopic for q ~ !n(n + 5) + 1. This extends with the above discussion to (ffJo)', r ~ 1, thus showing the h-principle (B2) to be equivalent to the following

(Bz) Existence Theorem. Every Ck-manifold (V, g) admits a free isometric Ck-immersion into IRq, q = !(n + 2)(n + 3), for all k = 5, 6, ... , 00, an.

(b) The principle possibility of the realization of Riemannian manifolds in IRq was established by Nash (1956). He proved that a compact manifold of class Ck,


k = 3, 4, ... , 00, can be isometrically and freely immersed into ~q for q = 1n(n + 9) and for q = !n(n + l)(n + 9) in the non-compact case. Ten years later Nash (1966) solved the problem for compact Can-manifolds.

The methods invented by Nash have not lost their importance although his values for q have been improved since then. In fact, a slight modification of those provides Ck-immersions into ~q for q = n2 + 5n + 3 [including k = 3,4 not covered by (Bz)] of all (compact or not) Ck-manifolds V. Yet, one does not know whether every Riemannian C2-manifold admits an isometric C2-immersion into some ~q.

1.2 Homotopy and Approximation

1.2.1 Classification of Solutions by Homotopy and the Parametric h-Principle

Two solutions, fo and fl of a relation ~ c Xl'), are called C'-homotopic if they can be joined by a C'-continuous homotopy of solutions ft: V ~ X, 0 :::;; t :::;; 1, where a homotopy of sections it: V ~ X is, by definition, C'-continuous, if and only if the homotopy ofthejets Jlt: V ~ Xl') is continuous. [This is equivalent to the continuity of the map t~ it of [0,1] into the space of C'-sections V ~ X with the ordinary C'-topology, see (C) below.] We say that ~ and (or) solutions of ~ satisfy the one-parametric h-principle if the existence of a continuous homotopy of sections ({Jt: V ~ ~, 0 :::;; t :::;; 1, between ({Jo = Jlo and ({Jl = Jlt is sufficient (it is, obviously, necessary) for a C' -homotopy of solutions between fo and fl for all pairs of solutions fo and fl of ~.

The one-parametric h-principle holds true for the relations studied in 1.1.2-1.1.5 under the same assumptions that are used for the ordinary h-principle (with the same references to the proofs). For instance, [compare (B) in 1.1.2] holomorphic maps fo and fl of a Stein manifold V into a complex homogeneous space G/H can be joined by a homotopy of holomorphic maps ft: V ~ G/H if and only if there is a homotopy of continuous maps between fo and fl.

(A) Regular Homotopies. This is a name for C1-continuous homotopies of immersions it: V ~ W (immersions are often called regular maps). Let us assume V to be an n-dimensional stably parallelizable manifold and let W = ~n+l. Then any splitting of the tangent bundle T(V x ~1) = (V X ~1) X ~n+l (obviously) induces some splitting of the immersion relation, oF = V X ~n+1 X Stn ~n+l, which provides (with the one-parametric h-principle!) a one-to-one correspondence between the regular homotopy classes of immersions V ~ ~n+1 and the homotopy classes of maps V ~ St~ ~n+1 = SO(n + 1).

Examples. (A1) Immersions Sl ~ ~2. Here SO(2) = Sl and maps Sl ~ Sl are classified by degree d = ... , - 2, -1, 0, -1, .... This implies the following


Theorem of Whitney (1937). Every immersion Sl --+ 1R2 is regularly homotopic either to the figure 00 (which corresponds to d = 0), or to the immersion z f-+ Zd of the unit circle Sl c C into C = 1R2 if d # O.

Exercise. Give a direct geometric proof of this Theorem.

(A2) Immersions S2 --+ 1R3. The orthogonal group SO(3) is doubly covered by S3 and therefore 1t2(SO(3)) = 1t2 (S3) = O. Hence, every immersion S2 --+ 1R3 is regularly homotopic to the standard embedding S2 c; 1R3. In particular, the reflection in the center of the round sphere S2 c; 1R3 can be homotoped to the original embedding by a regular homotopy of immersions S2 --+ 1R3, and in the course of such a homotopy the inward looking normal field on S2 turns outward. This is the famous

Smale's "Paradox". The sphere S2 c 1R3 (unlike Sl c 1R2) can be turned inside out.

(A~) Immersions V 2 --+ 1R3. Since 1tl(SO(3)) = 7L/27L and 1t2(SO(3)) = 0, immersions of an oriented surface V --+ 1R3 are classified by the cohomology group Hi (V; 7L/27L). This gives 4m classes of immersions for closed surfaces of genus m.

(A3) The Degree of an Immersed Hypersurface V --+ IRft+1. Let Gf : V --+ Sft be the tangential (Gauss) map associated to an immersion f: V --+ IRft+l of a connected oriented n-dimensional manifold V. Since Gf induces the bundle T(V) from T(Sft), it sends the Euler class X(T(Sft»eHft(Sft;7L) ~ 7L to x(T(V)eHft(V; 7L) ~ 7L. If n is even, then X(T(Sft)) = 2 e 7L (for the standard orientation of Sft), which implies the following relation [due to Hopf(1925)] between the topological degree deg Gf and the Euler characteristic of V.

If f: V --+ ~ft+l is an immersed closed oriented hypersurface, then 2 deg Gf = x(V) for n even.

Now, let n be odd. Then deg Gf may depend on (the regular homotopy class of) the immersion f as well as on the topology of V. For instance, let V bound a compact (n + l)-dimensional manifold V' and let an immersionf': V' --+ ~ft+l extend f from the boundary av' = V to V'. Then

deg Gf = x(V') for n odd.

Proof Double the immersion f' and smooth (in an obvious way) the resulting map of the double V' Uy V' --+ ~ft+1 to an immersion f": V' U V' --+ IRft+ 2 ::::> IRft+1. Clearly, deg Gf = deg Gr and X(V' Uy V') = 2X(V'); hence, the above even-dimensional formula applies.

(A~) Subexample. Let V' be obtained by deleting an open ball from a closed stably parallelizable manifold V". Then V', being parallelizable, admits an immersion f': V' --+ ~ft+t, which gives the degree deg Gf = X(V") - 1 to the boundary sphere Sft = av' for f = f'lav'. If x(V") # 2, this degree # 1 = deg Gfo of the standard sphere fo: Sft c; ~ft+l. In particular, for V" = sm X sm, 2m = n + 1, the immersion


f: sn --+ IRn+1 has deg Gf = 3 for m even and deg Gf = -1 for m odd. In fact, one does not need Hirsch's theorem to immerse (the parallelizable) manifold V' = sm x sm\Bn+1 into IRn+l. Indeed, this V' can be isotoped to an arbitrarily small neighborhood of the wedge sm v sm C sm X sm. Then V' obviously goes to IRn+1 with the standard map sm v sm --+ IRn+1 which sends each copy of sm onto a round sphere in IRn+1, where the two round spheres meet at two points in IRn+1 with one of them receiving the joint point of the wedge. The resulting immersion f: sn --+ IRn+1 (with deg Gf = 3) has no triple self intersection point, while no (known) general theory insures immersions vn --+ IRn+1 without triple points (compare 2.1.1).

(A4) The Signature of a Hypersurface V --+ IRn+l. An immersionf: V --+ IRn+1 is called null-cobordant if V bounds an oriented (n + 1)-dimensional manifold V' which goes into the half-space 1R~+2 :=l IRn+1 by an immersion 1': V' --+ 1R~+2, such that f = f'18V' = V and which is orthogonal to IRn+1 along the boundary 8V'. For example, the above immersions V' --+ IRn+1 can be (obviously) pushed to such positions in 1R~+2 :=l IRn+l. Let 1" denote the Abelian group with the generating set {(V,f)}, for all closed oriented (connected or not) n-dimensional manifolds V and for all immersions f: V --+ IRn+l, and with the relations

{(VI U V2,JI U f2) = (VI ,JI) + (V2,J2)},

for all (JIi,.fi), i = 1,2, where VI U V2 denotes the disjoint union of JIi andfl U f21 JIi = .fi for i = 1, 2. Denote by 10 c 1" the subgroup generated by null-cobordant immersions. Then the factor group lIn = 10110 is called the cobordism group of (oriented one-codimensional) immersions. It is isomorphic [Wells (1966), compare (G) in 2.2.7] to the (stable) homotopy group,

lIn ~ 1l:N+n(SN) = 1l:N+n+1 (SN+1) = ... , for N ~ n + 2.

Hence, by the fundamental theorem of Serre (1953) the group lIn is finite for n ~ 1. [In fact, lIo ~ 7L, III ~ 7L2 = 7L127L, 1I2 ~ 7L2, 1I3 ~ 7L24 , 1I4 ~ lIs = 0, 1I6 ~ 7L2, 1I7 ~ 7L 240 , lIs = 7L2 EB 7L2, 1I9 = 7L2 EB 7L2 EB 7L2 , 1I1O = 7L6, IIll = 7L S04 , IIll = 0, 1I13 = 7L 3 , ... , see Toda (1962).] The Serre finiteness theorem insures for every immersion f: V --+ IRn+l, n ~ 1, the existence of a manifold V' whose boundary consists of d = ord lIn copies of V and of an immersion 1': V' --+ IR~+ 2 which equals f on each copy of V in 8V'. Now, for n = 4k - 1, we define the signature of f by a(f) = d-Ia(V') where a(V') stands for the signature (or index) of the (quadratic) intersection (of2k-cycles) form on the homology H2k(V') (The intersection form may be degenerate since V' has a boundary, but the signature is defined, as usual, by the difference between the plus and minus signs in the diagonalized form.) If fo: Vo --+

1R~+2 is another immersion bounded by d copies of V, then the reflection of fo in the hyperplane completes Vito a closed immersed hypersurface V' U Vo --+ IRn+2 which has zero signature being stably parallelizable (see 2.1.3). Hence,

a(V') - a(Vo) = a(V' U Vol = ° by the (obvious) additivity of the signature [Novikov (1965); the minus sign comes from the reversal of the orientation by the reflection of Va], which shows a(f) to be independent of V'. In fact, a(f) is a regular homotopy invariant of J, since every


regular homotopy it: V --+ IRn+1, t E [0, 1], can be perturbed to an immersed cylinder !: V x [0,1] --+ 1R~+2 with fw x ° = fo and fl V x 1 = f1 and since adding the "collar" V x [0,1] to V' does not change a(V').

Example. Let Y be a closed oriented manifold of dimension 2k and let V' c Y x Y be a closed tubular neighborhood of the diagonal A ~ Yin Y x Y. The homology H2k(V/) is generated by the fundamental class [A] whose self-intersections (for the natural orientation in V') equals X(Y) by the Hopf-Lefschetz fixed point formula. Thus a(V') = sign X(Y) (that is zero for X = ° and ± 1 for X ~ 0). If Y is stably parallelizable, then, clearly, V'is parallelizable. This allows an immersion f': V' --+ IRn+\ for which a(f) = signX(Y). In particular, we obtain an immersion f: SO(3) --+ 1R4 with a(f) = 1 for Y = S2.

Exercises. (a) Immersions S3 --+ 1R4. Relate immersions S3 --+ 1R4 to the group 11:3 (SO (4)) = 1I:3(S3 x SO(3» = lL EB lL and define with the above deg and a homomorphisms of lL EB lL onto lL and into Q respectively. [In fact, the values of a lie in ilL c Q by a theorem of Rohlin, see Kervaire-Milnor (1960).] Show immersions fo and f1 to be regularly homotopic if and only if deg Gfo = deg Gfl and a(fo) = a(f1).

(b) Immersions sn --+ IRq for q ~ n + 1. Observe that the immersion relation .Y --+ sn has simply connected fibers (~Stn IRq) for q ~ n + 2 and that it admits a section sn --+.Y. Establish with this a one-to-one correspondence between the homotopy classes of sections sn --+.Y with those of maps sn --+ .Yv ~ Stn IRq, v E sn, and conclude

Theorem of Smale. The regular homotopy classes of immersions sn --+ IRq are classified for q ~ n + 1 by the homotopy group 1I:n(Stn IRq).

(B) Isometric Immersions. Since the isometric immersion relation Jo c J --+ X = V x IRq is fiberwise homotopy equivalent to.Y, the h-principle of Nash-Kuipe.r (see 1.1.5) allows the regular homotopies in the above examples to be C1-isometric. For instance, the sphere S2 c 1R3 can be turned inside out by a regular homotopy of isometric C1-immersions S2 --+ 1R3.

Exercise. The Signature and the 1J-Invariant. Recall the invariant 1J(V, g) of a C2-smooth Riemannian manifold (V, g) (see Atiyah-Patodi-Singer 1975) and show every isometric C2-smooth immersion f: (V, g) --+ IRn+1, n = dim V = 4k - 1, to satisfy a(f) = 1J(V,g). Then express the degree of Gf for all n ~ 2 by the integral over Vof a certain curvature function of g. Apply this to an arbitrary Riemannian C2-metric 9 on S2 and show the regular homotopy class of any isometric C2-immersion f: (S3, g) --+ 1R4 to be uniquely determined by 9 [compare (C) in 3.3.4]. Observe that invariants of 9 impose no restriction on the regular homotopy class of an isometric C1-smooth immersion (V, g) --+ IRn+1.

(C) The (Multi-)Parametric h-Principle. Recall that a continuous map between topological spaces, say J.1.: A --+ A', is called a weak homotopy equivalence if either of the two following (obviously) equivalent conditions is satisfied.

(i) The map J1 is bijective on the homotopy groups, J1i: ni(A) ~ ni(A'), i = 0, 1, .... (ii) Let P be an arbitrary cell complex, let Po c P be a subcomplex and let lXo: Po -+ A

be an arbitrary continuous map. Then lXo extends to a continuous map IX: P -+ A if and only if IX~ = J1 0 lXo: Po -+ A' extends to a continuous map IX': P -+ A'.

Next, invoke the space C(X) of C-sections V -+ X (of a fibration X -+ V, see 1.1.1) with the topology of uniform convergence of sections along with their partial derivatives (jets) of order :s; r on compact subsets in V, and let Solr f7l c C(X) be the subspace of the solution of a given relation f7l c x(r). Say that f7l and (or) solutions of f7l satisfy the parametric h-principle if the map Jr: Solr f7l-+ CO(f7l) for fl--+J; is a weak homotopy equivalence. This amounts to the vanishing of the relative homotopy groups nJ CO (f7l), HoIO), i = 0, 1, ... , for the subspace Holo c CO(f7l) of holonomic sections V -+ f7l (which is homeomorphic to Solr f7l by f 1--+ Jf).

Example. The surjectivity of Jr on no expresses the ordinary h-principle, while the injectivity on no is the one-parametric h-principle, which refers to continuous maps (C-homotopies) P = [0,1] -+ Solr f7l.

The h-principles of 1.1.2-1.1.5 hold true in the parametric form with the same reference to the proofs. For example.

(1) (Compare 1.1.5.) The space of isometric C1-immersions IRn -+ IRq, q ~ n + 1, has the same homotopy groups as the Stiefel manifold Stn IRq. (2) The space of free isometric Ck-immersions IRn -+ IRq for k = 5, 6, ... , 00, an. has the same homotopy groups as Stm IRq for m = tn(n + 3) and for q ~ t(n + 2)(n + 3). [Even in this example of the Euclidean metric on IRn one does not know how to remove the assumptions k ~ 5 and (or) q ~ t(n + 2)(n + 3) for n ~ 2.]

Exercises. (a) Prove (2) for n = 1 and for all k ~ 2 and q ~ 3. (b) Let V be a compact simply connected Riemannian COO-manifold. Prove with

the parametric h-principle the space F of free isometric Coo-immersions V -+ IRq, q ~ t{n + 2)(n + 3), to have finitely generated homotopy groups [compare Serre (1953)]. Show that ni(F) = ° for 1 :s; i < j = q - tn(n + 5) and that the group niF) is cyclic.

(C') Remark. The parametric h-principle refers to continuous maps cp: P -+ Solr f7l which are, in fact, sections of the fibration X x P -+ V x P satisfying f7l over each submanifold V x p c V x P, pEP. If P is a smooth manifold and if the sections cp: V x P -+ X x P in question are C-smooth, then the relations f7l1 V x p, for all PEP are equivalent to a single relation f7l' c (X X p)(r). That is the pull-back of f7l under the natural projection (X x p)(r) -+ x(r). Thus, the parametric h-principle for f7l can be reduced in most cases to the ordinary h-principle for f7l' = f7l~.In fact, one needs P = Si to show the surjectivity of the map Jr on ni(Solr f7l); the ball Bi+l is used for the injectivity of J' on ni. In the latter case one needs the h-principle for extensions (see 1.4.2, 1.4.4) of solutions from V x Si to V X Bi+l for Si = oBi.

Strictly speaking, the h-principle for f7l' is not equivalent to the h-principle for f7l due to somewhat different smoothness assumptions on pertinent sections V x P -+ V x X used in the definitions of the respective h-principles. However, if

qj c x(r) is an open subset, then any section (fJ: V x P -+ X x P which is (only) C -continuous in p can be easily approximated by C -smooth sections which satisfy qj' in so far as (fJ satisfies qjl V x p for all pEP. Hence, the parametric h-principle for open relations qj does follow from the ordinary h-principle for extensions of solutions of qj'. A similar approximation argument applies to many non-open qj (see 2.3.2). Alternatively, one could slightly modify the notion of the h-principle for qj' by starting with C -continuous solutions. In any case, the parametric h-principle for most relations is not harder to prove than the ordinary h-principle. In fact, Smale (1959) originally obtained the one-parametric h-principle for immersions sn -+ ~q

by first proving the multi-parametric h-principle by induction on n [compare (C) in 1.4.2; no such induction is possible if one restricts the number of parameters.] We adopt a similar approach in §2.2. On the contrary, the techniques in 2.1 and 2.4 directly yield the ordinary h-principle for qj (as well as for qj'), without ever mentioning any homotopy groups.

1.2.2 Density of the h-Principle in the Fine Topologies

Define the fine CD-topology in the space CO(X) of continuous sections V -+ X by taking the subsets CO(U) c CO(X), for all open U c X, for a base of this topology. Then the fine cr-topology in cr(x) is induced from the fine CO-topology in CO(x(r» by the embeddingf 1---+ J; of C(X) onto the subspace ofholonomic sections in C(X).

Example. A family F of C1-functions f: ~ -+ ~ gives a fine C1-approximation to a given C1-function fo iff for every positive continuous function e(t) on ~ there exists

. I d!c (t) df(t) I an elementfEF for whIch Ifo(t) - f(t)1 + + -dt ~ e(t) for all tE~.

Definitions. Let fo: V -+ X be a continuous section and consider a neighborhood U c X of the image fo(V) c X. Intersect the pull-back of U under the projection Po: x(r) -+ X with a given relation qj c x(r) and write qju = (Potl(U) n qj c x(r). Say that qj satisfies the h.-principle CO-near fo if every section (fJo: V -+ qj which lies over fo (i.e. Po 0 (fJo = fo) can be brought to a holonomic section (fJl by a homotopy of sections (fJt: V -+ qju, t E [0,1], for an arbitrary (small) neighborhood U c X of fo(V). The h-principle is called everywhere dense [in CO(X)] ifit is holds true CO-near every section fo: V -+ X. Clearly, the h-principle is everywhere dense if and only if the relation qju satisfies the ordinary h-principle for all open subsets U c X.

(A) Examples. (1) The h-principle for the Cauchy-Riemann relation (see 1.1.2) is nowhere dense since CO-limit of hoi om orphic sections is holomorphic.

(2) Immersions V -+ Wenjoy the CO-dense h-principle for dim V < dim W. [Hirsch (1959), see 2.1.2, 2.2.2, 2.4.3.]

Corollary. If a continuous map fo: V -+ W is homotopic to an immersion, then it admits a fine CD-approximation by immersions V -+ W. In particular, every continuous map fo: ~n -+ ~n+1 can be CO-approximated by immersions f: ~n -+ ~n+1.

Proof Since the projection J --+ X = V x W is a fibration, the existence of a lift of a section fo: V --+ X to J depends only on the homotopy class of fo.

Exercises. (a) Let fo: ~2 --+ ~3 be given in the Euclidean coordinates by

fo: (U 1,U2)f--+(ui,U1U2,U2)'

Observe the map fo to be a Coo-immersion outside the origin (0,0) E 1R2 and show that no immersion f: 1R2 --+ 1R3 has bounded C1-distance from fo, which means

/I a(~~ fo) /I + /I a(~: fo) /I ~ const < 00.

(b) Show that neither one of the two immersions 1R2 --+ 1R2 given by (u 1, U2)f--+ (U1'U~) and by (U1,U2)f--+(U1U2,ui - u~) respectively admit CO-approximations by immersions ~2 --+ ~2.

(c) Consider the manifold V of pairs v = (I, x), where I is a straight line in 1R3 through the origin and x E l. Map V to 1R3 by v = (I, x) f--+ X E 1R3 and show this map to admit no CO-approximation by immersions V --+ ~3. In contrast, prove the existence of some immersion V --+ 1R3. [See Siebenmann (1972) for general results on limits of equidimensional immersions.]

(B) Short Maps and Isometric Immersions Between Riemannian Manifolds. Let (V, g) and (w, h) be Riemannian manifolds. A continuous map f: V --+ W is called short if it does not increase the Riemannian length of smooth curves in V. This is equivalent, for C1-smooth maps f, to the inequality (r, r)g ~ (Df(r),Df(r»h for all tangent vectors r E T(V). Call fo strictly short if there is a strictly positive continuous function e = e(v) on V such that

disth(f(v 1),f(v2» ~ (1 - e(v1»distg (v1, v2)

for those pairs of points v1, V2 E V which have distg(v1 , v2) ~ e(v1).lffo is C1-smooth, then this is equivalent to (r, r)g > (Df(r), Df(r) \ for all r i= 0 in T(V).

(B 1) Theorem. If dim W > dim V then strictly short differentiable immersions V --+ W satisfy the h-principle near every strictly short CO-map fo: V --+ W (See 2.4.5.)

Remarks. (a) Strictly short differentiable immersions are (the only) solutions of the differential relation in X(1) which consists of strictly short injective linear maps CT.,(V), gJ --+ (Tw(W), hw) for all (v, w) E X = V x W. Thus, one may speak of the h-principle.

(b) Since every C1-map betweem smooth manifolds becomes strictly short with appropriatly chosen Riemannian metrics, (B d sharpens the above CO-dense h-principle of Hirsch.

(c) Wi'P-approximation. Another refinement of Hirsch's approximation can be achieved for all q ~ n = dim V with the (Sobolev) Wi'P-metric in the space of Coo-maps f: V --+ IRq, defined for i = 0, 1, ... and p ~ 1 by

distijf1,f2) = (Iv IIJ}, - J},IIPdl1 yIP,

where II II is some norm in the vector bundle x(r) ~ V (associated to X = V x ~q ~ V) and dp, is some smooth positive measure on V.

(B~) Theorem. Let q - n > p(i - 1) and let V admit an immersion V ~ ~q. Then, for every COO-map fo: V ~ ~q and for each 6> 0 there exists a Coo-immersion f: V ~ ~q for which distijf,fo) ::;; 6.

See Gromov-Eliashberg (1970, 19701) and 2.2.1 for the proof.

Exercise. (Gromov-Eliashberg 1970). Find, for given numbers n ;::: 2, q > n, i ;::: 2 and for all p;::: (q - n)/(i - 1) a Coo-immersion fo: ~" ~ ~q which admits no Wi,P-approximation by Coo-immersions ~" ~ ~q.

(B2) Theorem. If dim W > dim V, then the h-principle for isometric C1-immersions (V, g) ~ (W,h) is CO-dense near every strictly short continuous map fo: V ~ W.

Remarks and Corollaries. (a) One could start with an approximation of fo by a strictly short immersion [see (B2)] and then deform to an isometric immersion by applying the geometric method of Nash and Kuiper. But the intermediate approximation is suppressed by the technique of 2.4.9, where the theorem is proved.

(b) If V and Ware parallelizable (for example contractible) Riemannian manifolds, then every strictly short map V ~ W can be CO-approximated by isometric C1-immersions V ~ W, provided dim W > dim V.

Proof. The parallelizability makes the fibration fa ~ X = V x W split. Then every section V ~ X lifts to fa and the dense h-principle applies.

Exercises. (i) Let V admit an immersion into ~q for q > dim V. Construct an isometric C1-immersion f: (V, g) ~ ~q such that the diameter of the image abides Diamf(V);::: Diamg V - 6 for a given 6 > O.

(ii) Let fo: ~" ~ ~q be a linear (non-strictly) short map which is isometric on a straight line ~1 c ~". Show that fo admits no fine CO-approximation by isometric C1-immersions unless fo is isometric to start with.

(iii) Let a compact manifold V admit some immersion into ~q for q > dim V. Show that a continuous map fo: V ~ ~q admits a CO-approximation by isometric C1-immersions V ~ ~q if and only if it is short.

(C) The Ci-Dense h-Principle for i;::: 1. Let i ::;; r, and consider a neighborhood U c Xli) of the image Jjo(V) c Xli) for a given Ci-section fo: V ~ X. Set 9lu = (pif1 n 9l c x(r) for the projection pi: x(r) ~ Xli). We say 9l satisfies the h-principle Ci-near fo if for every neighborhood U c Xli) of Jjo(V) ano for every section <Po: V ~ 9l which lies over the jet Jjo (i.e. pi 0 <Po = Jj) there exists a homotopy of sections <Pt: V ~ 9lu, t E [0,1], for which <P1 is holonomic. The h-principle is called Ci-dense in a subspace of Ci(X) ifit holds true Ci-near every section in this subspace.

Warning. The Ci-density of some subset of sections in C(X) is, of course, a stronger property than the Ci- 1-density. Yet the Ci-dense h-principle does not necessarily

yield the Ci-1-dense one, because the former applies only to those sections <Po: V ~ x(r) whose projections to Xli) are holonomic. In fact, a relation [Jt may satisfy the everywhere Ci-dense h-principle for some i ~ 1 and violate the ordinary h-principle at the same time. A (trivial) example is provided by [Jt = (pD-1 (8l') for any 8l' c Xli) which violates the h-principle.

Examples. (C1) Free Maps f: V ~ (w, h). Define the first (covariant) derivatives of f in local coordinates Ui on V, i = 1, ... , n, by VJ = Dj(o;ouJ These are vector fields in W along (the coordinate chart in) V, as VJ(v) E Tw(W) for w = f(v). Next we invoke the actual covariant derivative V in the Riemannian manifold (w, h), which applies to tangent fields in W, and put Vijf = Vu;~f The subspace Span(VJ(v), ViJ(V)) c

Tw(W), 1 :s; i,j :s; n, does not depend on the choice of coordinates around v E V and it is called the (second) osculating space 1?(V, v) c Tw(W) (compare 1.1.4).

Exercise. Let (W, h) be isometrically realized by a C2-submanifold W c [RN. Show

ij (o~ ) that VJ(v) = -0 (V)E Tw(W) c Tw([RN) and that Vijf(v) = P -0 -(v) E Tw(W) c ~ ~~

Tw([RN) for the orthogonal projection P: Tw([RN) ~ Tw(W),

Call a C2-map f: V ~ (w, h) free if dim 1/(V, v) = tn(n + 3) for all v E V.

T,.eorem. If dim W ~ tn(n + 3) + 1, then free maps V ~ (w, h) satisfy the everywhere C1-dense h-principle as well as the everywhere CO-dense h-principle (see 2.4.3).

Remark. The CO-dense h-principle is not very interesting here since every C1-map fo: V ~ W for dim W ~ 2n admits a fine C1-approximation by an immersion V ~ W according to a (quite simple) theorem by Whitney (see 1.3.2).

Corollary. If a C1-map fo: V ~ W is homotopic to a free map, then it also admits a fine C1-approximation by free maps V ~ W, provided dim W ~ tn(n + 3) + 1. (This is unknown for dim W = tn(n + 3) and n ~ 2.)

Proof We may assume, with the above remark, the map fo is an immersion. Since dim W ~ tn(n + 3) + 1 ~ 2n + 1, the homotopy class of the jet JJo: V ~ Y into the immersion relation .F c X(!) ~ X ~ V is determined by the homotopy class of fo alone, for the fiber Y x ~ Stn [Rq, x = (v, w) E X = V x W, q = dim W, is n-connected. This insures a lift of J10 to a section <Po: V ~:#' for the freedom relation:#' c X(2)

and the C1-dense h-principle applies.

(C2) Free Isometric Immersions (V, g) ~ (w, h). Let the manifold (w, h) be COO-smooth (or can if Can-immersions are under consideration) and let (V, g) be Ck-smooth.

Theorem. If dim W ~ t(n + 2)(n + 3) and if k ~ 5 then the h-principle for free isometric Ck-immerions V ~ W is CO-dense in the space of those continuous maps V ~ W which are fine CO-limits of (i.e. can be finely CO-approximated by) strictly short C1-maps V ~ W Furthermore, this h-principle is C1-dense in the space of those

isometric CI-immersions which are fine CI-limits of strictly short immersions. (See 3.1.7.)

Exercises. (a) Derive from this h-principle the following

(C~) Approximation Theorem. A CO-smooth (CI-smooth) map fo: V -+ W admits under the above assumptions on k and on dim W a fine CO-approximation (respectively, CI-approximation) by free isometric Ck-immersions V -+ W if and only if fo is a uniform CO-limit of strictly short CI-maps (fo is an isometric CI-immersion which is a fine CI-limit of strictly short immersions).

(b) Let(W, g) have a non-positive sectional curvature and let Vbe compact. Show a short CI-map fo: V -+ W to be a CI-limit of strictly short maps if and only if fo is a CO-limit of such maps. (Relations between different strictly short approximations are unknown in the complete generality.)

(C;) Ci-Approximation for i ~ 2. Let fo: V -+ W be a free isometric immersion in some Holder class Ci. 1l (see 2.3.4; recall that Ci,O = Ci and that Ci ::) Ci,ll ::) Ci+ 1 for 0> IX ~ 1).

If j + IX > 3 and if k ~ 5, then fo admits a fine C2-approximation by free isometric Ck-immersions V -+ W (without any restriction on dim W).

This is shown by a purely analytic method [extending the implicit function theorem of Nash (1956)J in 2.2 where we also study Ci-approximation for i > 2.

Corollary. If a coo -smooth Riemannian manifold (V, g) admits a free isometric C4 -immersion (or, even, C 3 ,ll-immersion for IX > 0) fo(V, g) -+ IRq then it also admits an isometric Coo-immersion f: V -+ IRq which can be chosen arbitrarily C2-close to fo. (One does not know whether the existence of a free isometric C3 -immersion insures an isometric Coo-immersion into the same space IRq. Nor does one know whether the freedom assumption is essential for C4 -immersions.)

1.2.3 Functionally Closed Relations

A relation fJ1l c x(r) is called (functionally) Ci-closed for some i $; r if every C-limit f of C'-solutions V -+ X of f!Jl also satisfies f!Jl in so far as this limit f: V -+ X is C'-smooth, where the limits are understood for the ordinary (non-fine) Ci-topology.

One usually establishes this property by reducing (or "integrating") f!Jl to an equivalent possibly (non-differential) relation which involves no jet of order ~ i. For example, the shortness relation, <Df(T),Df(T»h $; <T,T)g, TET(V), for maps f: (V, g) -+ (W, h), is equivalent to the non-differential relation disth(f(v l ),f(v2» $;

distg(v l , v2), VI' V2 E V, which shows the shortness relation to be CO-closed. Another example is provided by the Cauchy-Riemann relation which can be expressed with the Cauchy integral formula by a non-differential relation, showing CO-limits of holomorphic maps to be hoI om orphic. On the other hand, the isometric immersion


relation is quite far from being CO-closed as the CO-dense h-principle allows all strictly short maps for CO-limits of isometric maps.

(A) Infinitesimally Enlarging Maps. A map f: V -+ Wbetween complete Riemannian manifolds is called enlarging if the image under f of a Riemannian ball B(v, p) c V with center v and radius p, for any v and p, contains the ball B(f(v), p) with the same radius p in W around f(v). A Cl-map f is called infinitesimally enlarging if the differential DJ : 7;,(V) -+ Tw(W), w = f(v) is enlarging on every tangent space 7;, (V).

If f is infinitesimally enlarging, then the differential DJ is a surjective map 7;,(V) -+ Tw(W) for all VE V and w = f(V)E W whose kernel Ker DJ c T(V) is a (q - n)-dimensional subbundle in T(V) for q = dim Wand n = dim V. Furthermore, if a smooth curve C c V is horizontal (that is everywhere normal to Ker DJ), then the length of the image satisfies lengthf(C) ~ length C. It follows that for every curve C c Wand for each point v E V over C, there exists a (unique iffis C2-smooth) horizontal curve C c V over C which passes through v. By applying this to geodesic segments C oflength p issuing from w = f(v), one shows the ball B(w, p) c W to be completely covered by the image f(B(v, p)) c W Therefore,

infinitesimally enlarging = enlarging.

With this one immediately sees the "infinitesimally enlarging" relation to be CO-closed.

(B) Exercises. (B l) Fill in the detail in the above argument and remove the completeness assumption in the final "CO-closed" conclusion. (B'l) Let f: V -+ W be an infinitesimally enlarging map of a complete manifold V into a connected manifold W Prove that, in fact, W is complete, and that f is a locally trivial fibration of V onto W (B2) Consider a Cl-function f: IRn -+ IR and denote by GrJ : IRn -+ IRn the map given by the functions of/oui on IRn, i = 1, ... , n. Show the differential relation GrJ(lRn) c A to be CO-closed for every closed subset A c IRn. [Hint: Reduce to the special case, A = {u E IRn III u II ~ I}, where solutions are infinitesimally enlarging maps IRn -+ R] (B~) Consider the trivial line bundle X = V x IR -+ V and show an arbitrary closed subset ~ c X(l) to be a (functionally!) CO-closed relation. (B3) Consider Cl-maps V -+ Wwhich do not increase the k-dimensional Riemannian volume of k-dimensional submanifolds in V. Show the corresponding relation is CO-closed for all k = 1,2, .... (B4) Study the CO-closure of Cl-maps f: IRn -+ IRn whose lacobians J = det(o/;jouj )

satisfy J ~ 1. Show every Cl-immersion in this closure has J ~ 1. Then prove the

relation det(o/;jouj ) ~ ° to be CO-closed. (o/;(V)). (B~) Show the quasi-conformality relation II DJ(v) lin ~ K Det ;u. ' V E IRn, IS CO-close for all K ~ 1. J

(Bs) Show the relation ~k imposed on C2-functions f: IRn -+ IR by requiring the

H · . (02f (V)) h k· . . 1 11· Ifl)n esstan matnx oUi

OUj to ave at most pOSItIve elgenva ues at a pomts V E tI\\

to be CO-closed for all k = 0, 1, ... , n.


(B6) Show, for every real IX, the following three relations imposed on C2-smooth Riemannian metrics g on V to be CO -closed.

(i) K(g) ~ IX; (ii) K(g) ~ IX; (iii) Ricci(t,r) ~ lX(r, t)g for all t E T(V), where K denotes the sectional curvature and Ricci stands for the Ricci tensor. (One does not know if Ricci ~ IX( t, t\ is CO-closed.)

(C) Convex Relations. Let X ..... V be a vector bundle and let Xl') - V have the associated vector bundle structure. A relation fIl c Xl') is called Ci-convex iffor each point in the complement, Xo E X('\fIl, there exists a fiberwise linear Ci-function L1: Xl') ..... IR such that L1 (x) ~ L1 (xo) for all points x E fIl which project to a sufficiently small neighborhood U c V of the point p'(Xo)E V. For example, a compact subset fIl c Xl') is COO-convex if and only if fIlv is a convex subset in X~) for all VE V. Furthermore, every Ci-smooth subbundle in Xl') is Ci-convex.

Proposition. If fIl is Ci-convex for some i ~ r then it is functionally cr-i-closed.

Proof Let ~(f) = L1(J;) for f: V ..... X and express the relation ~(f) ~ Co = L1(xo) over U c V by infinitely many inequalities

L ~(f)qJ du ~ Co L qJ du,

for all COO-smooth non-negative functions qJ on U with compact supports, where du = dU1 dU2, ... , dUn for some local coordinates in U. Since the coefficients of the differential operator ~ are Ci-smooth, one can integrate by parts i times and thus reduce (*) to

for some operators ~'(f) = L1'(J;-i) and ~1I(qJ) = L1"(J~). Since (**) depends only on J;-i it is cr-i-closed. Q.E.D.

(C1) Example. Consider the equation dd!! - a dg = ° in the functions f and g on IR, t dt

assume a = a(t) is C1-smooth and integrate by parts. The resulting equation

rt da(t) f(t) - a(t)g(t) + Jo Ttg(t)dt = const,

is obviously functionally CO-closed.

(C't> Counterexample. Let a(t) be a continuous nowhere differentiable function in t E [0,1]. That is, for every 8 > 0, there are disjoint intervals Ii = [ti' t;] c [0,1] for i = 1, ... , N = N(8), such that ~J=l t; - ti ~ 1 - 8 and such that Osc(aIIi) > 8-1(t; - ti), for all i = 1, ... , N, which means la(x) - a(y)1 > 8-1(t; - ti) for some points x and y in Ii. Then there obviously exist smooth non-negative functions qJ and qJ' on [0,1] whose supports lie in UJi' such that


f' cp(t)dt = f' cp'(t)dt = 1

and

for all i = 1, ... , N. Next, for a given C1-function fo(t), t E [0,1], define

{

c(fo(t~) ~ fo(t;)) It (cp'(r) - cp(r)) dr, for t E Ii' i = 1, ... , N, ( ) ti ti t

g t = •

0, for tEI\ U Ii' i

and

It dg(r) f(t) = fo(O) + ° a(r) Tr dr.

The functions f and g clearly are C1-smooth and

(+ ) df dg --a-=O. dt dt

Furthermore, II f - fo II cO ~ 0 and II g II cO ~ 0 for c ~ 0 as a straightforward computation reveals. Therefore, the relation (+) is not functionally CO-closed.

Exercises. (a) Show the C1-solutions of (+) to be CO-dense in the space of pairs of continuous functions fo(t) and go(t), for the above nowhere differentiable a(t), and

express this as a density of the graph of the operator g(t) 1---+ it a(r) dg(r) dr. Jo dr (b) Find further examples of integro-differential operators on functions on IRn

whose graphs are CO -dense. (c) Identify triples of C1-functions f, g and a on IR which satisfy ( + ) with Legendre

maps (compare 3.4.3) IR ~ 1R3 , which are by definition, everywhere tangent to the plane field Ker1] C T(1R 3 ) for the 1-form 1] = dz - xdy on 1R3. Prove Legendre C1-immersions IR ~ 1R3 to be CO-dense in the space of continuous map IR --+ 1R3.

(d) Consider a one-dimensional subbundle 1 c T(lRq) and study C1-maps f: IR --+ IRq which are everywhere tangent to 1 [i.e. Df T(IR) c I]. Prove the pertinent differential relation to be functionally CO-closed for all C1-smooth subbundles. Show for "sufficiently non-differentiable" CO-subbundles the maps IR --+ IRq, q ~ 2, tangent to 1 to be CO-dense in the space of continuous maps IR ~ IRq. Study the CO-closure of C1-immersions IR --+ IRq tangent to I.

(e) Consider C2-maps f: IRn --+ IRq, such that the mean curvature M of the graph If c IRn+q (which is a vector field M: If --+ T(lRn+q) I If normal to If) satisfies II M(y) II :::;; c, for all y E If. Show this condition to be functionally CO-closed for all c ~ O. Generalize this for maps between non-flat Riemannian manifolds f: V --+ W.


1.3 Singularities and Non-singular Maps

1.3.1 Singularities as Differential Relations

Let M = M(n, q) denote the space of linear maps IRn -+ IRq and let Ii c M, i = 0, 1, ... , m = min(n, q) consist of the linear maps of rank m - i. These are the orbits of the natural action of the group GLn x GLq on M; hence, they are Can-smooth (in fact real algebraic) submanifolds in M, and a straightforward computation shows codimIi = (n - m + i)(q - m + i).

Next, we turn to the jet space X(l) for X = V x W -+ V and we let Ii c X(l) consist of the linear maps T.,(v) -+ Tw(W) of rank m - i for m = min(n, q) where n = dim V and q = dim W. These Ii clearly are the orbits of the natural action of the group Diff V x Diff W on X(l) and codim Ii = (n - m + i)(q - m + i). The partition X(l) = U:"=o Ii is called the stratification of Whitney- Thom of the space X(l). Observe that the topological boundary oIi = CIIi\Ii equals the union

Ui>i Ii. Consider a Cl-map f: V -+ Wand denote by I} c V the pull-back (J) rl(Ii) of

the jet J}: V -+ X(l). The equivalent definition is I} = {VE Vlrankvf = m - i}. If the map f is Ck-smooth for k ~ 2 and if the jet J}: V -+ X(l) is transversal to Ii, then I} is a Ck-l-submanifold of codimension (n - m + i)(q - m + i) in Vby the implicit function theorem. Recall that a Cl-map between smooth manifolds, say cp: A -+ B, is transversal to a submanifold C c B if

codim[Djl(T,;(c)) c T,,(A)] = codim[T,;(c) c T,;(B)]

for all c E C and for all a E A, such that f(a) = c. The subset I} c V for i > ° is often called the Ii-singularity of f If I} = 0

for all i > 0, then the map f is called regular. If dim W ~ dim V, then the regular maps f: V -+ Ware immersions defined by rankf == n = dim V (see 1.1.3). If dim W:::;; dim V, then regular maps satisfy rankf == q = dim Wand are called submersions. The following result by Phillips (1967) generalizes the h-principle for equidimensional immersions.

(A) Theorem. If V is an open manifold, then submersions V -+ W satisfy the parametric h-principle.

Corollary. An open manifold V admits a submersion V -+ IRq if and only if there are q linearly independent vector fields on V.

Proof The independent fields define a fiberwise surjective homomorphism T(V)-+ T(lRq) which can be homo toped, according to (A), to the differential of a submersion V -+ IRq.

(Al) A Cl-map f: V -+ W is called a k-mersion if rankvf ~ k for all VE V. This can be expressed with the differential relation .1k = X(l) which is the union of

Ii for i ~ m - k. Clearly, every open subset in X(l) which is invariant under Diff V x Diff W equals the k-mersion relation for some k = 0, 1, ....

(Aid Theorem (Feit 1969). If k < dim W, then k-mersions V -+ W satisfy the parametric h-principle; moreover, the h-principle is CO-dense.

This generalizes Hirsch's immersion theorem in the extra dimension case. We prove (A'l) and (A) along with Hirsch's theorem in Sects. 2.1.2, 2.2.3, and in 2.4.3.

Remark. Regular maps f: V -+ W can be (obviously) characterized geometrically by the existence of local coordinates U1, ... , Un in V around every point v E V and of some coordinates U~, ... , u~ in W around W = f(v), such that f: (u 1, ... , un) H

(u 1, ... , Urn' 0, ... ,0) in these coordinates. This is equivalent to the transitivity of the action of the group Diff V x Diff W on the germs of regular maps V -+ W But this action is far from transitive on germs of k-mersions.

Example. Here are five 1-mersions [R2 -+ [R2 with completely different local behaviour. (1) (Immersion) f: (u 1, U2) H (u 1, u2). (2) (Folding) f: (u 1, u2) H (uf, u2). This map f folds along the line U1 = ° in [R2; it is an immersion on this line as well as on the complement to this line. (3) (Whitney's cusp) f: (u 1, u2) H (UI + U1 U2, U2). The singularity IJ is the parabola 3ui + U2 = 0, where the map fIIJ: IJ -+ [R2 is not regular at the point U1 = 0, U2 = 0. The singularity Ig1 C IJ of the restricted map g = flIJ is called the cusp IJ1 of the map f (4) (Blow-down) f: (u 1, U2) H (u 1 U2, U2). This map is regular outside the line U2 = ° and it collapses this line to (0,0) E [R2.

(5) (Totally degenerate map) f: (u 1, U2) H (Ulo 0). Here IJ = [R2.

Observe the maps (1), (2) and (3) to be stable under small Cro-perturbations of maps, while the geometry of (4) and (5) can be completely destroyed by small perturbations.

(B) The h-Principle for Folded Maps. A C2-map f: V -+ W is said to fold along a submanifold Vo c V if f is regular on V\ Yo, the map fl Vo: Vo -+ W is an immersion, the jet: iJ: V -+ X(l) is transversal to I1 c X(l) and Ij = yo. This description refers to the second jets of f and, hence, it is expressed by certain open relation g;l c X(2)

which is invariant under Diff W as well as under the diffeomorphisms of the pair (V, Yo)· Furthermore, the implicit function theorem allows, for dim V = dim W = n, local coordinates u1 , ••• , Un near each point v E Vo and some coordinates in W near f(v) E W such that the map f becomes f: (u 1, U2' ... ' un) H (ui, U2'·.·' Un) for Vo given by U1 = 0.

(B1) The Folding Theorem (Eliashberg 1970). If dim W = dim V ~ 2 and if each connected component of V contains a component of Vo then C2-maps f: V -+ W folded along Vo (and only along Vol) satisfy the everywhere CO-dense h-principle. (See 2.1.3.)

Fig. I

Remarks and Corollaries. (a) The parametric h-principle may fail to be true for folded maps. For instance, there exists an "exotic" map f: S2 -+ 1R2 folded along the equator Sl c S2 which admits no folded C2-homotopy to the standard folded map fo: S2 -+ 1R2 obtained by the linear projection S2 c 1R3 -+ 1R2. This is seen in the following amazing picture (Fig. 1) discovered by Milnor and featuring two different (!) immersed discs E0 2 -+ 1R2 which are bounded by the same immersed circle.

In fact, the folding theorem predicts a great variety of such examples as the dense h-principle yields a map S2 -+ 1R2 folded along Sl and CO-approximating a given continuous map S2 -+ 1R2. Moreover, any two exotic folded maps S2 -+ 1R2 can be joined by a C2-homotopy of folded maps (Eli ash berg 1972).

(b) The parallelizability of the sphere S3 implies (an exercise left to the reader) the existence of a section S3 -+.f7t = .f7tvo for an arbitrary closed surface Vo c S3. It follows, for example, that there exists a C2-map f: S3 -+ 1R3 which folds along a pair of concentric 2-spheres in S3 (and has no singularity anywhere else!).

(c) Consider two immersions of a closed oriented n-dimensional manifold into IRn+l, say 10 and 11: V -+ IRn+1. Write 10 -<11 in case there is an immersion of the cylinder, I V x [0, IJ -+ IRn+1 such that 11 V x ° = fo and 11 V x 1 = fl ' For example, two round embedded spheres fo and f1: sn -+ IRn+1 satisfy fo -< f1' if and only if fo(sn) lies in the open ball bounded by f1 (sn) c IRn+1. Thus, we obtain a partial order in the set of immersions f: V -+ IRn+1.

The relative version (see 1.4.2) of Eliashberg's theorem allows cylinders I V x [0, IJ -+ IRn+1 between given regularly homotopic immersions fo and f1 : V -+

IRn+1, such that 1 folds along the hypersurface V x ~ c V x [0,1]. Therefore,

there exists an immersion f2 : V -+ 1R"+1 such that f2 >- fo and f2 >- f1 for any given pair of regularly homotopic immersions fo and fl: V -+ 1R"+1.

Little is known about this order besides Eliashberg's theorem. A happy exception is an effective condition [due to Blank (1967)J for an immersed circle SI -+ 1R2 to bound an immersed disk (see Poenaru 1968).

(C) Totally Degenerate Maps. A map f: V -+ W is called totally degenerate if rankJ < m = min (dim V, dim W) for all VE V. The corresponding relation rYt c X(1)

is the union Ui>oIi.

Exercise. Show the relation Ui?:k Ii c X(l) to be functionally CO-closed for all k = 0, 1, ... , and find counterexamples to the h-principle for all k ~ 1.

(C') One does not know how to refine the h-principle (in order to make it valid) for maps f: V ---* W with k1 ~ rankJ ~ k2, VE V, where k1 < k2 < m. However, such a refinement is indicated in (B) of 2.2.7 for maps of open manifolds, f: V ---* Wof constant rank = k < m. For example

If V is an open n-dimensional manifold, then Coo-maps f: V ---* W of rankf = n - 1 satisfy the h-principle. In particular, there always exists a Coo-map f: V ---* V homotopic to the identity, such that rankf = n - 1.

(D) Invariant Relations fll c X(1). Every subset fll c X(l) invariant under the action of Diff V x Diff W is the union of some subsets Ii c X(1). No h-principle is known for such an fll except for the above-mentioned results.

Exercises. (a) Study Coo-maps f: sn ---* sn which collapse the equator sn-1 C sn to a single point and are regular outside the equator. Show that the equality I;-l = sn-1 for such a map f implies degf = ±(l + (-lr1). Then provide examples of Can-maps with I;-l = sn-1. Finally, construct can-maps f of degrees d = 0 and d = 2, such that If = sn-1.

(b) Find a can-map f of the projective space pn onto sn whose only singularity is the subspace pn-1 = I;-l.

(c) Let V be a closed oriented manifold. Construct a Coo-map f: V ---* sn, n = dim V, of a given degree degJ = d, and such that IJ is empty for 0 < i < n.

(d) Let V and W be connected manifolds without boundary, such that dim W ~ dim V, and let f: V ---* W be a Coo-map for which IJ is empty (e.g. a holomorphic map between complex manifolds). Assume f to be a proper map (e.g. Vis compact) and show that either f is onto or is totally degenerate. In particular, if V is compact and the subgroup f*(n1(V» c n1(W) has infinite index, then IJ = 0 implies the total degeneracy of f

(e) Let V and W be closed connected orient able n-dimensional manifolds and let f: V ---* W be a Coo-map which is not totally degenerate. Assume I} to be empty for i = 1, ... , k, and show the homomorphism f*: Hi(V; JR) ---* Hi(W; JR) to be surjective for i = 1, ... , k. Moreover, assume n 1 (W) = 0 and prove f*: Hi(V; £:) ---* Hi(W; £:) to be surjective for i = 1, ... , k.

Question. Under what assumptions does there exist a Coo-map f in a given homotopy class of maps V ---* W, such that Il is empty for a given subset of indices (i1' i2,···, ij , ... , ik )? For example, when can one find f (looking like a holomorphic map) for which I} is empty for all odd i?

(E) Ramified Maps. A map between equidimensional manifolds, f: V ---* W, is said to ramify along a codimension 2 submanifold Vo c V ifin suitable local coordinates u1, ... , Un near each point Vo E Vo and for some coordinates in W near f(vo) the map fbecomes


f: (U 1' U2 , U3"'" Un)H (Rez', Imz', U3,···' Un),

for z = U 1 + .J=lu2 and for some integer s ~ 1. (This s may be different for different components of Vo.)

Example. Take a codimension 2 submanifold Vo c Wand let W' be a finite covering of the complement W\ Yo. Then the metric completion Vof W' (for the metric induced from some Riemannian metric in W) admits a smooth structure, such that the covering map W' -+ W\ Vo extends to a Coo-map V -+ W which ramifies along V\ W' c V, and has no singularity on W'.

Question. Does every closed n-dimensional parallelizable manifold V admit a COO -map f: V -+ S" which ramifies along some Vo c V and has no singularity outside Vo? The positive answer is known for n = 3 [Alexander (1920), see Rolfsen (1976)].

(F) Invariant Relations in x(r) for r ~ 2. The action ofthe group Diff V x Diff W on x(r) for X = V x W -+ V may have infinitely many orbits for r ~ 2. In fact, the number of orbits is infinite for r ~ 2 and dim V ~ 2, unless r = 2 and dim W = 1. (This is an easy exercise for the reader.) Any union of such orbits gives an invariant relation t1l c x(r) which may be tested for the h-principle. According to Levin (1965) and Poenaru (1966) maps f: V -+ W without cusps Ell (i.e. E} = 0 for i ~ 2 and the map fiE}: E} -+ W is regular) satisfy the h-principle.

The h-principle for a general class of open invariant relations t1l c x(r) is due to Du Plessis (1976). Apparently, all known h-principles for open invariant relations t1l c x(r) follow from the generalized folding theorem (Eliashberg 1972) which applies to non-equidimensional maps (see 2.1.3).

1.3.2 Genericity, Transversality and Thom's Equisingularity Theorem

Define the COO -dimension dim S of an arbitrary subset S in a smooth manifold Y to be the lower bound of the integers m, such that S is contained in a countable union of Coo -submanifolds of dimension m in Y.

Examples. (A1) Take a Coo-function f: Iij" -+ Iij and consider the set E = E} C 1ij"

where the differential (or the gradient) df of f vanishes. Let Ek c E, k = 1, 2, ... ,

00, be the subset where the partial derivatives oIf = OUi, o~::. . . OUi, vanish for

1 :$; III ~ il + i2 + ... + il = I:$; k [here I stands for the multi-index (il"'" il )]. ThenSk = Ek\Ek+lliesin theunionofd = (n + k - 1)!f(n - 1)!k! COO-hypersurfaces HI c Iij", for III = k, where each HI is defined by the equation oIf(v) = 0 and by the non-equality dolf(v) =F 0 for VE Iijll. Hence, dim(E\Ek ) :$; n - 1 for all k = 1,2, ... ,00.

(A2) Stratified Sets. A stratification of S is a partition of S into finitely many locally closed COO-submanifolds (Strata) Si c Y, i = 0, ... , k, such that the topological

boundary iJSi = (S n CI Si)\ Si lies in the union U j>i sj C S for all i = 0, 1, ... , k. Thus, SO is an open subset in S, that is SO = S n U for some open U c Y, and SO c U is a Coo-submanifold which is a closed subset in U. Next, S1 is open in S\SO, S2 is open in S\(SO U S1) and so on. Clearly, dimS:::;; SUPi dim Si. In fact, the Coo-dimension of S (obviously) equals the topological dimension.

(A~) Semi-algebraic Sets. A subset S c IRn is called semi-algebraic if it is a finite union, U j S(j), j = 1, ... , m, where ech S(j) is defined by a finite system of polynomial equations and (strict or not) inequalities in IRn. Every such S, as well as the image of S under. an arbitrary Coo-diffeomorphism IRn ~ IRn, can be canonically stratified as follows. Define SO = Reg S c S to be the maximal open subset in S which is a (locally closed) Coo-submanifold in IRn. Then take S1 = Reg(S\SO), S2 = Reg(S\(SO U S1)) and so on. It is not hard to show that Si is empty for i > dim S (See Levin 1971; Wall 1971; Whitney 1957; Lojasievicz 1965), which makes the partition U i Si a stratification of S. Moreover, an arbitrary partition of S into semi algebraic subsets SI' c S, Jl. = 1, ... , M, can be canonically refined to a stratification of S by first taking the minimal Jl. = Jl.o for which dim SI' = dim S and then by defining SO to be the maximal open subset in S which is contained in SI'O and is a Coo-submanifold in IRn. The same applies to the complement S\So partitioned by (S\So) n SI" thus giving the stratum S1 c S\So; then one gets S2 in the subset S\(SO U S1) partitioned by [S\(SO U S1)] n SI' and so on. Eventually, one stratifies S by at most M dim S strata Si.

(B) Generic Points, Maps and Sections. One says that a property of a point x (e.g. of a function or a section) in a Bair space f£ is generic if it holds for x E f£' c f£, where f£' is a residual subset (i.e. a countable intersection of open dense subsets in f£). This f£' can be defined any time you like in a discussion. It can become smaller in the course of an argument when we need it, but it must remain residual. The expression: "a property A is satisfied by a generic Ck-section (or sections) V ~ X", is often used instead of "the property A is generic in the space f£ = Ck(X) with the fine (or the ordinary if so indicated) Ck-topology".

Remark. The notion of genericity could be based on any class of "large" subsets in a given space (like subsets offull measure in a Lebesgue space) which is stable under unions and countable intersections of subsets. In fact, a deeper analysis of this notion belongs with the mathematical logic which is not discussed here.

Example. Let f£ be an infinite dimensional Banach space. Call a subset OJJ c f£ thin if, for every s > 0, there is a sequence of balls Bi c f£, i = 1,2, ... whose radii satisfy ri < sand ri ~ ° for iH 00, such that UiBi::::J OJJ. Show countable unions of thin subsets to be thin and prove that no residual subset is thin. (Thus, one could associate a genericity notion to the complements of thin subsets.)

(C) Theorem (A.P. Morse 1939). Let S be an m-dimensional manifold and let f: S ~ IR be a Coo -function. Then f is transversal to a generic point x E IR.

Proof Since 8 is a countable union of balls, we may assume 8 to be such a ball B c ~m to start with. The function fIEJ: EJ --+ ~ for EJ = 8\EJ is transversal [see the definitions in (Ai) and in 1.3.1] to all x E~. Furthermore, we may assume by induction in m the map fIH/: H/--+ ~ for III < m to be transversal to generic points x E ~, which implies the desired transversality on 8\Em. Finally, the subset Em C 8 = B c ~m can be covered (as any other subset in B c ~m) by N ~ const e-m balls with the centers in Em and of radii ~ e, for any given e > O. The f-image of such a ball is an interval in ~ of length ~ const' em+i by the Taylor remainder theorem. Hence, the (compact!) image f(Em) c ~ can be covered by N intervals of total length ~ const" e for all e > 0, and so no generic point x E ~ lies in f(Em). Thus, f has the required transversality on all of 8 as well as on 8\Em• Q.E.D.

(C') Corollary (Sard 1942). Take a Coo-submanifold in a product, 8 c U X ~d. Then 8 is transversal to the submanifold U x Z c U X ~d for generic points Z E ~d.

Proof. If d = 1 this is equivalent to (C) applied to the projection 8 --+ ~i. Then, for d ~ 2, we split ~d = ~d-i X ~ and establish with (C) the transversality of 8 to U X ~d-i X X c U X ~d-i X ~ for generic points x E ~. For these, the intersection 8x = 8 n (U x ~d-i X x) is an (m - I)-dimensional manifold, which may be assumed, by induction in m = dim 8, to be transversal to U x y x x c U X ~d-i X x for generic y E ~d-i. Hence, the subset Z' c ~d-i X ~ ofthe pairs z = (y, x) for which 8 is transversal to U x y x x is dense in ~d = ~d-i X ~. Since the complement ~d\Z' (obviously) is a countable union of compact subsets in ~d, the subset Z' is residual as well as dense. Q.E.D.

Exercise. Prove Sard's theorem for Ck-submanifolds 8 for k ~ min(l, m - d).

(D) Transversality Theorem (Thorn 1955). Let 8 c x(r) be a Coo-submanifold. Then the jet J;: V --+ x(r) is transversal to 8 for generic Coo -sections f: V --+ X. In particular, codim8f = codim8 for 8f = (J;ri(8) c V.

Proof Since 8 can be covered by countably many small compact balls, one may assume 8 to be such a ball, which projects to a small (coordinate) neighborhood ucv.

The subspace PI' c Coo(X) of sections f: V --+ X whose jets J;: V --+ x(r) are transversal to such a (compact) 8 is obviously open. To approximate a given Coo-section fo: V --+ X by those in PI', take a holonomic splitting y(r) = U X ~d, C

x(r) around J;o(U) c x(r) [see (c) in 1.1.1] and apply (C') to 80 = 8 c y(,) c U X ~d,

and to U x z = J;.<U) c y(r). This yields the transversality of J;. to 8 for generic ZE ~d" which allows such a "transversal" fz arbitrary Coo-close to fol U. Finally, the sectionfz: U --+ Y c X extends to all of Vby a small perturbation outside p'(8) c U. [Take a Coo -function ((): U --+ ~ with a compact support, which equals one near pr(8), and extend fz by fo + (()(fz - fo) outside p'(8).] Hence, f!{' c C'°(X) is dense as well as open and Thorn's theorem follows.

(D') Corollary. If S c x(r) is an arbitrary (e.g. stratified) subset whose Coo-codimension abides codim S ~ dim V + 1, then the jet J;: V --+ x(r) misses S for generic COO-sections f: V --+X.

Remark. In fact one does not need Sard's theorem for the proof of (D') but rather the following obvious fact: Every Coo-map S --+ ~d misses a generic point z E ~d, provided dim S < d. Furthermore, in many cases one can reduce (D) to (D') which applies to the subset S' c x(r+1) of the I-jets of those holonomic Coo-sections V --+ x(r) which are not transversal to S c x(r). This works, for example, for S c X(O) = X and also for real analytic subsets S c x(r) for all r = 0, 1, ....

Exercises. (a) Apply (D') to the Thorn-Whitney strata 1: i c X(1), for X = V x W and i ~ 1 (see 1.3.1) and prove

Whitney's Theorem (1936). A generic Coo-map V --+ W is an immersion for dim W~ 2n, where n = dim V.

(b) (Nash 1956). Show generic Coo-maps V --+ ~q to be free for q ~ tn(n + 5). (c) Generalize (D) and (D') in order to obtain the following theorem.

(Whitney 1936). A generic Coo-map f: V --+ W has not double points for dim W ~ 2n + 1. Furthermore, the subset of k-multiple points, Mk = {(Vi'"'' Vk) E V x V x ... x Vlf(v i) = f(V2) = ... = f(vk), for Vi =I- V2 =I- ••• =I- Vk} has dimMk ~ n(k - l)(dim W - n) for generic maps f: V --+ W.

(E) Equisingular Maps and Sections. Consider a point zeX(l) and let y = p5(z)eX and V = pi (z) = p(y). Take a linear subspace r E T.,(V) and denote by A(z, r) c X~i) the (affine) subspace of those linear maps x: T.,(V) --+ T,(X) for which xlr = zlr. Notice that dim A(x, r) = n(q - dim r) for n = dim V and q = dim Xv' Next, take a stratified subset 1: = Ui1:i c X(1), i = 0, ... , k, and call a Coo-section f: V--+X 1:-equisingular along a submanifold S c V if

(i) the image Jj(S) lies in a single stratum 1:i for some i = i(S,f). (ii) The Coo-dimension of the intersection 1:i n A(Jj(s), T.(S» is constant in SE S for

allj = 0,1, ... , k.

Example. If 1: = Ui1:i c X(i) is the Whitney-Thorn stratification for X = V x W, then a Coo-map f: V --+ W is equisingular along some S c V if and only if the ranks rank.f and rank.(fIS) for flS: S --+ Ware constant in SES.

Remark. It would be natural to strengthen (ii) by requiring all topological invariants of the pertinent intersection to be constant in s, but for our applications the weak "equidimensional" definition suffices.

Next, call a stratification UiSj = S, of the subset S = Ef = (J}r i (E) c Vequisingular if the section f is E-equisingular along Sj for all j = 0, 1, ... , m.

Finally, a stratified subset E = UiEi c X(1) is called locally semi-algebraic if there exists, for each x E E, a split neighborhood Y = U x Wo c X with local coordinates Ul"'" Un and Yt> ... , Yn [see (c) in 1.1.1] such that y(l)3X and the subset Ei n y(l) is semi-algebraic in y(1) for all i = 0, 1, ... , k, where y(l) is identified with ~n+qn by means of the coordinates Ul , ... , Un' Yl' ... , Yq [compare (c) in 1.1.1]. For example, the Whitney-Thorn stratification (obviously) is locally semi-algebraic.

(E l ) Theorem. If E = UiEi c X(l) is locally semi-algebraic, then the subset Ef c V admits an equisingular stratification for generic Coo -sections f: V -+ X.

This fact (and the proof which follows) is an abstract version of the

(E2 ) Thom Equisingularity Theorem (1955). If f: V -+ W is a generic Coo-map, then there exists a stratification U i Si = V, such that the ranks of the maps f and flSj: Sj -+ W is constant on each stratum Sj.

Proof. (E2) follows from (El) which applies to the Whitney-Thorn stratification. To prove (E l ) we need the following

Canonical Partition. Consider a submanifold Z c x(r) and let Z = (p;+lfl(Z) C

x(r+1). Take a point Z E Z represented by the differential Dip: Tv(V) -+ ~(x(r») of a germ of a holonomic Coo-section cp: V -+ x(r) for v = pr+l(Z) and Z = p;+l(Z) EZ c x(r) (see 1.1.1). Set -r(Z) = D;l(~(Z)) c T,,(V) and let di(Z) = dim(Ei n A (p1(z), -r(Z)). Then partition Z into the subsets Z(i,<5) = {zEZldi(z) = <5}, for i = 0, ... , k, <5 = 0, 1, ... , dim Ei and canonically refine the partition {Z(i, <5)} [see (A2)]. This gives us a stratification of Z in case the partition {Z(i, <5)} is semi-algebraic in some local coordinates around each point Z E Z.

Now, we apply the partition and the refinement procedures to each stratum Ei c X(l) in place of Z in order to stratify the pull-back (pIfl(Ei) c X(2) for all i = 1, ... , k. Thus, we obtain some stratification, say UiEi(2) c X(2) of(pIfl(E) which refines the lift to X(2) of the stratification UiEi = E. We do the same to each stratum Ei(2), thus getting a stratification of (pD-l(E) c X(3), then we pass to X(4)

and so on. The resulting tower of stratifications of (piTl (E) c x(r) for r = 1, 2, ... , enjoys the following stability which is immediate with the definition of i(z): if some stratum Z' c z(r) lies over Z c x(r-l) and if codim Z' = codim Z, then di(z) is constant in ZEZ' = (p;+1)-l(Z') C x(r+1). Hence, no stratum in x(r) of co dimension ::;;,n is stratified further when lifted to x(r+l) for r ~ n + 1.

Now the pull-back of the stratification in x(n+1) for n = dim V under the jet Jj+1 of a generic section f: V -+ X clearly is the required equisingular stratification of Ef·Q·E.D.

Remark. Return to a generic map f: V -+ W, assume dim W ~ dim V and show that no smooth curve C c V goes to a single point in W. In fact, the vanishing of the first r derivatives of f along C at a fixed point c E C is given by rq independent equations for q = dim W. On the other hand, the r-jets of germs of non-parametrized curves C in V form an n + r(n - 1) dimensional manifold for n = dim V. Hence, the

1.4 Localization and Extension of Solutions 35

condition fYtr c x(r) which expresses the vanishing of the first r-derivative of f on some curve C satisfies codim fYtr = rq - nr(n - 1) ~ r - n. Therefore, the r-jet of a generic f misses fYtr for r > 2n. Q.E.D.

(E~) Corollary. A generic map f is an immerion on each stratum Si of the equisingular stratification insured by (E2)' That is the sub bundle Ker DJISi is transversal to Si for all strata Si.

Exercises. (a) Show for dim W < dim Vas well as for dim W ~ dim Vthe subbundle Ker DJISj c T(V)ISi is transversal to Si for codim Si > 0 (assuming f is generic, of course).

(b) Generalize (E 1) to stratified subsets Xc x(r) for r ~ 2, and apply this to X = X(2\$' for the freedom relation $' c X(2).

(c) Observe with (E 2) that the COO-dimension is monotone non-increasing under generic Coo-maps (This is unlikely for non-generic Coo-maps. Probably, there exists a Coo-map f: [R --+ [R2 whose image is not contained in a countable union of Coo-smooth curves in [R2).1

(d) Generalize (E 1) and (E2 ) to generic Ck-sections (and maps) for k ~ n + 2. Observe the set Xl for a generic C2-function f: V --+ [R to be discrete. Show Xl for a generic C1-function f is a Cantor set (i.e. dim(op Xl = 0 and no point in Xl is isolated). Study generic C 1 _ and C2-maps [R2 --+ [R2.

References. See Boardman (1967) for a detailed study of equisingular stratifications of generic maps f: V --+ W See Golubitsky-Guillemin (1973) for an introduction to singularities of smooth maps.

1.4 Localization and Extension of Solutions

1.4.1 Local Solutions of Differential Relations

Take a subset C c V and study solutions of a given relation f7l c x(r) over a small neighborhood U c V of C. These are holonomic sections U --+ fYt. In the following consideration we often use an arbitrarily small but non-specified neighborhood of C, denoted by {l)fiC c V (opening of C in V). This is a small neighborhood of C which may become even smaller in the course of the argument. The following dictionary helps to avoid any ambiguity in dealing with these openings.

The space of Ck-sections (l)fiC --+ x, by definition, is the direct (inductive) limit of the spaces of Ck-sections U --+ X over all neighborhoods U c V of C. This is also called the space of germs of sections (defined) near C. There is no useful natural topology in this space; however, there is a weaker structure, called quasi-topology, which nicely behaves under direct limits.

1 Such a map f [with f(lIl1) ::J A x A for some Cantor set A c 1Il1] was constructed by A.G. D'Farrell (1986).

36

A function or a section Ion (!J/tC c V.

An extension of I from (!J/tCI to (!J/tC2 for C1 C C2 C V.

Two sections 11' 12: (!J/tC -+ f!l are homotopic.

A Ck-continuous family of Ck-section Ip: m/tc -+ X for pEP.

1. A Survey of Basic Problems and Results

Such a function or a section on some neighborhood U c V of C.

There exists a neighborhood U' c V of C such that both functions are defined and equal on U'.

This is a function f' on some neighborhood U' c V of C2 , which equals Ion a sufficiently small neighborhood U" c V of C1 where I and f' are simultaneously defined.

There is a neighborhood U of C on which 11 and 12 are defined and homotopic.

such a family on some neighborhood of C.

(A) Definition (Spanier-Whithead 1957). A quasi-topological structure in a set A is given by distinguishing a subset in the set of all point-set maps of topological spaces P into A, such that these distinguished maps, called "continuous" for the moment, enjoy the following formal properties of ordinary continuous maps.

(i) If J1.: P ~ A is "continuous" and if cp: Q ~ P is an ordinary continuous map, then the composed map J1. 0 cp: Q ~ A is "continuous".

(ii) If a map J1.: P ~ A is locally "continuous", then it is "continuous" where the local "continuity" requires a neighborhood U c P of every point in P such that the map J1.1 U: U ~ A is "continuous".

(iii) Let P be covered by two closed subsets PI and P2 in P. If a map J1. is "continuous" on Pl and on P2 , then it is continuous on all of P. Therefore, if U~=l Pi = P is a covering of P by finitely many closed subsets, then a map J1.: P ~ A is "continuous" if and only if J1.IPi: Pi ~ A is "continuous" for all i = 1, ... , k. [The above (ii) implies a similar property for finite a well as for infinite coverings of P by open subsets.]

Next, a map between quasi-topological spaces, a: A ~ B, is called continuous if a 0 J1.: P ~ B is "continuous" for all continuous maps J1.: P ~ A and for all topological spaces P.

We will from now on write continuous instead of "continuous" if the meaning is clear from the context.

Now, the space of Ck-sections &/tC ~ X is endowed with the quasi-topology which is the direct limit of the quasi-topologies associated to the Ck-topologies in the spaces of Ck-sections U ~ X for all neighborhoods U c V of C. Thus, the notion of continuity for maps J1.: p 1-+ Jp : &/tC ~ X, PEP, agrees with the above Ck-continuity of families.

The standard definitions of homotopy theory (e.g. the weak homotopy equivalence) obviously generalize to quasi-topological spaces. With this we formulate the following


(B) Local h-Principles. A relation flI c x(r) is said to satisfy the h-principle near a subset Vo c V (or the h-principle on (9 It Yo c V) if every section cP: {9 It Vo -+ flI is homotopic to a holonomic section. That is, according to the dictionary, for every neighborhood U c Vof Vo and for every section cP: U -+ flI there exists a neighborhood U' c U of V such that cP I U' is homotopic to a holonomic section U' -+ flI. Furthermore, the parametric h-principle near Vo claims that the map fHJ; of the space of solutions of flI on {91t Vo to the space of sections {91t Vo -+ flI a weak homotopy equivalence. Finally, the local (near Vo) h-principle is called CO-dense in a subspace O!f c CO(XI Vo) (compare 1.2.2) if for every section fo E O!f, for every neighborhood U c X of fo(Vo) c XI Vo = p-t(Vo) c X, and for every section CPo: {91t Vo -+ flI, such that Po 0 CPo I Vo = fo, there exists a homotopy of CPo to a holonomic section CPt by a homotopy of sections CPt: (91t Vo -+ flI, t E [0, 1], such that Po 0 CPt(Vo) c U for all t E [0,1].

Exercise. Give a consistent definition of the Ci-dense local h-principle for i ~ 1.

Remarks and Examples. (B t ) Localization of (DiffV)-Invariant Relations. Suppose there is a natural action of diffeomorphisms of Von X, and let flI be invariant under the associated action of DiffV on x(r). For example, immersions and free maps V -+ W. are defined by (Diff V)-invariant relations while isometric maps are not. If U c V is a regular neighborhood of a piecewise smooth subpolyhedron Vo c V (e.g. a tubular neighborhood of a submanifold Vo c V), then U can be brought to {91t Vo by a diffeotopy in U which is constant on Vo. Therefore, the local h-principles on {91t Vo are equivalent to the corresponding global h-principles on U. If, for instance, V is diffeomorphic to ~n, then the global h-principle on V reduces to the local one near a single point Vo E V. Furthermore, let V be an open manifold. Then there exists a codimension one subpolyhedron Vo c V, such that the h-principle on V localizes to (91t Vo c V by a diffeotopy of V which brings V arbitrarily close to Vo.

Indeed, every open manifold admits a positive Moore function Jl without local maxima. The Morse complex ~ c V of such a Jl clearly has codim ~ ~ 1 and the gradient flow of J-l brings V to (9jt(~).

Exercises. (a) Prove the existence of the above Jl. (b) Construct Vo c V by using some triangulation of Jl. Hint. Since V is open it

embeds to the complement of the set of barycenters of the n-simplices. (c) Show every open manifolds V admits a (non-Morse) COO-function without

critical points.

(B2) Solutions of flI Near a Point Vo E V. If flI c x(r) is an open subset, then the local h-principle on (9ltvo c V obviously holds true. Indeed, every jet x E x(r) over Vo is represented by a holonomic germ f = fx: (9ltvo -+ x(r), such that f(vo) = x. Therefore, f( (9jtvo) c flI for any fixed open subset flI3 x. Moreover, the germ Ix can be easily made continuous in XEx(r) which insures the parametric h-principle near Vo.

Now, let flI be a locally closed submanifold of codimension s in x(r) which is given as the zero set, flI = {x E U c x(r) I 'I'(x) = O}, for some Ck-map, '1': U -+ ~. transversal to 0 E ~', for an open subset U c x(r). If s equals the dimension q of the

fiber Xv C X, V E V, then the h-principle may easily fail to be true. This is seen, for instance, in the well-known examples (see Hormander 1963) oflocally non-solvable linear differential equations. However, if s -# q [which makes the corresponding P.D.E. system 'P(J;) = 0 overdetermined for s > q and underdetermined for s < q] then the local h-principle generically holds true near each point v E V. The following two theorems make this claim meaningful.

(B2) Frobenius' Integrability Theorem. Let A: x(r) -+ IRs be a generic Coo-map for s> q. Then Coo-solutions of the (over-determined) system A{Jf) = 9 satisfy the h-principle on (!)jtv c V for all VE V and for all COO-maps g: V -+ IRS.

Remark. This h-principle insures local solutions f of A (J;) = 9 for the maps 9 which satisfy necessary consistency conditions [see (F) in 2.3.8].

(B~) Let A: x(r) -+ IRs be a generic Coo-map for s < q. Then there exists a stratified subset E c x(r') for some r' = r'(dim x(r» ~ r of positive codimension such that COO -sections f: V -+ X for which

A{.Tf) = 9 and J;'{V) c Xr\E

satisfy the h-principle on (!)jtVE V for all VE V and all Coo-maps g: V -+ IRs.

This is shown in (E) of 2.3.8 for generic fiberwise linear maps A by a purely algebraic argument. Then the non-linear case follows by Nash's implicit function theorem (see 2.3.2).

Remarks. (a) The analytic techniques in 2.3 also delivers Ck-solutions of (*) for k ~ ko = ko(dim x(r», provided the map 9 is Ck+ko-smooth. But the analytic method does not apply to maps 9 of low smoothness. However, the geometric method of convex integration yields Cr-solutions of(*) for all continuous maps 9 (see 2.4.6). Yet, the local h-principle for C+1-solutions of (*) is unknown unless 9 is Cko-smooth for a sufficiently large ko.

(b) If A and 9 are real analytic, then the local h-principle for (*) holds true for all s (including s ~ q) by the classical Cauchy-Kovalevskaya theorem.

(B3) Local Isometric Immersions. The local h-principle is especially interesting when the global one is not available. For example, the h-principle is unknown for (free) isometric Coo -immersions f: (V, g) -+ IRq for q ::;; !(n + 2)(n + 3); however, the local h-principle does hold true by the following

(B~) Theorem. If a Coo-submanifold Vo c V has codim Vo ~ 2, then free isometric Coo-immersions ((!)jtVo,gl(!)jtVo) -+ IRq satisfy the parametric h-principle for all q. This h-principle is also valid for codim Vo = 1 unless q = !n(n + 3) (where the h-principle is unknown). Moreover, the h-principle is CO-dense in the space of strictly short maps Vo -+ ~q for such codim Vo and q.

This is derived in 3.1.6 and 3.1.7 from the global h-principle for isometric immersions Vo -+ IRq.


Observe that (B~) agrees with (B;) when Vo reduces to a single point Vo E V. The

pertinent map L1 is given in local coordinates by s = tn(n + 1) functions, / of , Of), \ oU; oUj

for 1 ~ i ~j ~ n = dim V and L: = X(2)\ff, where ff is the freedom relation. In this case, the existence of a free Coo-immersion ((DjZvo,g!(DjZvo) --+ IRq for q = tn(n + 3) is due to R. Green (1970); moreover, one has with (B~) a weak homotopy equivalence between the space of free isometric immersions (DjZvo --+ IRq and the Stiefel manifold Stm IRq for m = tn(n + 3). Yet, for no small (but fixed) ball B in V around v E V one knows how to construct a Coo -homotopy of isometric immersions between two given free isometric Coo-immersions of B into IRq, unless q z t(n + 2)(n + 3), where the global h-principle of 1.1.5 applies.

Exercise. Assume V to be a parallelizable manifold with a Coo-metric g and derive from (B~) the following

Local Coo-Immersion Theorem. A sufficiently small neighborhood U c V of every Coo -submanifold Vo c V of positive codimension admits a free isometric Coo -immersion into IRq for q = tn(n + 3) + 1, and for q = tn(n + 3), provided codim Vo z 2.

Then find examples of non-parallelizable manifolds V for which no free map (DjZ Vo --+ IRq, q = tn(n + 3) + 1 exists.

(A;) Let (V, g) be a Can-manifold. Then the Cauchy-Kovalevskaya theorem allows a non-free isometric extension of immersions from Vo to (DjZ Vo c V (see 3.1.6). This yields the following generalization of Janet's theorem (which applies to Vo = Vo E V, (see 1.1.5).

The Can-Immersion Theorem. If V is parallelizable and if a Can-submanifold Vo c V has the trivial normal bundle, then, for codim Vo z 2, some sufficiently small neighborhood U c V of Vo admits an isometric Cn-immersion into IRq for q = tn(n + 1). [Compare Gromov (1970).]

Exercise. Let Vo be the projective line in the projective plane p 2 1R with the metric of constant curvature + 1. Show that no neighborhood U c p 2 1R of Vo admits an isometric C2-immersion into 1R3.

1.4.2 The h-Principle for Extensions; Flexibility and Micro-flexibility

The h-principle for extensions of Ck-solutions of ~ c x(r), for some k z r, from a subset e c V to a subset C:::J e in V claims, for every Ck-section CPo: (DjZC --+ ~ which is holonomic on (DjZe, there exists a Ck-homotopy to a holonomic Ck-section CPl by a homotopy of sections CPt: (DjZC --+ /Jl, t E [0, 1], such that CPt! (DjZe is constant in t. This is also called the h-principle over (DjZC relative to (DjZe, or the h-principle over the pair (C, e).

Exercise. Define the parametric and the dense h-principles over (C, e).


The relative h-principle allows one to build global solutions of a relation with the following

(A) Lemma. Let V be a triangulated manifold and let qj satisfy the h-principle over (S, as) for all simplices S of the triangulation. Then qj satisfies the ordinary h-principle (over all of V).

Proof Use the standard induction by skeletons along with the

(A') Flexibility Sublemma. Consider a pair (C, C') of compact subsets in V for C' c C. Let ({)o: {9ftC-+qj be a section and ({);: {9ftC-+qj, tE[0,1], be a homotopy of ({)o = ({)1{9ftC'· Then ({); extends to a homotopy ({)t: {9ftC -+ &l of ({)o (i.e. ({)tl{9ftC' = ({);).

Proof Take a continuous function 15: {9ftC -+ IR+ with a compact support, such that 15 == 1 in a smaller neighborhood U' c {9ftC' of C' c (9ftC'. Let r = tc5(v) and set: ({)t(v) = ({);(v) for v E {9ftC' and ({)t(v) = ({)o(v) for v E {9ftC\{9ftC'.

Exercise. Let qj satisfy the h-principle over (B\ aBk) for all smooth embedded balls Bk c V with smooth boundaries aBk = Sk-l c V for k = 0, 1, ... , dim V - 1. Assume V to be an open manifold and let qj be (DiffV)-invariant [compare (B l ) in 1.4.1]. Show qj satisfies the h-principle.

The flexibility of sections V -+ qj expressed by (A') can be strengthened with the following

(B) Definitions. Let a.: A -+ A' be a continuous map between quasi-topological spaces. Consider a continuous map of a compact polyhedron into A, say ({): P -+ A, and let CP': P x [0,1] -+ A' satisfy CP'IP x 0= ({)' for ({)' = a 0 ({): P -+ A'.

The map a. is called a (Serre) fibration if cP' lifts to a map CP: P x [0,1] -+ A such that CPIP x 0= ({) and a.o cP = CP', for all polyhedra P, maps ({): P -+ A and homotopies cP' of q/.

Calla. a micro-fibration if for all P, ({) and cP' there exists a positive 8 ::5; 1 and a map CP: P x [0,8] -+ A (where e may depend on P, (() and CP') such that CPIP x 0= (() and a. 0 cP = CP'IP x [0, e].

Examples. (a) A submersion between smooth manifolds, a: A -+ A', is a microfibration. This a, if a proper map, is necessarily a fibration. Another condition which insures the Serre fibration property of a submersion a is the contractibility of a.-l(a) c A for all a E A', where a.-l(a) is assumed non-contractible in case it is empty (see 3.3.1).

(b) If A is a topological space and a: A -+ A' is a micro-fibration, then alB: B -+ A (obviously) is a micro-fibration for all open subsets Be A. However, the restriction of a fibration a. to B, is usually not a fibration, but only a micro-fibration B -+ A'. For example, the restriction of the identity map A -+ A on B is a fibration if and only if B is a path component of A.

(B') The fibration property of a map 0:: A --+ A' allows lifts of polyhedral homotopies (/J' from A' to A, while for a micro-fibration the initial phase of (/J' lifts to A. If a homotopy (/J' is constant in t on some subpolyhedron Po c P, then the lift to A can be chosen constant on Po as well. Moreover, let P be a sub polyhedron in another polyhedron Q ::::J P and let qJ: P --+ A and (/J': Q --+ A' be continuous maps, such that O:OqJ = (/J'IP. Then

(i) if 0: is a micro-fibration there exists a lift (/J: (!)jzP --+ A, such that (/JIP = qJ and 0: 0 (/J = (/J'I (!)jzP ,for (!)jzP c Q;

(ii) if 0: is a fibration and P is a homotopy retract in Q [or, equivalently, 1!i(Q, P) = 0, i = 0, 1, ... ], then there exists a lift (/J: Q --+ A, such that (/J I P = qJ and qJ 0 (/J = (/J'.

The proof (which is easy and well known) is left to the reader.

Now, we observe that the proof of (A') applies to continuous families of sections qJp for PEP, thus showing the restriction map qJ--+qJl(!)jzC for qJECO(9PI(9jzC) to be a Serre fibration CO(9f!I(!)jzC) --+ CO(9PI(!)jzC). We express this by calling CO-sections V --+.tJll flexible (over all pairs of compact subsets C and C c C in V). A similar flexibility property is satisfied by Ck-sections of an arbitrary Ck-fibration X --+ V, as the restriction map Ck(XI(!)jzC) --+ Ck(XI(!)jzC) is obviously [compare the proof of (A')] a Serre fibration for k = 0, 1, ... , 00 (but not for can-sections).

Next, we call Ck-solutions V --+ X of.tJll flexible (micro-flexible) over (C, C) if the restriction map of the space of Ck-solutions (!)jzC --+ X to the space of those over (!)jzC is a fibration (micro-fibration).

Examples. (B 1) Open Relations. If 9f! c x(r) is an open subset, then, clearly, [compare the above (b)] Ck-solutions of 9f! are micro-flexible over all pairs of compact subsets in Vfor k = r, r + 1, ... ,00. However, the flexibility may fail to be true. This happens, for instance, to immersions of the disk D2 --+ 1R2, as a regular homotopy near the boundary aD 2 = Sl brings an embedded circle Sl c 1R2 to the figure in 1R2 which extends to no immersion D2 --+ 1R2. Yet, immersions V --+ Ware flexible over all pairs of compact subsets in V in the extra dimension case, that is for dim W > dim V. [See (C) below.]

(B2 ) Generic Underdetermined Systems of P.D.E. If a relation 9P c x(r) of co dim 9f! < q = dim Xv is given by the system (*) in (B;) of 1.4.1, then COO-solutions, as well as C -solutions of 9f! are micro-flexible over all compact pairs in V. This is proven along with the local h-principle [see (E) in 2.3.8 and 2.4.6].

(B~) Linear Equations. Let X and Y be vector bundles over V and let L1: x(r) --+ Y be a homomorphism. Then the flexibility of solutions f of the linear P.D.E. system ~f = g for ~: ff--+ L1(JJ) and for any given section g: V --+ Y is equivalent to the micro-flexibility. In fact, both properties are obviously equivalent to the existence of an extension of every solution of the homogeneous equation ~f = ° from (!)jzC to all of V for all closed subsets C c V. Such an extension is, as a rule, impossible for determined and overdetermined systems, where dim Y ~ dim X. But if dim Y < dim X, and if L1 is a generic COO-homomorphism, then these extensions do


exist [see (E) in 2.3:8] and insure the flexibility of COO-solutions f to flfif = 9 for all COO-sections g: V ~ Y.

(B;) Free Isometric Coo-Immersions (V, g) ~ (w, h). These are micro-flexible over all pairs of compact subsets in V (see 2.3.2).

(C) Flexibility and the h-Principle. Suppose a relation fll c x(r) satisfies the parametric h-principle over a compact set C c V and also over a smaller compact subset C c C. That is the map f,--d'j is a weak homotopy equivalence of the space A of C'-solutions (!}ftC ~ X to the space B = CO(fll\ C) and the space A' of solutions over (!}ftC is w.h. equivalent to B' = CO(fll\ C).

(C1) Lemma (Smale 1959). If C'-solutions of fll are flexible over (C, C), then they satisfy the (parametric) h-principle over (C, C).

Proof Consider the commutative diagram of continuous maps

A~B at til A'~B'

J'

where the horizontal arrows are f 1----+ J'j and where rx and f3 are restrictions of sections from C to C. Take an arbitrary point a'EA', let b' = J'(a')EB', and consider the fibers rx-1(a') c A and P-1(b') c B. Then the exact homotopy sequences of the fib rations rx and f3 form the commutative diagram

... ~ 7r;(A) ~ 7ri(A') ~ 7ri-1 (rx- 1 (a')) ~ 7ri- 1 (A) ~ 7ri-1 (A') ~ ...

t t t t t ... ~ 7ri(B) ~ 7ri(B') ~ 7r i - 1 (P-1 (b')) ~ 7r i - 1 (B) ~ 7r i - 1 (B') ~ ...

where the four non-central vertical arrows are isomorphisms. Hence, by the five homomorphism lemma, the vertical arrow in the middle also is an isomorphism for all a' E A'. This is clearly equivalent to the (parametric) h-principle over (C, C).

(C2 ) Corollary [Smale-Hirsch, compare (B 1) in 1.4.1]. Let flJl c x(r) be a (DiffV)invariant relation which satisfies the parametric h-principle over {!}ftv c V for all v E V and which is flexible over (B\ aBk) for all balls Bk c V with smooth boundaries Bk = Sk-1 C V for k = 0, 1, ... , dim V. Then flJl satisfies the h-principle (over V). Furthermore if V is an open manifold, then the flexibility is only needed for k = 0, 1, ... ,dimV-1.

Proof Since flJl is (Diff V)-invariant the local h-principle implies that over {!}ftBk c V for all balls Bk c V. Hence, by (C1), the parametric h-principle over {!}ftSk-1 implies that over (B\ aBk). Since the sphere Sk is covered by two balls Bk which meet over the equator Sk-1 c S\ we conclude with the proof of the above (A) (which generalizes to families of sections) to the parametric h-principle over {!}ftSk for all Sk c V. Thus, by induction, we obtain the h-principle over the pairs (B\ aBk). Finally, we triangulate V and apply (A) to the balls obtained by smoothing the boundaries of the

simplices of the triangulation. The details of this proof and the study of an open V are left to the reader.

(e~) Remark. We shall later generalize this corollary (see 2.2.2) to all relations PA with no Diff-invariance assumption.

(el ) In order to prove the h-principle with (e2) one needs the flexibility of ~ over (B\ aBk). Smale (1958, 1959) proved this for the immersion relation in the extra dimension case by a direct geometric argument which was generalized to submersions by Phillips (1967) and to k-mersions by Feit (1969). Since these relations are open, the local h-principle is obvious, and so the flexibility does imply the h-principle. The following general fact reduces the flexibility to the micro-flexibility for relations ~ c x(r) which are invariant under "sufficently many" diffeomorphisms of V. Namely, we assume V = Vo x IR and we suppose that ~ is invariant under some (natural) action on x(r) of the group of those diffeomorphisms of V x IR which preserve each line v x IR c V x IR, v E V.

(el ) Theorem (see 2.2.2). If Ck-solutions of ~, for some k ;:?: r, are micro-flexible over all pairs of compact subsets in V, then these solutions are flexible over (C, C') for all compact subsets C c Vo x 0 c V and C' c C.

(e;) Corollary. If ~ c x(r) is open and (Diff V)-invariant, then it satisfies the h-principle over (!J/t Vo c V for Vo = Vo x 0 c V.

Proof Since ~ is open, the solutions of ~ are micro-flexible, and hence, flexible over the pairs in Vo x O. Furthermore, the openness of ~ also yields the local h-principle over (!J/tv E V, v E V, and (the proof of) (e2) applies.

Exercise. Prove with (e;) the h-principle for k-mersions (in particular for immersions) V --+ W for open manifolds V and for all k = 0, 1, .... Then derive the extra dimensional case (dim V < dim W) of Hirsch's immersion theorem for non-open manifolds V. [Feit's k-mersion theorem for non-open manifolds V and k < dim W also follows from the case of V open, but the argument is more difficult than the one for immersions. See Feit (1969) and 2.2.4. A short proof of Feit's theorem is given in 2.1.1.]

(e4 ) Isometric Immersions into Pseudo-Euclidean Spaces. Let h be the (indefinite) form ~],::;1 dxf - L:++l dxf on IRq and let 9 be an arbitrary quadratic differential COO-form on a manifold Vo. A C 1-map f: Vo --+ IRq is called isometric if f*(h) = g, which is expressed in local coordinates Ui, i = 1, ... , n = dim Vo on Vo by the system

/ af , af ) = gij [compare (Is) in 1.1.5]. The pertinent differential relation is in\aui aUj h variant under the isometry group of (vo, g) which is a very small (in fact, generically trivial) subgroup in Diff Vo. But the manifold V = (Vo x IR, 9 ~ 0) has all v x IR preserving diffeomorphisms for isometries! Furthermore, free isometric Coo-immersions V --+ (IRq, h) are micro-flexible and they satisfy the parametric


h-principle over {!}jzv c V for all v E V, as we shall see in 2.2.2 with Nash's implicit function theorem. This implies flexibility and hence, the h-principle [see (C~)] for free isometric Coo -immersions {!} jz Vo -+ ~ for Vo = Vo x 0 c Vo x ~. Finall, we construct a section of our (free isometric) relation over {!}jzVo (see 3.3.1) for

(*) q+ ~ 2n + 1, q_ = q - q+ ~ 2n + 1, q ~ t(n + 1)(n + 8)

and thus conclude to the following

(C~) Isometric Immersion Theorem. The inequalities (*) insure an isometric COO-immersion (VO, g) -+ (~q, h) for all COO-forms g on Yo. (See 3.3 for sharper results.)

Remark. A similar approach applies to more general "isometric" immersions f: V -+ W which induce given tensors (e.g. symplectic and contact forms, see 3.4) on V.

1.4.3 Ordinary Differential Equations and "Zero-Dimensional" Relations

An ordinary equation on V is associated to a vector field L on V and it is expressed with the Lie derivative Lfby P"(v,f(v), Lf(v)) = g(v), where fis the unknown Cl-map V -+ ~q, and where P": V x ~q x ~q -+ ~s and g: V -+ ~s are given maps.

Example. The simplest linear equation Lf = g for functions f, g: V -+ ~ may easily violate the h-principle. In fact, if V is a closed manifold, then every g = Lf necessarily satisfies Jv g dp, = 0 for all probability measures dp, on V which are invariant under the flow on V generated by L. This provides a (non-vacuous!) condition for solvability of Lf = g which is not accountable for by the h-principle. For example, the inequality Lf > 0 is never solvable on closed manifolds V (which, of course, is obvious without any dj1.), while the h-principle predicts such an f for non-vanishing fields L.

Exercise. Let V be a compact connected manifold with a non-empty boundary and take a smooth non-vanishing field L on V whose every integral curve is- a segment in Vending in aVo Prove the h-principle for the inequality Lf> 0 and for the equation Lf = g for all smooth functions g on V.

We shall see in 2.3 that many non-linear under-determined (i.e. q > s) ordinary equations do satisfy the h-principle, and then we shall derive the h-principle for a class of partial differential equations on V (e.g. for isometric Cl-immersions V -+ ~q). Now, we concentrate on easier relations concerning the behaviour of f on subsets Vo c V whose intersections with the integral curves of L are zero-dimensional.

Example. (AI) Let L be a non-vanishing field on V and let Vo c V be a submanifold transversal (i.e. nowhere tangent) to L. Then the relation Lfl Vo -# 0, which requires Lf(v) =F 0, for all VE VO (and claims nothing what-so-ever outside Yo), satisfies the h-principle.

Proof This is obvious with functions f on V which satisfy fl Va = fa and Lfl Va = fl for given functions fa and f1 on Va.

(A'd Remark. The relation Lfl Va # 0 obviously satisfies the h-principles we have met so far. In particular, the h-principle holds true over each pair of closed subsets C c V and C' c C.

(A2 ) Semi- Transversality. A field L is called semi-transversal to a closed subset Va c V if there is a stratification of Va by smooth submanifolds Ii c Vo (see 1.3.2) i = 0, ... , k, which are transversal to L.

If L is semi-transversal to Va then the relation Lfl Va # 0 satisfies the h-principle.

Proof Let fj,-1 = Un~iLj, assume, by induction in i = 0, 1, ... , the h-principle for the relation Lfl fj,-i+1 # 0 and observe that fj,-i = Ii U fj,-i+1 and that the topological boundary aIi = CIIi\Ii lies in fj,-i+1. Then (A~) insures the h-principle over the pair (I;, Ii n CI(@jZfj,-i+1)) for all @jZ"Yk -i+1 c V This allows an extension ofthe h-principle from @jZ fj,-i+1 to @jZ fj,+1 C V Q.E.D.

(A~) Remark. The definition of the semi-transversality and the above proof obviously generalizes to an arbitrary line subbundle L c T(V)I VA' for which our relation can be expressed by dflL # 0 for the differential df of f Moreover, one easily obtains the parametric and the dense h-principles as well as the h-principle over the pairs of closed subsets in V

(A~) Corollary. Let F': V -4 IRq-1 be a Coo-map. Impose a differential condition f!Jl on f: V -4 IR by requiring the map F = F' EEl f: V -4 IRq to be an immersion. If F' is generic and q - 1 ~ dim V then fYt satisfies the h-principle.

Proof Let Va = IF' = {VE VlrankvF' < dim V}. Then the relation fYt-4 V reads dJIKer Dr # 0 (i.e. dJ should not vanish on the non-zero vectors in Ker Dr c

T(V)I Va). If f!Jl admits a section V -4 fYt, then, obviously, Ker DF' is a line subbundle in T(V)I Va. Since F' is generic, this subbundle is semi-transversal to Va [see (E~) in 1.3.2] and (A~) applies.

Remark. We shall see in 2.1.1 how (a slight generalization of) (A~) yields the h-principle for immersions, submersions, k-mersions and for free maps into IRq.

Exercises. (a) Let L be a non-vanishing vector field on V and let a closed subset Va c V contain no open interval in any integral curve C of L [i.e. dimtop(Vo n C) = OJ. Show the relation Lfl Va = g satisfies the CO-dense h-principle for all CO-functions gon V

(b) (Hirsch 1961). Let V be an open manifold. Prove, by localizing to some Va c V of codim Va = 1 [compare (Bd in 1.4.1] that Coo-functions V -4 IR without critical points (i.e. submersions V -4 IR) satisfy the parametric h-principle.


(B) Remarks and Open Questions. One usually proves the h-principle for a relation f!It by reducing the problem to an auxiliary elementary relation like those considered in (AI)' The reduction process uses standard (often tedious but straightforward) methods of soft analysis like the partition of unity or an induction by strata of some stratification, etc. But the resulting elementary problem may require a specific geometric construction for its solution. In many cases, however, the elementary relation becomes very simple, like the "zero-dimensional" relation in (AI) or like an ordinary "one-dimensional" differential equation (see 2.4), and no geometry is needed at all. Yet, there are many interesting relations which do not (seem to) reduce to anything "one-dimensional" an where the geometric intuition fails to provide a 2-dimensional construction.

(B')· Example. Directed immersions. Let V be an oriented n-dimensional manifold and let A c S" C 1R"+1 be an arbitrary subset. An immersion f: V -+ 1R"+1 is, by definition, directed by A if the tangential (Gauss) map Gf : V -+ S" sends V into A. If V is an open manifold and if A is an open subset in S", then the h-principle for directed immersions is immediate with (C2 ) in 1.4.2. Now, let the manifold V be closed. Then, for every vector s E S" c 1R"+1 there obviously exists a hyperplane H c 1R"+1 orthogonal to s which is tangent to a given closed immersed hypersurface f: V -+ 1R"+1 at some point f(v) Ef(V) c 1R"+I. Therefore, Gf(V)U( -Gf(V)) = S", for the reflection - Gf(V) of the image Gf(V) c S" in the center. [Iff: V -+ 1R"+1 is an embedding then, clearly, Gf(V) = S".]

Question. Let A c S" be an open connected subset, such that A U ( - A) = S". Do immersions directed by A satisfy the h-principle? In particular, does there exist a single closed immersed hypersurface in 1R"+1 directed by A? (If A =F S", such a hypersurface is, necessarily, parallelizable.)

If n = 1, then the answer is an easy "yes" (an exercise for the reader). But for n = 2 and for V = T2 (the torus T2 is the only parallelizable closed surface) the answer is unknown for most subsets A c S2. For example, the "yes" is obtained (with the convex integration, see 2.4.4) for those A c S", whose complement is a finite subset, or, more generally, a disjoint union of small balls. Yet, the question is open for A = S2 \ C, where C is a simple arc in S2 which is e-dense in S2 for a small e > 0, say for e = 0.1.

1.4.4 The h-Principle for the Cauchy Extension Problem

Consider a relation f1l c x(r) and let Vo c V be a closed subset. Call the Cauchy (initial value) data of order i a section qJo: Vo -+ X(il such that there exists a holonomic section qJ~: l!!fiVo -+ X(i) for which qJ~1 Vo = qJo. We always assume i::; r; otherwise, we lift f1l to f1li-r -+ X(i) (see 1.1.1). Furthermore, when dealing with Ck-solutions of f1l, we assume qJo = J}b for a Ck-section fo: l!!fiVo -+ X. We associate with given Cauchy data qJo on Vo the Cauchy relation f1lrpo c f1l which equals f1l over V\ Yo, while f1lrpol Vo equals (pitl(qJO(V)) n f1l. Thus, solutions f of f1lrpo are those solutions


of ~ which satisfy J}I VO = CPo. The h-principle for the Cauchy problem for given ~ and CPo is defined as that for the Cauchy relation ~qJo' We distinguish the local Cauchy problem, which concerns the h-principle for ~qJo over {r)jz Vo c V, and the global one, where we extend CPo to a solution f of ~qJo on all of V. If the h-principle for extensions of solutions of ~ from {r)jz Vo to V is available, then the local h-principle (for ~qJo on (r)jz Vol yields the global one.

Examples. (A) If ~ is an open relation of the first order, ~ c X(1), then ~qJo obviously satisfies the local h-principle for all initial data CfJo of order zero or one. However, the local extension may run into a non-trivial global problem on Vo for relations of order ;:::: 2. Consider, for instance, the freedom relation § c X(2) over V = Vo x IR, and let CPo: Vo --+ W, for p = t(n - l)(n + 2) and n - 1 = dim Vo, be a free map that we want to extend to a free map f: V --+ IRP x IRq ::::> IRP x 0 = IRP for Vo = Vo x 0 c V. Denote by 1': V --+ IRq the orthogonal projection to IRq of the derivative (df/dt): V --+ W x IRq, t E IR. If the map f is free on {r)jz Vo c V, then a straightforward computation shows 1'1 Vo: Vo --+ IRq to be an immersion. Thus, the extension of free maps from Vo to {r)jz Vo is at least as difficult as producing immersions Vo --+ IRq. In fact, the h-principle for free extensions from Vo to {r)jz V is valid by the techniques of 2.2.

(B) Isometric C1-Immersions. Any extension of such an immersion from a submanifold Vo must be preceded by an extension which is strictly short outside Vo, in order to apply the Nash-Kuiper techniques. For example, if Vo c V is a nongeodesic line and if fo: Vo --+ IRq is an isometric map onto a straight line in IRq, then no short (in particular isometric) extension to {r)jz Vo exists.

(C) Free Isometric COO-Immersions. The local Cauchy problem for these satisfies the h-principle (see 3.1.6). This is used in the proof of the h-principle for free isometric immersions {r)jzVo --+ IRq [see (B3) in 1.4.1].

(D) The Cauchy Problem with Can-Data. The local Cauchy problem is easier in the Can-case, as the solution can be often reduced to the Cauchy-Kovalevskaya theorem [compare (A;) in 1.4.1]. On the other hand, a can-extension of a can-section fo: Vo --+ X to a (global!) section f: V --+ X, which is not a subject to any relation, is a non-trivial problem solved by H. Cartan (1957). In fact, Cartan's result goes along with the analytic techniques of 2.3 which imply, in particular, the following

Approximation Theorem. Let f: (V, g) --+ IRq be a free isometric COO-immersion which is real analytic on some Can-submanifold Vo c V. If the Riemannian manifold (V, g) is real analytic, then f admits a fine Coo-approximation by isometric Can-immersions 1': (V, g) --+ IRq, such that 1'1 Vo = fl Vo·

Part 2. Methods to Prove the h-Principle

2.1 Removal of Singularities

Consider a differential relation f!It c x(r) whose complement E = x(r)\f!It is a closed stratified subset in x(r) of codimension m ~ 1 and take a generic holonomic COO-section I: V -+ x(r) whose singularity Ef = I-1(E) c V may be non-empty (compare 1.3). Let us try to solve f!It by deforming I to a holonomic E-non-singular section 1: V -+ x(r). Such a deformation can not be, in general, localized near Ef [see Exercise (a) below] but one can find in some cases an auxiliary subset E' = E'(f) ::::> Ef in V of codimension m - 1, such that the desired deformation does exist in an arbitrarily small neighbourhood of E'. The major difficulty in the construction of 1 comes from the holonomy condition. In fact, the problem becomes quite easy without this condition, as one can see in the following

Exercise. (a) Let E be an m-codimensional submanifold in x(r) such that the projection E -+ V is a proper map. Let a (holonomic or not) COO-section 10: V -+ x(r) be transversal to E and let Efo be a non-empty submanifold (of codimension m) in V. Consider an arbitrary homotopy of continuous sections J;: V -+ x(r) which are equal to 10 outside a sufficiently small neighbourhood of Ef 0 in V and show the subsets Eft = J;-1(E) C V to be non-empty for all t ~ O.

(b) Assume the existence of a continuous section iii: V -+ f!It = x(r\E and construct a smooth family of (possibly non-holonomic) sections J;: V -+ x(r), such that Ef 1 is empty and the sections J; are equal to 10 outside an arbitrarily small neighbourhood of some stratified subsetE' = E'(Io, <pdin Vofcodimensionm-1.

We are interested, of course, in holonomic E-non-singular sections and these cannot be constructed without additional assumptions on E. The method of removal of singularities which we present in this section applies, roughly speaking, to those differential relations rYt = x(r\E which are "semi-transversal" (compare 1.4.3) to sufficiently many subsets E' in V.

2.1.1 Immersions and k-Mersions V -+ ~q for q > k

Start with a Coo-map F: V -+ ~q = ~q-1 Ee ~ which is split into the orthogonal sum, F = F' Ee I for the projections F' and I of V to ~q-1 and ~ respectively. Our first objective is a COO-function 1: V -+ ~ for which the map F = F' Ee 1: V -+ ~q

2.1 Removal of Singularities 49

is an immersion. The pertinent singularity here is the subset EF C V where rank F < n = dim V. Clearly codim EF = q - n + 1 for generic maps F (see 1.3.1). A natural candidate for E' is the subset Er C V where rank F' < n. Indeed, the sum P = F' EEl 1 is an immersion outside this E' for any 1. since E' => Eli'

Now, recall (A;) of 1.4.3.

(A) Lemma. If q > n and if the map F' is generic then COO-functions 1 for which the sumP = F' EEl 1 is an immersion satisfy (all forms of) the h-principle. In particular the following conditions (a) and (a') are necessary and sufficient for the existence of J (a) The map F' has rank ~ n - 1 everywhere on V. Hence the kernel of the differential Dr is a one dimensional subbundle of the tangent bundle T(V)IE'. (a') The bundle Ker Dr on E' is trivial. This is equivalent to the existence of a non-vanishing vector field L on V near E', such that Dr(L)IE' = O.

(A') Remark. Let tP = tPr denote the space of those Cl-smooth 1-forms iii on V which do not vanish on Ker Dr C T(V). Then an arbitrary form iiio e tP admits a Cl-homotopy iiit in tP, t e [0,1], to an exact form iiil in tP. In fact, this is equivalent to the above h-principle, since Ll = iii(L) for the (exact) form iii = df

Furthermore, a map F = (fl" .. ,/q): V -+ IRq is an immersion if and only if the differentials of the coordinate functions, say ({Ji = d/;, i = 1, ... , q, span the cotangent space T.,*(V) for all ve V. This suggests the following generalization of (A). Denote by tPq the space of the q-tuples of 1-forms «((Jl, ... ,({Jq) which span T.,*(v) for all ve V. Fix a (q - 1)-tuple "" = «({Jl"'" ({Jj-l' ({Jj+1"'" ({Jq) for some j between 1 and q and let tP"" consist of those Cl-smooth forms iii on V for which the q-tuple «({Jl' ... , ({Jj-l' iii, ({Jj+1' ... , ({Jq) is contained in tPq.

(B) Let ({Ji' i = 1, ... ,j - 1, be generic exact COO-smooth 1-forms on V (i.e. ({Ji = d/; for generic COO-functions /; on V) and let ({Ji for i = j + 1, ... , q be generic (non-exact) Coo -smooth 1-forms. If q > n, then an arbitrary form iiio e tP"" admits a homotopy iiit in tP"", t e [0, 1], to an exact form iiil = dl e tP"".

Proof The pertinent differential relation (on 1) is concentrated on the subset E' c V, where the forms ({Jl' ... , ({Jj-l' ({Jj+1"'" ({Jq fail to span the cotangent bundle T.,*(V). Then there is a vector field L near E', such that ({Ji(L)IE' = 0 for i oF j and iiio(L) does not vanish on E'. The proof of (E~) in 1.3.2 shows that field L is semitransversal to E' under our genericity condition on "" = «({Jl"'" ({Jj-l, ({Jj+1,"" ({Jq), for q - 1 ~ n. Hence, the proof of (A) applies.

(C) The h-Principle of Smale-Hirsch for Immersions V -+ IRq. If q > n then an arbitrary q-tuple '" = «({Jl, ... ,({Jq)etPq admits a homotopy in tPq to a q-tuple If/etPq

of exact forms iii; = d"/;, (i = 1, ... , q).

Proof The q-tuple '" is made component-wise exact in q steps with lemma (B). The genericity assumption in (B) is satisfied at each step with a small generic perturbation of the pertinent function fl' ... ,fj-l and the forms ({Jj+l' ... , ({Jq. Q.E.D.

50 2. Methods to Prove the h-Principle

Remark. This h-principle is equivalent to the one stated in 1.1.3, since q-tuples of 1-forms on V define homomorphisms T(V) ~ T(~q) modulo translations in ~q.

(C') The Parametric h-Principle and the h-Principle for Extensions. The proof of (C) equally applies to families of immersions (compare 1.2.1) thus providing the parametric h-principle. Furthermore, the same argument delivers the h-principle for extensions (see 1.4.2) as well as the CO-dense h-principle. In fact, the reduction to the zero-dimensional case allows the W;'P-approximation (see 1.2.2) which is left as an exercise to the reader.

(D) Further Exercises. (a) Prove the h-principle for k-mersions V ~ ~q for q > k. Namely, take an arbitrary q-tuple '" of 1-forms ({J;, i = 1, ... , q, which span a subspace of dimension ~k in the cotangent space T,,*(V) for all VE V and then deform this", to a q-tuple of exact forms with the same k-spanning property.

(a') State and prove a similar result for q-tuples of exterior forms of degree d ~ 1 on V. Then derive the following

Corollary. Denote by A:(V) the space of exterior d-forms on T,,(V) [which has dim Ad = (d) = n!jd!(n - d)!] and show that every parallelizable manifold V admits Gi) + 1 exact d-forms which span ~(V) at all points v E V.

Remark. If 2 :::; d :::; n - 1 then there are Gi) forms which span ~(V), v E Vbut the proof (see 2.4.3) exploits one-dimensional (rather than zero-dimensional) techniques.

(b) Consider two vector bundles Y and Z over V and let ~ be a Coo -smooth linear differential operator on sections, P): roo(y) ~ roo(Z). Fix two integers, k ~ 0 and q > k and prove the h-principle for those q-tuples of sections f1, ... , h: V ~ Y for which the sections !?}f: V ~ Z, i = 1, ... , q, span a subspace of dimension ~ k in each fiber Zv E Z, V E V. [Compare Gromov-Eliasberg (1971), Burlet (1976). See 2.4.3 for a stronger result.]

(b') Derive (a) and (a') from (b). Then prove the h-principle for free maps V ~ ~q for q ~ [n(n - 3)/2] + 1, by considering the operator ~:f ~ JJlconst for the 2-jet J2 on functions f: V ~ ~.

(c) Consider a manifold V with a given k-dimensional Coo-subbundle 't c T(V) and prove the h-principle for those Coo-maps f: V ~ ~q, q > k, whose differential ~ i V ~ T(~q) is injective on 't. Then prove this h-principle for an arbitrary continuous subbundle 't c T(V).

Hint. Study the pertinent singularities of generic C1-maps V ~ IRq-i.

(E) Embeddings V ~ W. If V is compact, then embeddings f: V ~ Ware characterized by the absence of double points. That is Vi =F V2 implies f(vd =F f(v 2 ). This is not a differential relation, of course, but even so a suitably chosen h-principle makes good sense. Denote by S = S(V, W) the space of those continuous maps s: V x V ~ W x W commuting with the involutions (Vi' V2) ~ (V2' Vi) on V x V and (Wi' W2) ~ (W2' Wi) on W x Wand having s(vi , V2) =F S(V2' vd for all pairs of distinct points V1 and V2 =F V1 in V. The Cartesian square Sf: V x V ~ W x Wof

any map f: V --+ W commutes with the above infolutions, while the second property of s = sf expresses the "no double point" condition. Now we state the h-principle as the possibility to deform an arbitrary continuous map in S inside S to a smooth map of the form sf. Haefliger (1962) proved this h-principle for smooth proper embeddings under the assumption q > 1(n + 1) for q = dim Wand n = dim V. He also found counterexamples below this dimension. He removes double points by a more geometric (and more complicated) method than our step by step argument [see the proof of (C)]. In fact the step by step removal procedure also yields Haefliger's h-principle, at least for embeddings V --+ ~q (see Sziics 1980). Moreover, this method applies to other comparable classes of maps, such as maps without triple points (see Sziics 1982, 1983, 1984).

Exercises. (a) Express the "no double point" condition for maps F = (fl'··· .!2): V --+

~q in terms of the (linear difference) operator which sends functions f on V to anti symmetric function on V x V by the rule f --+ f(v 1 ) - f(V2).

(b) Show a C1-map F: V --+ W to be an immersion if and only if every sufficiently small CI-perturbation of F is a local embedding, which means no double points which are close in V.

(c) Derive from Haefliger's h-principle the following corollaries (see Haefliger 1962; Haefliger-Hirsch 1962).

(c') Every topological embedding V --+ W for q > 1(n + 1) admits a Coapproximation by Coo-embeddings.

(c") If V is a k-connected manifold [i.e. n;(V) = 0, i = 1, ... , k] for k < n12, then it admits a COO-embedding into ~2n-k.

Question. Consider a vector bundle Y --+ V and take its mth Cartesian power (yt --+ (Vt· The power (f)m: (vt --+ (y)m of a sectionf: V --+ Yis defined by the rule

(ft(Vl' V2,···, vm) = f(vd EB f(v2) EB ... EB f(vm)·

Observe that the permutation group Sm naturally acts on (Yt and on (Vt. Take another bundle Z --+ (vt with an action of Sm which covers the action on (v)m and consider a differential operator on sections, ~: roo(y)m --+ roo(Z) which commutes with Sm. We are interested in those q-tuples of sections fl' ... , f2: V --+ Y for which the sections ~(fIt, ... , ~(hf: (Vt --+ Z span in each fiber of Z a subspace of dimension ~ k. What conditions on n, q, m, k and ~ would imply the h-principle for such q-tuples (f1'··· ,h)?

Example. Let m = 2 and let Y and Z be trivial line bundles, say Y = V x ~ --+ V and Z = V x V x ~ --+ V x V, where (VI' V2' t) goes to (V2' VI' -t) under the generator of S2 ~ 71.1271.. Let ~ be the zero order operator which acts on sections (f1.J2): V x V --+ (V x V) x ~ x ~ by ~: (fl (VI' V2).!2(V I, V2» --+ fl(VI, V2) -f2(V I,V2). Then the q-tuples in question (for k = 1) are embeddings F = (fl, ... ,/q): V --+ ~q [compare the above (a)].

Remark. To make the general situation completely consistent with the case of embeddings one must augment the requirement dimension ~ k by some condition

at the diagonals of the manifold (Vr = V x V x ... x V. For example, the "no m-multiple points" requirement for Coo -maps F: V ~ ~q brings forth (among others) the differential condition (at the principal diagonal L1 ~ V in v<m») which claims "no local m-multiple points" for small Coo-perturbations of F [compare the above (b)].

Another Example. Consider those Coo-immersions F of V ~ Sl into ~q for which

the vectors dF (vd, ... , ddF (vm) in ~q are linearly independent for all m-tuples of ds s

distinct points Vl' ••• , Vm in V. Then the associated differential condition is the mth order freedom of F (see 1.1.4).

(E') Further Questions and Exercises. The conception of "multiplicity" generalizes to maps f: V" ~ W q, for q :::;; n, by introducing some measure of the "topological complexity" of the pull-backs f-l(W) E V", WE W. For example, let P(X) denote the sum of the Betti numbers of a space X with the coefficients 7L/27L and let P(f) = SUPweW P(f-l(W)). Then one would like to evaluate inf P(!) over some class (e.g. a homotopy class) of maps f: V ~ W.

(a) Find a map f: V ~ ~q which minimizes P(f) among all smooth maps for a given surface V and a given integer q ~ 1.

(b) Let Vbe a closed n-dimensional manifold. Find a (generic) Coo-mapf: V ~~" for which P(f) :::;; N for some universal constant N = N(n).

Hint. Construct a cobordism W"+1 between V and a disjoint union V' of certain standard n-dimensional manifolds such that W"+l is obtained from V' by attaching [en + 1)-thickenings ofJ k-handles for k:::;; n - 1. [Compare Gromov-Lawson (1979)].

(b') Construct, for a given q ~ n, a Coo-map f: V ~ ~q which meets each affine (q - n)-dimensional subspace A in ~q at no more than N = N(n,q) points [which means P(f-l(A)) :::;; N for generic maps fJ.

(b") Let 1tl(V) = 0 and n = dim V ~ 5. Find a (generic) Coo-map f: V ~ ~"-l for which P(!) :::;; N = N(n).

2.1.2 Immersions and Submersions V ~ W

Fix a submanifold Vo c V of positive codimension, take a generic Coo-map F': V ~ ~q-l and look for a Coo-function 1: V ~ ~ for which the map F' EB 1: V ~ ~q is a submersion near Yo.

(A) Lemma. The above functions 1 satisfy all forms of the h-principle.

Proof An induction by strata of some equisingular stratification of F' (see 1.3.2, 1.4.1,1.4.2) reduces the lemma to the equisingular case where the map F'I Yo: Vo ~ ~q-l has constant rank roo Furthermore, the kernel bundle Ker Drl Vo is not contained in the tangent bundle T(Vo) c T(v)1 Yo for generic maps F'. [In fact, the inclusion relation Ker Dr c T(Vo) imposes a differential condition on F' which has infinite codimension in the space of jets of infinite order, compare 1.3.2.J


Take an arbitrary I-form iiio on V which does not vanish on Ker Dr. Then there exists a vector field L on V near Vo such that LI Vo is contained in Ker Dr c T(V)I Vo and such that the function iiio(L) does not vanish. Since Ker Dr 1- T(Vo), generic fields L with the above properties are semitransversal to yo. [This is obtained with an obvious computation ofthe codimension of the "bad" jets of fields, compare (E~) in 1.3.2.J Now we may assume L semitransversal to Vo and then we obtain [see (Az) in 1.4.3J a homotopy of forms ifit, t E [0, IJ, which do not vanish on L, and such that the form iii! is exact, say iii! = dJ But the non-equality iii! (L)lVo =F 0 implies iii!IKer Dr =F 0 which is equivalent to the sujectivity of the differential of the map F' EB J Thus we homotopied iiio to the desired exact form as required by the h-principle. Furthermore, this argument automatically gives the CO-dense hprinciple for extensions and it also applies to families of functions which amounts to the parametric h-principle.

(A') Coronary to the Proof. Submersions (!)/t Vo --+ IRq satisfy all forms of the h-principle.

Indeed, an arbitrary q-tuple of I-forms ({Ji on V, i = 1, ... , q, which are linearly independent on T(V)I Vo can be deformed in q steps [compare (B) and the proof of (C) in 2.1.1J to a q-tuple of exact forms ifit = i[; which are independent on T(V)I Vo as well. Then the map F = (!t, ... ,h): V --+ IRq is a submersion on a small neighborhood (!)/t Vo c V. Q.E.D.

(B) The h-Principle for Submersions. If V is an open manifold, then submersions of V into an arbitrary manifold W satisfy the parametric h-principle.

Proof The manifold V can be identified with an arbitrarily small regular neighborhood of some smooth subpolyhedron Vo of positive codimension in V (see 1.4.1). Then the (non-parametric) h-principle reduces to the CO-dense h-principle for submersions (!)/t Vo --+ W To prove the dense h-principle, subdivide Vo into small simplices L1 such that the image fo(L1) lies in a small neighborhood U = U(fo, L1) ~ IRq, q = dim W for all ,,1. Then the h-principle for (extensions of) submersions (()Id --+ U [see (A')J implies (with the standard induction by skeletons) the CO-dense h-principle for submersions (()/t Vo --+ W Furthermore, this argument applies to families of submersions (!)/t Vo --+ W, thus yielding the parametric h-principle.

(B') Remarks. The above globalization argument which reduces submersions V --+ W to those into IRq applies to all differential relations which are invariant under diffeomorphisms of the target manifold. In particular, Hirsch's theorem for immersions vn --+ wq, q > n, follows from (C) in 2.1.1. Furthermore, the h-principle [due to A. Phillips (1969)] for maps V --+ W transversal to a given foliation of codimension p in W reduces to the above (A') provided V is open. Indeed, the problem localizes to the maps V --+ IRq which project to submersions V --+ IRP = IRq/lRq-p. However, this localization does not work for maps V --+ W transversal to a non-integrable subbundle 1" E T(V), where the techniques of 2.2 are needed.

Exercise. Prove the h-principle for free maps V --+ W, where W is a Riemannian manifold of constant sectional curvature and dim W ~ [n(n + 3)/2J + 1.


Hint. This W is locally projectively isomorphic to ~q and the freedom is a projective invariant of maps V -+ ~q.

2.1.3 Folded Maps vn -+ WI for q ~ n

The h-principle is no longer true for immersions V -+ W between equidimensional manifolds, if V is a closed (i.e. compact without boundary) manifold. In fact, no immersion f of a closed manifold V into an open W is possible for q = n. This is, of course, obvious, but the following three proofs illustrate different kinds of obstructions to the h-principle.

(1) Compose f with a COO-function cp: W -+ ~ without critical points [see (Btl in 1.4.1]. Then, the map f fails to be an immersion at each maximum point of cpo f: V -+~.

(2) Assume V and W to be orientable (if not pass to the oriented double coverings) and let w be a (non-vanishing!) oriented volume form on W Since Hn(w, ~) = 0, the pull-back formf*(w) has Jvf*(w) = 0 by the De Rham-Stokes theorem and so f fails to be an immersion at the zero set of the induced form f*(w) on V.

ol Since V has no boundary, every equidimensional immersion f: V -+ W is an open map, and since V is compact, the image f(V) c W is compact as well as open. This is absurd, for W is open.

Exercises. (a) Show that a closed surface with a metric of negative curvature admits no isometric C2-immersion f: V -+ ~3.

Hint. Study maximum points of the function II f II on V.

(b) Let W be a (possibly closed) simply connected (n + I)-dimensional manifold with a fixed volume form wand let L be a non-vanishing divergence free (i.e. Lw = 0) vector field on W Show that no closed n-dimensional manifold V admits an immersion f: V -+ W which is everywhere transversal to L.

(c) Let V and W be connected complex analytic manifolds of the same dimension and let V be closed. Suppose there is a holomorphic map f: V -+ W whose Jacobian is not identically zero. Show that W is also closed and that the map f is onto.

(d) Let fo: V -+ W be a submersion, where q ~ n and where V is a closed manifold. Show (assuming W is connected) that every map f: V -+ W homotopic to fo sends V onto W

(e) Prove for an arbitrary submersionf: ~n -+ W n- 1 that sUPwewlengthf-l(w) = 00. Generalize this to submersions ~n -+ W n- k•

Now we turn to those maps f: yn -+ wq, q ~ n, whose singularity is as simple as possible. Recall (see 1.3.1) that the singularity Ef c V (where f fails to be a submersion) has codim Ef = q - 1 for generic maps f

Basic Example. Let Uo = Vo X ~q-n+l for dim Vo = q - 1 and let fo: ~q-n+1 -+ ~ be the polynomial fo(x) = fo(x 1 ,··. ,xq- n+1 ) = If=l xl - I7~::t xl. Then the singu-

larity of the map j: Vo ~ Vo x IR given by j(v, x) = (v,fo(x)) is supported on Vo = Vo x ° c V and this is called the standard folding (along Vo) of type 10"1 = In - q + 1 - 2pl.

An arbitrary smooth map f: V ~ W is said to fold with type 10"1 along a (q - 1)dimensional submanifold V' c V if each point v E V' admits a (small) neighborhood Vo c V' and a split neighborhood Vo = Vo x IRq-n+1 c V such that the map fl Vo decomposes into f = 10 0 j, where j: Vo ~ Vo x IR is the standard folding of type 10"1 and where 10: Vo x IR ~ W is a Coo-immersion.

Remark. One can often take Vo = V'. This is possible, for example, for q = n if the submanifold V' c V is normally orientable (compare 1.3.1).

Theorem (Eliashberg 1972). Let V; c V, for i = 0, ... , 0"0' be disjoint (non-empty!) (q - I)-dimensional properly embedded submanifolds where 0"0 is the greatest integer ~ (n - q + 1)/2. If q ~ 2, then the Coo -maps f: V ~ W which fold along V; with type n - q + 1 - 2i for i = 0, ... ,0"0' and which are non-singular outside Ui V; satisfy the CO -dense h-principle.

We shall sketch below the proof for q = n and we refer to Eliashberg's 1972-paper for the general case. [Compare Yoshifumi (1982).J

Remark. Consider a smooth family of functions fv: IRq-n-l ~ IR, v E Vo, and define F: Vo ~ Vo x IR by F(v, x) = (v,f,,(x)). Then the singularity of the map F consists of those pairs (v, x) E Vo, for which the point x is critical (i.e. singular) for the function f". This connection between singularities of maps and critical points of families of functions plays a crucial role in the study of both. In particular, the equidimensional folding theorem is obtained with a simple (one-dimensional) analysis of functions fv: [0, IJ ~ IR as follows.

(A) Lemma. Let Q+ be an open subset in Vo x [0, 1J and let qJo = qJo{v) be an arbitrary non-negative Crnjunction on Vo. If the projection Vo x [0, IJ ~ Vo sends Q+ onto Vo, then there exists a non-negative crn junction qJ on Vo x [0, 1 J whose support lies in Q+ and such that J6 qJ(v, t)dt = qJo(V) for all v E Vo. [Compare (B) in 2.4.1.J

Proof. With a partition of unity in Vo, the lemma reduces to the obvious case where the support S c Vo of qJo satisfies S x t c Q+ for some t E [0, 1].

Let Vb = Vo x [0,1J and let I = I(t/J) c Vb be the zero set of a COO-function t/J: Vb ~ IR. Denote by Q+ c Vb the subset where t/J > ° and let Q_ be the subset where t/J < 0. Let fo: (!)/toVb ~ IR be a COO-function, such that dfo/dt = t/J1(!)/toVb·

(A') Lemma. If the subset Q+ as well as Q_ goes onto Vo under the projection Vb = Vo x [0, 1J ~ Vo, then there exists a function f: Vb ~ IR such that


(b) ~ \mftL' = ifJ,

(c) the derivative df/dt is positive on Q+ and negative on Q_.

Proof One easily constructs with (A) a COO-function cp on Uo which equals ifJ on mftL', whose derivative dcp/dt equals fo on mftauo and such that

(i) cp is positive on Q+ and negative on Q_,

(ii) SA cp(v, t) dt = fo(v, 1) - fo(v, 0).

Then the function f(v, t) = fo(v, 0) + S~ cp(v, r) dr is the required one.

Remark. This lemma establishes, in fact, the h-principle for the functions f in question, since the linear homotopy of ifJ to df /dt does not destory the pertinent properties of ifJ.

(B) Exercises. (a) Consider a non-vanishing COO-vector field L ona manifold V. Let ifJ be a COO-function on V such that every orbit of L meets Q+ (where ifJ > 0) as well as Q_. Prove the existence of a COO-function f on V whose Lie derivative Lf is positive on Q+, negative on Q_ and Lfl mftL' = ifJ I mftL' for the zero set L' of ifJ.

(a' ) Define by induction Lkf = L(Lk-1f) and say that critical points of fare k-nondegenerate along L if the equality Lf(v) = 0 (which characterizes critical points v of f along L) implies Lif(v) =f. 0 for some i ~ k. Show that functions f with k-nondegenerate critical points along V satisfy the h-principle for every k ~ 3. In particular, prove the existence of an f whose critical points along L are 3-nondegenerate.

(b) Study counterexamples for the above h-principle for k = 2. Consider, in particular, V = s2m+1 where the vector field L is the infinitesimal generator of the standard S1-action on s2m+1. Prove for an arbitrary COO-function f on s2m+1 that the subset L'1 C s2m+1 where Lf and L 2 f vanish has codimension :s; 3 in s2m+1. Denote by K1 C cpm = S2m+1/S1 the image of L' 1 under the quotient (Hopf) maph: s2m+1 -+ cpm and prove the restriction homomorphism H2m-2(cpm)-+ H2m-2(K1) to be non-zero.

(b') Denote by Ki C cpm the subset of those points x E cpm for which the function flS; has at least 2i critical points on the circle S; = h-1(x), where f is an arbitrary smooth function on s2m+1. Show the homomorphism H2m-2i+2(cpm)-+ H2m-2i+2(Ki) to be non zero. In particular, the function flS; has at least 2m + 2 critical points for some circle S;.

(b") Let C°(V) be the space of continuous functions on a compact n-dimensional manifold V and let X be a k-dimensional linear subspace in CO (V). Prove the existence of a non-zero function cp E X whose maximum set Mtp C V (where cp assumes the absolute maximum) contains at least s points, where s is the first integer ~(k - 1)/(n + 1).

Hint. Study the convex hull of Fx(V) c ~k, where the map Fx: V -+ ~k is given by k independent functions in L; alternatively, study the set valued map cp t-+ Mtp for all cp E L, whose pertinent property is Mtp n M _tp = rP for cp =f. O.

(C) Poenaru's Pleating Lemma. Consider a Coo-map F: Uh = Vo x [0,1] -+ W, where dim W = n = dim Uh, such that Ft = FI Yr = Vo x t is an immersion Yr -+ W for all t E [0,1]. If the map F is also an immersion on (I)/t Vo c Uh then the (non-

vanishing) field a; I Vo induces an orientation in the (one-dimensional) normal

bundle of the immersion Fo: Vo -+ W which uniquely extends to a normal orientation of Ft : Yr -+ W for all t E [0, 1]. Now we assume the manifold Vo to be connected and the map F to be an immersion on (I) /t V1 c Uh as well as on (I) /t Vo c Uh. Thus we get two normal orientations of the immersions Ft. We call the map F even if these orientations coincide; otherwise F is odd. For example, if F is an immersion on all of Uh, then it is even. If the only singularity of F is a folding along Yr for a single value t E [0, 1], then F is odd.

Take some points ° = to < t1 < ... < tN+1 = 1, where the parity of N equals that of F.

If Vo is compact and if ti+1 - t; :s;; e for i = 0, ... , N and for some sufficiently small e = e(F) > 0, then there exists a Coo-map F": Uh -+ W whose only singularity is a folding along Yr1 U Yr2 U ... U YrN and such that F" equals F on (I)/t(Vo U V1) c Uh· Furthermore, the maps F" CO-converge to F as e -+ 0.

Proof Let t; = t(t; + ti+1) for i = 0, ... , N. There obviously exists a Coo-map F': Uh -+ Wo, such that F; is arbitrarily Coo-close to Ft for all t E [0,1] and F' is an immersion near Vta U Yr\ U ... Yr" in Uh. Moreover, this F' can be made odd on Vo x [t;, t;+1] for all i = 0, ... , N, and such that F'I(I)ji(Vo U V1) = FI(I)ji(Vo U V1). If e is small, then the map F; is close to F~H1 for tE [t;, t;+1], and using this we perturb F' outside (I)/t(Yro U ... U Yr,,) to the required map F' as follows. Since the perturbation we are after separately applies to each interval [t;, t;+1], the problem reduces to the proof of the lemma for N = 1, where the maps Ft are assumed Coo-close to Fo for all t E [0,1]. Then there obviously exist a Coo-immersion F: Vo x IR -+ Wand a Coo-map F: Uh -+ Vo x IR such that F 0 F = F and F(v, t) = (v,fo(v)) for some function fo on Uh. Now we apply (A') to I = Yr1 and to a suitable function t/I on Uh whose derivative dt/l /dt does not vanish on I. Then the function f insured by (A') gives us the map F': (v, t)t---+(v,f(t)) which folds along I and for which the composition F 0 F': Uh -+ W is the required map with the only folding along Yr1 = I. Q.E.D.

Remark. The maps Ft' obtained by this construction are Coo-close to Ft for all t E [0,1]. This is stronger than the mere CO-closeness between F" and F (compare C.l-approximation in 2.4.1).

(C') The h-Principle for Folded Maps (Poenaru 1966). Let V be a connected n-dimensional manifold, let Vo be a normally oriented closed hypersurface in V and let fo: V -+ W, where dim W = n, be a continuous map which lifts to a homomorphism of the tangent bundles, say <p: T(V) -+ T(W), such that

(a) the homomorphism q> is fiberwise injective outside a small neighborhood lPftVo c V;

(b) there is a split tubular neighborhood Uo = Vo x ~ c V, which properly contains the above lPft Yo, such that q> is injective on the tangent bundle T(J-;) c T(V)I J-; for J-; = Vo x t and for all t E ~.

Then there exists a Coo-map f: V -+ W homotopic to fo whose only singularity is a folding along finitely many submanifolds J-;, c V where ti = i/N + 1 for i = 1, ... , N and for some N ~ No(f). Moreover, the differential Dr: T(V) -+ T(W) (for a suitable f) can be deformed to q> by a homotopy of homomorphisms which is injective on T(V)I V\ Vo x [0, 1] and on T(J-;) for all t E ~.

Proof The h-principle holds true for immersions V\ Vo -+ W (since V\ Vo is open) and for immersions Vo -+ W (since dim Vo < dim W). Thus, we obtain a Coo-map F: V -+ W which is an immersion outside U~ = Vo x [0,1] in V and such that Ft = FI J-; is an immersion for all tE~. Then this F is modified on U~ to the desired folded map f: V -+ W.

Corollary. If the manifold V is stably parallelizable (i.e. V x ~ is parallelizable) then there is a Coo-map f: V -+ ~n whose only singularity is a folding along some closed (possibly disconnected) normally oriented hypersurface in V.

An Application to the Signature Theorem. Let V be a closed oriented 4k-dimensional manifold and let u(V) denote the signature of the intersection form on the homology H2k(V; ~). The famous theorem of Thom-Hirzebruch claims the existence of a universal polynomial L in the Pontryagin classes Pi of V, such that (L(Pi), [V] > = a(V). In particular, a(V) = 0 for stably parallelizable manifolds V. Let us reduce this vanishing theorem to the above corollary with the following

(e") Lemma. Suppose there exists a Riemannian Coo-metric g on V and an open subset U c V which admits an orientation reversing isometric involution I: (U,g) -+ (U,g) and such that the complement V\ U is a stratified subset of codimension one in V. Then u(V) = o. Warning. An involution I on U which preserves gl U may be discontinuous on V=>U.

Proof Since the signature is the index of some elliptic differential operator on V associated to g, it can be expressed by an integral over the (oriented!) manifold V of some universal polynomial in covariant derivatives of the curvature tensor of g, say u(V) = SyPdv. Then SyPdv = SuI*(P)(-du) = -SuPdu = O. [See Atiya,h (1976) for a conceptional explanation of this "locality" of u.]

Now, if V is stably parallelizable, it admits a Coo-map f: V -+ ~n which folds along some normally oriented hypersurface V' c V. Then there exists a COO-function fo on V with support in a small neighborhood U' c V of V', such that the map f' = fEe fo: V -+ ~n+1 = ~n X ~ is an immersion which is symmetric on U' in the


hyperplane IRn x ° c IRn+l. Let U c V be the maximal open subset whose image feU) c IRn does not meet fey'). Then the induced metric g on V c 1R"+1 obviously has the required involution on U. Q.E.D. [See (G) in 2.2.7 for a similar study of non stably paralellizable manifolds.J

(D) The Equidimensional Folding Theorem of Eliashberg. Take a normally oriented hypersurface I in V and let I x [ -1, IJ c V be the tubular neighborhood of I. Let the space V be obtained from V by identifying the points (u, t) and (u, - t) for all (u, t) E I x [ -1, IJ and let p: V ~ V be the obvious map. There is a unique vector bundle T ~ V such that the induced bundle f = p*(T) ~ V is canonically isomorphic on V\I to the tangent bundle T(v)1 V\I.

Example. If I is an equator in the sphere S" then f is the trivial bundle which is induced from the tangent bundle of the ball B" by the obvious map S" ~ B" which sends I onto iJE".

If a smooth map f: V ~ W, for dim V = dim W folds along I, then the differential of f naturally defines a homomorphism Df : f ~ T(W), which equals Df outside I and which is a fiberwise isomorphism over I.

Theorem. Let a continuous map fo: V ~ W lift to a fiberwise isomorphic homomorphism cp: f ~ T(W). Then for n ~ 2 there exists a Coo-map f: V ~ W which folds along I and such that the homomorphism Df is homotopic to cp by a homotopy of fiberwise isomorphic homomorphisms.

Warning. This theorem is false for n = 1 and also for n ~ 2 if I is empty.

Proof Take an arbitrary normally oriented closed hypersurface Vo c V which transversally meets I over a non-empty (n - I)-dimensional manifold In Yo. Fix a split tubular neighborhood Uo = Vo x IR of Vo such that the projection Uo ~ Vo is an immersion of In Uo into Yo. Then we reduce the proof [compare the proofs of (C) and (C')] to the following special case.

Let fo be a smooth function on U~ = Vo x [0, IJ, such that the map F: U~ ~ Uo given by F(v, t) = (v,fo(t» satisfies the following two conditions, (1) the only singularity of F near Vo U V1 C U~ is a folding along In lDfi(Vo U Vd, (2) there exists a Coo-function r/J on U~, whose zero set equals I, whose derivative dr/J/dt does not vanish on I and such that r/J equals fo near Vo U V1 [compare (A')].

If the projection U~ ~ Vo were surjective on Q+ and Q_ we could prove the theorem by modifying F inside U~ with an appropriate function f [compare the proof of (C)]. To achieve this surjectivity we start with a small embedded circle Sl c V which transversally meets I at two points. Then we consider a small tubular neighborhood U1 of Sl in V and take Vo = aU1 ~ Sl X S,,-2, such that the intersection I n U~ becomes the union of two disjoint copies of S,,-2 x [O,IJ embedded into U~ = Sl X S,,-2 X [0, IJ, whose projections to Vo = Sl X S,,-2 are immersions. Furthermore, we can (and we do) arrange this Vo such that each segment s x [0, 1 J in E n U~ goes to Sl under either projection by an orientation preserving immersion.

60 2. Methods to Prove the h-PrincipJe

Now, let .: [0,1] _ Sl be a (surjective) Coo-map, such that,

(i) .(t) == e for t E [0, t] u [1, 1], where e is the neutral element ofthe additive group s1,

(ii) • is an orientation preserving immersion on the open interval (t,1). Define the following diffeomorphism i of V which is the identity outside Ub and

which acts on Ub = Sl X sn-2 X [0,1] by i: (s, s', t) H (s + .(t), s', t). The manifold r = i(I) n Ub retains the properties (1) and (2) we insisted upon earlier, but now ~ goes onto Vo under the projection Ub - Yo. Hence, the pertinent sets Q+ and Q_ also go onto Vo and then (A') applies. Q.E.D.

(D') Exercise. Let V be a connected manifold with a non-empty boundary av. Fix a collar C = av x [0,1] c V and let a finite group G freely act on C preserving avo x 0 c C. Prove the h-principle for those Coo-immersions f: V - W for dim V ~ 2, which satisfy f(gv) = f(v) for all pairs (g, V)E G x C. [See (G) in 2.2.7 for a general result of this kind.]

(D") Additional Remarks and Exercises. The h-principle of Eliashberg fails for dim W = 1 since the Morse theory gives additional restrictions (which are not accounted for by the h-principle) on the critical points of maps of V into IR and into Sl. A similar situation arises for mapsf: V - wq, q ~ 2, if the topology of the map flI: I - wq is brought forward. [Compare Serf (1984).]

(a) Consider a Coo-map f of a closed manifold V into 1R2 which folds along a union of circles I = U~=l Si' and such that the immersion flI: I - 1R2 has d transversal double points and no triple point. Establish the following bound on the sum of the Betti numbers P(f-l(W))

P(f-l(W)) ~ k + d for all WE 1R2.

(b) Prove that P{V) ~ (k + d)(2k + 3d).

(E) Lagrange and Legendre Immersions. Let 0( denote the standard (linear differential) form Li'=l Xi dYi on 1R2n whose differential h = dO( = Li'=l dXi 1\ dYi is the standard symplectic 2-form. A Cl-map f: V - 1R2n is called Lagrange if f*(h) == 0 on V, which amounts to the identity dLi'=l XidY; = 0 for the coordinate functions Xi> Y;: V-IR off

A Lagrange map f = (X, Y) = (Xl' ... ,Xn, Yl , ... , Y,,): V - 1R2n is called exact if the induced form f*(O() = Li'=l Xi dY; on V is exact. This can be expressed with the contact form P = Li'=l Xi dYi + dz on 1R2n+1 by defining the exactness of f as the existence of a lift off to a Legendre map F = (f, Z): V _ 1R2n+1 for some Cl-function Z on V, where Legendre maps F: V _ 1R2n+1, by definition, are those for which F*(P) = L;'=l XidY; + dZ == 0 on V.

Examples. An arbitrary smooth map f = (X, Y): V - 1R2n is the sum of two exact Lagrange maps, f = (0, Y) + (X, 0).

Let Tn be the n-torus with the cyclic coordinates t i , i = 1, ... , n. Then the map (Xi = sin ti, Y; = cos til: Tn - 1R2n is a Lagrange Coo-embedding but it is not exact.


Proposition. Let Y: V --+ IRn, n = dim V, be a Coo-map whose only singularity is a folding along a normally oriented hypersurface Vo c V. Then the map Y lifts to an exact Lagrange Coo-immersion f = (X, Y): V --+ 1R2n for some COO-map X = (Xl' ... ,Xn ): V --+ IRn.

Proof Let U = Vo x [ -1,1] c V be a tubular neighborhood of Vo which admits a Coo-immersion Y: U --+ IRn such that Y(v, t 2 ) = Y(v, t) for all points (v, t) E U, and let Z be a Coo-function on V, such that ZI U = t 3 . Since the map Y is an immersion outside Yo, the differentials d¥;, i = 1, ... , n, span the cotangent bundle T*(V)I V\ Yo. Hence, there exist unique Coo-functions Xi' i = 1, ... , n on V\ Yo, for which 2::1=1 Xi d ¥; = - dZ. Furthermore, there are unique Coo -functions Xi on U, such that 2::1=1 Xi dY; = dt. Then the functions X;(v, t) = -~tXJv, t2 ) satisfy 2::1=1 X; d¥; =

- 3t2 dt = dt 3 on U. Hence, X; I U\ Vo = Xii U\ Vo, which shows the functions Xi to be Coo-smooth near Vo as well as outside yo.

Since theform dt = 2::1=1 XidY; does not vanish, the map X = (Xl"" ,Xn ): U --+

IRn also does not vanish, and hence, the derivative

ax(v t) - --c:---' - = - ~ X(v t2 ) - ~t dX(v t2 ) at 2, 2 ,

does not vanish on Vo = Vo x 0 c U. Therefore, the exact Lagrange Coo-map f = (X, Y): V --+ 1R2n is an immersion. Q.E.D.

Corollary. An arbitrary stably parallelizable manifold V (for example V = sn) admits an exact Lagrange Coo-immersion f: V --+ 1R2n.

Indeed, such a V admits the required folded map Y: V --+ IRn by Poenaru's theorem.

Exercises. (a) Show the normal bundle of an arbitrary Lagrange immersion f: V --+

1R2n to be isomorphic to the tangent bundle T(V). (This in particular implies the vanishing of the rational Pontryagin classes of V.)

(a') Show that the existence of a Lagrange embedding of a closed orientable manifold V into 1R2 n implies the vanishing of the Euler characteristic of V. (One knows that no closed manifold V = vn admits an exact Lagrange embedding into 1R2n, compare 3.4.4.)

(b) Prove the h-principle for exact Lagrange immersions V --+ 1R2n by lifting to 1R2n generic Coo-maps Y: V --+ IRn. (See 3.4.2 for a different approach to this h-principle.)

2.1.4 Singularities and the Curvature of Smooth Maps

Consider an oriented n-dimensional manifold V and an immersion f: V --+ IRn+1. The orientation of V defines a unit normal vector v at each point v E V and thus we get the normal (Gauss) map Gf : V --+ sn c IRn+1. The Jacobian of Gf equals the product of the principal curvatures of the hypersurface V --+ IRn+t, say

J(v) = TIi=l a;(v, v), where the choice of the unit normal v at v determines the signs of the curvatures ai(v), such that ai(v, -v) = -ai(v, v). If the manifold is closed, then, obviously, fvJ(v) dv = (deg Gf ) Vol sn. Furthermore, ifn is even then deg Gf = h(V) for the Euler characteristic X(V) (compare 1.2.1).

Exercise. Let V c IRn+1 be a smooth closed embedded hypersurface and let Xl"'" xn+1 be linearly independent vectors in IRn+1. Define

Ci = {v + txi,tlvE V c IRn +1 , tE [0, oo)} c IRn+1 x [0, (0)

for i = 1, ... , n + 1 and let Co = V x [0, (0) c IRn+l x [0, (0). The cylinders Ci have no common point for small t > ° and they are disjoint for large t ~ 00. Hence, small generic perturbations C; of Ci transversally meet at finitely many points away from t = 0. Prove the algebraic (i.e. counted with properly chosen ± signs) number of points of the intersection n i,!J C; to be equal to the degree d of the normal map V ~ sn (that is the Euler characteristic of the compact domain in IRn+1 bounded by V, see 1.2.1). Assume d =I ° and prove the existence of some points Vi E V, i = 0, ... , n + 1, such that Vi - Vo = tXi' i = 1, ... , n + 1, for some t > 0. Show, in particular, the existence of a regular (n + I)-simplex in IRn+1 whose vertices lie in V.

Question. Do embeddings V ~ IRn+1 for which the intersection ni,!J Ci is empty away from t = ° (for fixed vectors Xi) satisfy some h-principle?

(A) A Lower Bound on the Total Curvature. Let fx: V ~ IR denote the orthogonal projection to the line in IRn+1 through a given point x E sn, that is fAv) = <f(v), x). Then the singular set of the function fx (i.e. the set of critical points of fx) equals the pull-back Gil {x U - x} c V. Furthermore, the critical points of fx are nondegenerate if and only if the points x and - x E sn are non-critical values of Gf , that is ±x ESn\ Gf(L') for the zero setL' ofthe Jacobian ofGf . Then 2 deg Gf = Lv( _1)indv, where v runs over the critical point of fx. This yields the equality deg Gf = h(V) for n even.

If the immersionf: V ~ IRn+l is COO-smooth, then the map Gf : V ~ sn also is Coo and by Sard's theorem almost all x E sn are non-critical values of Gf . Thus, critical points of fx are discrete and non-degenerate for almost all x and their number c(x) obviously satisfies fsn c(x) dx = fv IJ(v)1 dv. Therefore, Morse inequalities bound from below the total curvature fv IJ(v)1 dv of closed manifolds V ~ IRn +1 (see ChernLashof 1957; Kuiper 1958),

Iv IJ(v)1 dv :2: tP(V) Vol sn

for the sum of the Betti numbers P(V) = 2::i=0 MV) with an arbitrary coefficient field. Observe that (*) provides an obstruction to the h-principle for immersions

V ~ IRn+1 with a prescribed curvature J = J(v).

(A') Non-orientable Manifolds V. The inequality (*) makes perfect sense without any orientation in V and the above proof immediately extends to non-orientable manifolds V. Furthermore, the normal map is naturally defined (regardless of an

orientation in V) on the unit normal bundle of V in IRn+1 which is (in the codimension one case) a double covering V --+ V. Thus Hopf's formula for X(V) generalizes to Jv J(v) dv = (VoISn)X(V), where J(v) is the Jacobian of the normal map V --+ sn which equals the (signed!) curvature at the point v under v.

Exercises. (a) Generalize the above to immersions V --+ IRq, q > n, by considering the normal map of the unit normal bundle of V 4 IRq to Sq-l.

(b) Suppose some m among n + 1 Betti numbers bi(V) are non-zero. Show that the complement V\I to the singularity I of the normal map G/ V --+ sn has k ~ ml2 components. (This indicates an obstruction to the h-principle for immersions f whose normal map has a prescribed singularity.)

(c) Let Vt be the number of cusps of the orthogonal projection of V c IRq onto the 2-plane r c IRq, for all r E Gr = Gr 2 IRq. Find a density function f.l on V expressible in terms of the curvature of V c IRq and first derivatives of the curvature, such that Jvf.ldv = JGr vtdr. [Consult Morin (1965).]

(B) Let us generalize (A) by allowing an arbitrary smooth map f: V --+ IRn+l whose singularities Ii = {VE Vlrankvf = n - i} are stratified subsets in V. The curvature J(v) is now defined on the non-singular locus I O of f only.

If dim Ii.::; n - i-I for i = 1, ... , n, (which is the case for generic maps f) then JEoIJ(v)ldv ~ tf3(V)VoISn• Furthermore, if JEoIJ(v)ldv < 00 and n is even, then JED J(v) dv = tx(V) Vol sn. [If Io is non-orientable then HID J(v) dv is used instead of J J(v)dv, compare (A').]

Proof Since dim Ii .::; n - i-I the critical set of the function f does not meet Ii for almost all x E sn and the argument in (A) applies.

(B') Maps f: V --+ IRq for 2 .::; q .::; n. Let the singularity Ii = {v E VlrankJ .::; q -i} be a stratified subset for all i = 0, 1, ... , q, and let dim Ii .::; q - i - 2 (which is a generic condition on f). Let If c Il be a stratified subset of dimension .::; q - 2, such that the complement IJ = Il \If is a smooth (q - I)-dimensional submanifold in V on which the map f is an immersion IJ --+ IRq. The critical set of the function fx: V --+ IR now lies in IJ for almost all x E Sq-l, which implies the following bound on the curvature J of the hyper surface IJ in IRq,

f IJ(v)ldv ~ tf3(V)VoISq-l, E~

Hence, the geometry of I J (unlike the topology, see 2.1.3) is not accountable for by the h-principle.

(B") Hopf's Formula. Let K --+ IJ denote the kernel Ker DJ c T(V)IIJ (which is a (n - q + I)-dimensional bundle over IJ) and let N --+ IJ be the (one-dimensional) normal bundle of the immersion fl IJ: IJ --+ IRq. Assume the second differential D}: K --+ N to be non-singular (as a quadratic form) outside a stratified subset I' of dimension .::; q - 2 in IJ (which is a generic condition on f) and let Io = IJ \I'.

Take a unit normal vector iJ E Nv, v E I o, and define s(iJ) = ± 1 as follows. The vector iJ defines an isomorphism a: Nv"Z IR for a(iJ) = 1 and then 15f = a 0 15flKv is a quadratic polynomial Kv ~ IR, that is Lf=l yr - Li~::i yr for some basis in Kv' Then we put s(iJ) = (-1)P, and we have the following integral formula for the curvature J(iJ) of Io for n = dim V even.

If SEo IJ(v)1 dv < 00, then

f s(iJ)J(iJ) diJ = X(V) Vol sq-l, 1:0

(+ )

where Io is the (zero-dimensional) unit normal bundle of Io ~ IRq with the (naturally oriented) volume element diJ.

Proof. The index of each critical point v E Io C V of the function fx: V ~ IR is the sum of indv (fxIIo) and of the index of 15f : Kv ~ IR which corresponds in the chosen normal iJ at v. Hence, the argument in (A) applies.

(C) On the Convergence of SEo I (J(v) I dv. Let c(x) denote the number of critical points of the function fx I Io. Then

f IJ(v)1 dv = r c(x)dx. Eo JSq~l

Hence, SEo IJ(v)1 dv < OCJ provided the function c(x) is bounded on Sq-l.

Corollary. If V is a closed real analytic manifold and the map f: V ~ IRq is can, then ho IJ(v)11 dv < 00.

Proof. This is obvious in the real algebraic case as c(x) is bounded (according to Bezout's theorem) by the algebraic degrees of V and X. Moreover, the results of Hironaka (1973) on subanalytic sets imply the following general fact which suffices for our purpose.

Let X and Y be compact real analytic spaces and let h: X ~ Y be a real analytic map. Then for each i = 0, 1, ... , the Betti number bi(h-1(y)) (with 7L/27L coefficients) is bounded. Namely, bi(h-1(y)) ~ const = const(X, Y,h),for all yE Y.

Observe that the curvature J is (obviously) expressible in terms of the vector bundles homomorphism D/ T(V) ~ T(lRq) and that Coo-automorphisms of the bundles T(V) and T(lRq) change the integral S IJ(v)1 dv by a bounded factor. According to Mather (1973), a homomorphism T(V) ~ T(lRq) transversal to Whitney-Thorn singularities is locally reducible by some automorphisms of the bundles to a canonical Can-form. This implies the convergence of S IJ(v)1 dv for generic Coo-maps f: V ~ IRq. [See Burago (1968) and Levin (1971) for another approach to Hopf's formula in the presence of singularities.]


(D) Exercises. (a) Construct a Coo-map f: S2 ~ ~2 which folds along the equator Sl c S2 and such that the curvature J of f1S 1 : Sl ~ ~2 vanishes at exactly two points in S1, and SS! IJ(v)1 dv = C for a given constant C > 2n.

(a/) Find a necessary and sufficient condition on a 1-form w on Sl for the existence of a Coo-map f: S2 ~ ~2 with the only fold along Sl and such that J(v) dv on Sl equals w.

(a") Divide a closed surface V of genus 2 by a circle Sl c V into two punctured tori. Study the curvature J on Sl of Coo-maps V ~ ~2 which folds along Sl and have no other singularities.

(b) Let V be a closed 3-dimensional manifold and let f: V ~ ~3 be a generic map such that rankJ ~ 2 for all v E V. Then the singularity IJ c V is a smooth surface, while the map fIIJ: IJ ~ ~3 may be singular along some curve Ij1 c IJ. Bound the Betti numbers of I in terms of the curvature of IJ and IJ 1 and of the number of singular points IJll c Ij1 of the map fIIJ1.

(c) Let V be a closed oriented 4-dimensional manifold and let f: V ~ ~5 be a generic Coo-map. Then the singularity I = IJ is a smooth closed surface in V such that rankvf = 3 for all v E I. Let g: I ~ Gr3 ~5 be the map which assigns the image DAT,,) c ~5 (which is a 3-dimensional subspace in ~5) to each point VE V. Prove, for properly normalized Euler form w in Gr 3 ~5, [which is a closed SO( 5) invariant 2-form on the Grassmann manifold Gr3~5 = Gr2~5], the equality h'g*(w) = Pl(V) for a natural orientation in I and for the first Pontryagin number Pi (V) of V.

Hint. Express Pi (V) by the I2-singularity ofthe projection of V onto a hyperplane ~4 in ~5 and then average over the Grassmannian Gr4~5 = S4 c ~5.

(b /) Find similar formulae for the Pontryagin numbers of closed manifolds V mapped into ~q with generic singularities.

(b") Generalize (b /) to homomorphisms of vector bundles X ~ Yover V, where Y is a trivial bundle. Allow complex vector bundles and express Chern numbers by integrals over pertinent singularities.

Question. Let g be a COO-smooth positive semidefinite quadratic differential form on a closed manifold V. If g is definite then the Euler characteristic equals the Gauss-Bonnet integral of a certain polynomial in curvature of g. Furthermore, the Gauss-Bonnet formula holds true according to (B) for forms g induced by generic smooth maps V ~ ~q, q ~ n + 1. However, these forms g have no (?) simple intrinsic description. For example, one does not know under what condition the Gauss-Bonnet integral absolutely converges and whether the convergence implies [under suitable assumptions, compare (B)] the Gauss-Bonnet theorem for singular metrics g on V.

2.1.5 Holomorphic Immersions of Stein Manifolds

Let V be a complex analytic manifold and let X ~ V be a holomorphic fiber bundle. Then one defines in a natural way the bundle x(r) ~ V of rth order jets of germs of


holomorphic sections V -+ x(r). The holomorphic h-principle for a locally closed complex analytic subset (relation) ~ c x(r) is defined as the possibility to deform an arbitrary holomorphic section CPo: V -+ ~ to a holonomic section CPl = Jf: V -+ ~ by a homotopy of holomorphic sections V -+~. Similarly one defines the parametric h-principle and the h-principle for extensions from analytic subsets in V. One must be careful, however, with the approximation problem since the (ordinary) Ci-topologies are equivalent for all i = 0, 1, ... , 00, for holomorphic sections, while the fine CO-topology is discrete for non-compact connected manifolds V.

The holomorphic h-principle, when it holds true for a given relation ~, does not immediately yield holomorphic solutions of ~, but rather reduces the problem to Oka's principle that is the ordinary h-principle for the Cauchy-Riemann relation (compare 1.1.2).

(A) Immersions of Stein Manifolds V into eq• Let V be a Stein manifold of complex dimension n. Recall that the Stein property is equivalent to the existence of a proper holomorphic embedding V -+ eN for some sufficiently large N.

Theorem (Gromov-Eliashberg 1971). If q > n = dime V, then holomorphic immersions f: V -+ eq satisfy the holomorphic h-principle.

Proof We mimic the COO-argument in 2.1.1 with the following

(A') "Zero-Dimensional" h-Principle. Consider a closed analytic subset Vo in V and let L be a non-vanishing holomorphic vector field along Vo that is a holomorphic section Vo -+ T(v)1 Yo. Fix a point Vo E Vo and let Ul"'" Un be local coordinates near Vo such that L = a/aUl near Vo. We say that L has finite tangency to Vo near Vo of order ::;; N, if there exists a holomorphic function F = F(u l , ... , un) whose zero set contains the germ of Vo near Vo and such that the restriction of F to each line {U2 = const2 , U3 = const3 , ••• , Un = constn } in the coordinate domain has each zero (in the variable u l ) of order ::;;N + 1. This definition of the tangency is clearly independent of the local coordinates.

The field L is called semi-transversal to Vo if it is tangent to V near each point v E V with order at most N for some integer N which does not depend on v.

Denote by 10 is ideal of holomorphic functions on V which equal zero on Vo and let Ii, i = 1, 2, ... , be the ideal generated by Ii-! and the derivatives If for the functions f E I i - l , where I is some holomorphic extension of L to V. It is clear that the ideals Ii do not depend on the choice of this extension. If L is tangent to Vo with order ::;;N, then the Weierstrass preparation theorem allows one to choose the above function Fofthe form F = I;:'o FiUL where Fi are holomorphic functions in U2' ... , Un and FN == 1. Then the local ideal IN near Vo contains all germs of holomorphic functions at vo. This applies to all points in Vo and, since V is Stein, the (global) ideal IN contains all holomorphic functions on V.

Lemma. Let L be semi-transversal to Vo and let g be an arbitrary holomorphic function on V. Then there exists a holomorphic function 1 on V, such that III Vo = g I Vo.

Proof If g E 10 then the solution is obvious with 1 == O. Furthermore, an arbitrary function g Elk by definition is g = fo + L~,;;-f rt.iL/; for some holomorphic functions rt.i on V and for some /; E Ii. Then the function j = L~,;;-f rt.i/; satisfies the equation Ll = g - g' for the function g' = fo + L~,;;-f (Lrt.;)/; in Ik- 1 • We assume, by induction in k, the existence of a function 1', for which L1'I VO = g'l Vo, and then we solve the equation Ll = g with 1 = j + 1'. Since each function g lies in IN' the proof follows with k = N.

Now let 2 be a line subbundle of the tangent bundle of V restricted to Vo, that is 2 c T(v)1 Vo, and suppose there is a holomorphic I-form qJ on V which does not vanish on 2. Then there exists a unique holomorphic section L: Vo ~ 2, such that qJ(L) == 1 on Vo. We call 2 semi-transversal to Vo if the field (section) L is semi-transversal to Vo and we observe this property of 2 to be independent of a particular form qJ. The above lemma immediately implies the following

Corollary. If 2 is semi-transversal to Vo, then holomorphic functions f on V whose differentials df do not vanish on 2 satisfy the holomorphic h-principle.

Exercise. Generalize the above to the fields L which have finite tangency of order N = N(v) near each point VE VO, where N(v) is unbounded for v ~ 00.

(A") Generic Properties of Holomorphic Maps and Sections. Start with an arbitrary holomorphic fibration X ~ V and let cP be an arbitrary subspace in the space r(X), of holomorphic sections V ~ X, with the topology of uniform convergence on the compact subsets in V. A p-dimensional holomorphic homotopy in CP, by definition, is a map 1/1: cP ~ cP c r(X) for which the associated map V x cq ~ X is hoi om orphic. A set '1' of such homotopies is called composible if for every two homotopies in '1', say 1/11: CPt ~ cP and 1/12: CP2 ~ cP there exists a homotopy 1/1: CPt x cP2 ~ cP in '1', called a composition of 1/11 and 1/12, such that I/IICPt x 0 = 1/11 and 1/110 x CP2 = 1/12.

(An Example. Let X be the trivial fibration X = V x U ~ V and let L"(i) c X(l), i = 1, ... , q consist of the I-jets of (germs of) those sections qJ: V ~ X, represented by q-tuples of holomorphic functions fl' ... , /q on V, for which the map F; = (fl' ... '-/;-1'/;+1' ... ,/q): V ~ Cq-l fails to be an immersion. That is J~(v) EL"(i) if and onlyifrankv F; < n = dim V. Denote by J'(i) the sheaf of ideals of hoi om orphic functions on X(l) which vanish on .E'(i) and fix an integer k ~ O. A homotopy 1/1: C1 ~r(X) represented by functions /;(v,z) on V for i = 1, ... , q and ZEC 1 is called (I'(iW-stable ifthe functions fl' ... ,/;-1 ,fi+l' ... ,/q are constant in the variable Z and ifthe function/;(v, z) - (/;(v,O) is contained in the pull-back of the ideal (I'(i))k under the map V x C1 ~X(l) given by the I-jets of sections 1/1 (z): V~X, ZEC 1.

Next, we by induction in p = 1,2, ... , define a homotopy 1/1: cP x C1 ~ r(X) to be k-regular if the homotopy I/IlcP x 0: CP ~ r(X) is k-regular and if the homotopy I/Ilz x ct: C1 ~ r(X) is (I'(i))k-stablefor some i = 1, ... , q and for all ZECP where every "homotopy" for p = 0, by definition, is regular. The k-regular homotopies for k ~ 2 preserve immersions V ~ cq. Namely, if cP c r(X) is the subset of those


sections which correspond to immersions V --. cq and if a k-regular homotopy 1/1: cP --. reX) for k ~ 2 has 1/1 (zo) E cP for some Zo E CP, then I/I(z) E cP for all z E CPo Furthermore, if V is a Stein manifold, then k-regular homotopies constitute a composible set for each k. In fact, this remains true for an arbitrary sheaf of ideals in the space of jets of any order, say for I(i)Ex(r), i = 1, ... , q, since the basic theorems A and B of H. Cartan (see Gunning and Rossi 1965) allow extensions of holomorphic functions on V x CP which lie in a given idea1.

(A2) Lemma. Let 'fF be a composible set of holomorphic homotopies in cP and let E be an analytic subset in the jet space x(r). Suppose that for each section qJ E cP and for each point VE V there exists a neighborhood U c V of v and a homotopy I/Iv: C --. cP in 'fF such that 1/1(0) = qJ and such that the jet J:'. sends U to the complement x(r)\E

't'(a:)

for all z =f 0 in C which are close to zero. Then each section, say qJo E cP, admits a continuous homotopy qJtE cP, tE [0,1], such that the jet J~l sends V to the complement x(r)\E.

Proof Compose the homotopies I/Iv for all points v E V and restrict the composition to an appropriate continuous path in the resulting space Coo. This is straightforward and the detail is left to the reader.

(A;) Corollary. Let V be Stein and let, for the above qJ E cP and v E V, there exist a holomorphic vector bundle Y --. V, a fiber preserving holomorphic map IX: Y --. X and a sheaf of ideals I on V (i.e. I is a subsheaf of the structure sheaf of V), such that

(i) the zero section of Y is sent by IX to qJ; (ii) the zero set of I does not contain the point v;

(iii) if 13 E F(Y) and Y E I, then the map IX sends the one-dimensional homotopy z f-+ zyf3, z E C, in F(Y) to 'fF;

(iv) let IXr : y(r) --. x(r) denote the natural map induced by IX on the jet spaces. Then the codimension of the pull-back 1X-1(E) c y(r) near v E V = V x 0 c y(r) satisfies codimvlX-1(E) > n = dim V.

Then, each section qJo E cP admits the above desingularizing homotopy qJt E cPo

Proof If 13' is a generic holomorphic section of Y over a small neighborhood U c V of v, then by (a simple local analytic version of) Thorn's transversality theorem the rlh order jet of the section zf3': U --. Y, z E C, misses 1X-1(E) for all small z =f O. Since V is Stein, there exists a global section 13: V --. Yand some y E I for which the section yf3 approximates 13' on U, such that the jet of zyf3 misses E as well. The image I/Iv of the homotopy z --. zyf3 under the map IX: Y --. X meets the assumptions of(A2) which insure the homotopy qJt.

(A~) Let holomorphic I-forms qJ 1, ... , qJn-l on V be linearly independent at a given point v E V and let u1 , ••• , Un be local coordinates in a small domain U c V around v, such that qJi(fJ/fJud == 0 for i = 1, ... , n - 1. Denote by Yu the cotangent bundle of U and let E(r) c YIf) be the analytic subset which controls the tangency of order ~r of the field fJ/au l to the zero set of the functions F", = qJ(a/aUl) for


forms ({J: U -+ Yu [compare (A')]. Namely, J~(V)E,E(r) if and only ifakF:(v) = 0 for aUl

k = 0, 1, ... , r. It is clear that codim ,E(r) = r + 1. Furthermore, if YJr) c YI/) denotes the subspace of the jets of exact forms ({J on U, then codim(,E(r) n YI/) c YJr») = r + 1 as well.

Fix a form iii on V and some i = 1, ... , q. Denote by (/J = (/J?-l(iii) the space of those (q - 1)-tuples of holomorphic forms ({Jl' ... , ({Ji-l, ({Ji+l, ... , ({Jq on V, where ({Jl' ... , ({Ji-l are exact and such that the forms ({Jl' ... , ({Ji-l' iii, ({Ji+l, ... , ({Jq span the cotangent bundle of V (compare 2.1.1). Denote by ,E' c V, j = 1, ... , i - 1, i + 1, ... , q the subset where the forms ({Jj fail to span the cotangent bundle T*(V) and let 2 c T(V)I,E' c T(V)I,E' be the line subbundle on which these forms vanish.

Lemma. If V is Stein and q > n = dim V, then an arbitrary (q - 1)-tuple in (/J admits a continuous homotopy in (/J to a (generic) (q - 1)-tuple for which the corresponding bundle 2 is semi-transversal to the new ,E'.

Proof For an arbitrary (q - 1)-tuple {({Jj} in (/J and for each point vE,E' c V there are n - 1 forms among ({Jj' which span together with iii the cotangent space T,,*(V). To avoid a mess in the notations, we assume these to be ({Jl'···' ({Ji-l, ({Ji+l,···, ({In.

Let ,E" c V be the subset, where the forms ({Jl' ... , ({Ji-l, iii, ({Ji+l, ... , ({In fail to span the cotangent bundle T*(V). This ,E" is the pullback of some subset,El in the sum of n copies of the cotangent bundle T*(V) under the section (({Jl' ... , iii, ... ,((In): V -+

® T*(V). Denote by 11 the pull-back to V of the ideal of (functions vanishing on) ,El. (Observe that the zero set of 11 equals ,E", but, yet, 11 may be properly contained in the ideal of ,E".) Now, we have a distinguished set of holomorphic homotopies in (/J which only move the last component ({Jq (which is not among the chosen forms) by adding to ({Jq forms zyp, ZEC, where YEll and where p is an arbitrary form on V. [If ({Jq happens to be among the chosen forms and the extra form ({J was among the exact ones, say ({J = ({Jl = dfl, then we would use the homotopies ({Jl + d(zyP) for YE(Id2 and for functions p on v.] These homotopies, for all (q - 1)-tuples in (/J and all v E V, do not form a composible set. However, since V is Stein, there exists a composible set 'P which contains all these homotopies, and which is built up by generalizing (A~) in an obvious way. Now the lemma follows from (A;) with the above relation codim ,E(r) = r + 1 for r = n.

Finally, this lemma reduces the holomorphic h-principle for immersions V -+ cq to the 1-dimensional h-principle (A') (compare 2.1.1) and the theorem follows.

(B) Immersions V -+ 0 (Continuation). If V is an n-dimensional Stein manifold and if q > n then holomorphic immersions f: V -+ cq satisfy the (ordinary) h-principle.

Proof The holomorphic h-principle in (A) reduces this h-principle to Oka's principle for holomorphic sections of the bundle J -+ V whose fiber f", v E V, consists of injective complex linear maps T,,(v) -+ cq. Since J v is transitively acted upon by the linear group GLqC, Grauert's theorem [see 1.1.2 and (C) below] applies.

Corollary. If 2q ~ 3n - 1, then every n-dimensional Stein manifold V admits a holomorphic immersion into eq•

Proof Since the fiber J v is (2q - 2n)-connected and since V is homotopy equivalent to a polyhedron of real dimension n (see 1.1.2), the fibration J -+ V admits a continuous section V -+ J for 2q ~ 3n - 1. Then the above h-principle yields immersions V -+ eq unless n = q = 1. If n = q = 1, then our proof ofthe h-principle does not apply, but a holomorphic immersion V -+ e 1 does exist for an arbitrary open one-dimensional manifold V. [See Gunning and Narasimhan (1967) and 3.2.4.]

Questions. Does the h-principle hold true for holomorphic immersions of ndimensional Stein manifolds into en? Does every Stein manifold admit a holomorphic function f whose differential does not vanish?

The key difficulty can be seen if one takes a non-vanishing holomorphic vector field L = Li'=l ({J;(%z;) on en and looks for a holomorphic function f whose derivative Lf = Li'=l ((J;(of/oz;) does not vanish. Does such an f exist for every field L?

Exercises. (a) Show holomorphic immersions V -+ eq, q > n = dim V, to enjoy Thorn's transversality theorem. Namely, let x(r) be the space of rth order jets of holomorphic maps f: V -+ eq and let E c x(r) be an analytic subset of codimension ~n + 1 for n = dim V. Then immersions f: V -+ eq whose rlh order jets miss E constitute a residual subset in the space of immersions (with the topology of uniform convergence on compact subsets in V).

Hint. Prove the transversality theorem for functions on f whose differentials do not vanish on a subbundle !l' c T(V)I VO semi-transversal to Yo.

(b) Prove the parametric h-principle for holomorphic immersions V -+ eq,

q~n+1.

(c) Prove the holomorphic h-principle for holomorphic k-mersions V -+ eq for k < q (compare 2.1.1).

Remark. The ordinary h-principle for these k-mersions in unknown, unless k = n or k = 1. The difficulty arises in Ok a's principle for the fibration qt -+ V, whose fiber qtv consists oflinear maps T,,(v) -+ eq of rank ~ k. This qtv is not homogeneous (at least in an obvious way) for 2 :s; k :s; n - 1 and so Grauert's theorem does not apply [compare (C) below]. Yet, one can produce holomorphic sections V -+ qt under suitable assumptions on n, q and k (see Gromov-Eliashberg 1971) thus making the holomorphic h-principle for k-mersions non-vacuous.

(d) Fix a proper holomorphic map Fo: V -+ Oo>n and consider the holomorphic maps f: V -+ ep for which F = Fo EE> f: V -+ eq = eqo x ep is an immersion. Prove the holomorphic h-principle for these maps J, provided the underlying map Fo is generic [i.e. the jet of Fo of a sufficiently high order r = r(n) misses a pertinent singularity of codimension > n in the jet space]. Then prove Oka's principle for


holomorphic sections of the corresponding differential relation !7l --+ Vand conclude for a generic Fo: V --+ en +1 ) to the following

Theorem. If 2q ~ 3n + 1 then every n-dimensional Stein manifold V admits a proper holomorphic immersion into Cq•

Question. Do proper holomorphic immersions V --+ 0, q > n, satisfy the h-principle?

(d') Study holomorphic maps f for which Fo Ef> f: V --+ Cq is a holomorphic embedding (i.e. an immersion without double points). Express this property of fin terms of the map sf: V x V --+ cP defined by sf(v1 , V2) = f(v 1 ) - f(V2) [compare (E) in 2.1.1]. Formulate the holomorphic h-principle for these maps f Prove this h-principle, provided Fo is a proper immersion with normal (i.e. transversal) crossings and 2q ~ 3n + 3. Then prove the pertinent Oka's principle and the (ordinary) h-principle for such maps f.

Remark. The above h-principle also holds for generic maps Fo which are not necessarily immersions [This is announced in Gromov-Eliashberg (1971) but the proof has not been published.] This yields proper holomorphic embeddings V --+ cq

for all n-dimensional Stein manifolds V, provided 3n :s; 2q - 3. [See Forster (1970) for a direct construction of a proper holomorphic embedding of every V into Cq for q = ent(in) + 2, compare Schaft (1984).]

(e) Let V be a smooth affine algebraic variety over an algebraically closed field K of characteristic zero. Formulate the algebraic h-principle for regular (i.e. given by polynomials on V c KN) immersions V --+ Kq and prove this h-principle for q > n = dim V. Show that every n-dimensional affine group V over K admits a regular immersion V --+ en+1 . Give further examples of algebraically parallelizable manifolds V to which this h-principle applies.

(f) Prove the holomorphic counterparts of the h-principles in (D) of 2.1.1. In particular establish the h-principle for free holomorphic maps V --+ 0 for q ~ (n + 2)(n + 3)/2. [Compare Gromov-Eliashberg (1971).]

(g) Study holomorphic Lagrange immersions [compare (E) in 2.1.3] of Stein manifolds V into cP x C P with the (exact holomorphic) 2-form h = Lf=l dx; A dy;. In particular, construct a holomorphic immersionf: V --+ c2 n, n = dim V, such that f*(h) == 0 on V for all parallelizable (for example, topologically contractible) Stein manifolds V.

(C) Holomorphic Maps into Elliptic Spaces. Consider a complex manif~ld W which satisfies the following condition

(Elld For each Stein mainfold V and for each holomorphic map f: V --+ W there exists a holomorphic map (homotopy) ((J: V x cP --+ W for some p ~ dim W such that ((J I V = V x 0 = f and such that the differential d((J sends each tangent space T,,(cP) ~ cP c T,,(v x CP), v = (V,O)E V X cP onto Tw(W), w = f(v).


Examples and Exercises. (a) Let Wbe a homogeneous space under some holomorphic action of a complex analytic Lie group G. Then the corresponding action of the Lie algebra ~ ~ CP, P = dim G, on W integrates to the (exponential) map W x ~ --+ W which induces the required homotopy for all f: V --+ W. Hence, W satisfies (Ell1)'

(a/) Let X --+ Wo be a holomorphic fibration with the fiber Wand the structure group G. If the base Wo satisfies (Ell1), then so does X. Indeed, homotopies V x CP --+ Wo lift to horizontal homotopies in X with some holomorphic connection in the induced fibration over V x CPo These are complemented by vertical (fiberwise) homotopies in X induced by the exponential maps in the fibers.

Observe that the above X may be not homogeneous for homogeneous Wo and W. For instance, the total space of a negative line bundle over Cpl is not homogeneous.

(b) Let al> ... , ap be holomorphic actions of C on W whose generating (holomorphic) vector fields span the tangent space Tw(W) for all WE W. Then the composition of the actions al' ... , ap gives a holomorphic map W x CP --+ W which insures (Elll)'

(b/) Let A c 0 be a Zariski closed subset such that codim A ~ 2. Then the complement W = Cq\A admits the above actions al"'" ap- Indeed let Z be a field L1=1 Ci(fJ/fJZi) for some constants Ci. Then there exists a non-zero polynomial P on o which vanishes on A and is constant on the orbits of Z. The field PZI W clearly integrates to an action of C in Wand there are sufficiently many of such fields to span T(W).

(b") Take a Zariski closed subset A c Cpq, assume codim A ~ 2 and prove (Elld for CPq\A. Study W = Wo \A for algebraic homogeneous spaces Woo

(bill) Let W be covered by open subsets Wi c W, i = 1, ... , k, whose complements Ai = W\ Wi are analytic subsets in W. Let Wi satisfy (Elld for all i = 1, ... , k and show W to satisfy (Ell1) as well.

Question. Is (Elld a birational invariant of projective algebraic manifolds?

(c) Let W admit a geodesically complete holomorphic connection. Then the geodesic spray exp: T(W) --+ W insures (Elld. For instance, the quotient W = cq/r for an arbitrary discrete group r of affine transformations of 0 satisfies (Elld.

(c/) Show the Hopf manifold W = cq\ {O}/Z satisfies (Elll)' [Here Z acts on cq by (z,x)r-+zx for (Z,X)EZ x Cq.]

(d) A domain U c V is called Runge if every holomorphic function on U can be approximated by functions which holomorphically extend to V. Let W satisfy (Elld and let fo: V --+ Wand fl: U --+ W be holomorphic maps, where U is Runge and such that fl can be joined with fo I U by a continuous homotopy of holomorphic maps Ji: U --+ W, t E [0, 1]. Show that f1 can be approximated by maps which holomorphically extend to V.

(e) Let W satisfy (Elld and let W 1, ••• , Wk, ••• , be an arbitrary sequence of points in a connected component of W. Prove the existence of a holomorphic map f: C --+ W for which f(Zi) = Wi' i = 1, ... , where Zi E C are arbitrary points, such that IZil --+ 00 for i --+ 00.

(e/) Let V be an arbitrary Stein manifold and let fo: V --+ W be a holomorphic


map. Construct a holomorphicmapf: V ---+ Whomotopic to fo, such thatf(zi) = Wi

for a given divergent sequence of points Zi E V.

(C') The Holomorphic h-Principle for Immersions. If W satisfies (Ell1 ) and if V is Stein, then holomorphic maps V ---+ W satisfy Thom's transversality theorem for analytic subsets L in the jet space x(r) for X = W x V ---+ V and all r = 0, 1, ....

The proof is straightforward and left to the reader.

Next, the argument in the above (A) and (B) easily generalizes to those W (in place of Cq ) which split into products, say W = WI X Wz X .•• x ltk. Namely, if WI' ... , ltk satisfy (Elld and dim W - dim W; ~ dim V for all i = 1, ... , k, then immersions of Stein manifolds V to W satisfy the holomorphic h-principle.

Exercise. Let W be a homogeneous space under an action of a complex Lie group G which is the product G = G1 X Gz x ... X Gk • Assume that the orbits of Gi in W have dimension qi' i = 1, ... , k, such that L~=l qi = q = dim Wand such that q - qi ~ n for all i = 1, ... , k. Show holomorphic immersions of n-dimensional Stein manifolds V into W to satisfy the holomorphic h-principle. Apply this to immersions to an arbitrary commutative group G which has dim G ~ n + 1.

(C") Oka's Principle. Applications of the holomorphic h-principle depend on the ordinary (Oka's) h-principle which expresses the following property of hoI om orphic sections of a fibration X ---+ V.

(Ellz) Let Vo be an arbitrary analytic subset in V and let fo: {!!ft Vo ---+ X be a holomorphic section over a small neighborhood {!!ft Vo c V of Vo c V. Then holomorphic sections f: V ---+ X, which have 1;1 Vo = 1;01 Vo for a given r = 0,1, ... , satisfy the h-principle.

A fundamental theorem of Grauert (see 1.1.2) insures (EI12 ) for those fibrations over Stein manifolds V which admit a complex Lie group for the structure group which transitively acts on the fiber. This theorem immediately implies the following

Corollary. Let Y ---+ X be a holomorphic fibration whose structure group is a complex Lie group transitive on the fiber. If V is Stein and if a fibration X ---+ V satisfies (Ellz) then the (composed) fibration Y ---+ V also satisfies (Ellz).

Next, a manifold W is said to satisfy (Ellz) iff the trivial fibration W x V ---+ V satisfies (Ellz) for all Stein manifolds V.

Remarks. (a) Obviously, (Ellz) implies (Ell1) but the implication (Ell1) = (Ellz) is unknown.

(b) The above corollary provides many (EIl2 )-manifolds W: these are built out of homogeneous spaces by successive fibrations.


(c) The ellipticity axioms (Elld and (EU2) allow. "many" holomorphic maps C -+ W. This is contrary to the hyperbolicity of W which prohibits non-constant holomorphic maps C -+ W. Observe that every connected Riemannian surface W (i.e. dim W = 1) is either hyperbolic (if it is covered by the disc) or elliptic (if it is covered by C or by S2). One may also view the ellipticity as a dual to the Stein property of W which claims "many" holomorphic maps W -+ C.

Finally, the above corollary immediately yields the h-principle for holomorphic fiberwise injective homomorphisms T(V) -+ T(W), provided V is Stein and W satisfies (E1l2). Hence, the holomorphic h-principle [see (C')] for immersions V -+ W implies the ordinary h-principle in case W is (E1l2).

(D) Harmonic Immersions. Many properties of the Cauchy-Riemann system generalize to other systems of linear (and quasi-linear) systems of elliptic P.D. equations. Let, for example, V be an open Riemannian manifold. Green and Wu (1975) have pointed out that the harmonic structure of V is Stein. Namely, V satisfies Stein's axioms with harmonic functions in place of the holomorphic ones.

Corollary (Green-Wu). Every open n-dimensional Riemannian manifold V admits a proper harmonic embedding f: V -+ 1R2n+1.

Questions. Does the removal of singularities yield the h-principle for (proper) harmonic immersions V -+ IRq for q ~ n + 1? What happens for q ::;;; n? For example, does every open Riemannian manifold V admit a harmonic function f: V -+ IR whose gradient nowhere vanishes? Which Riemannian manifolds W satisfy (Elld and (E1l2) for harmonic maps of open Riemannian manifolds V -+ W?

2.2 Continuous Sheaves

Consider a relation .f1l c x(r) for a fibration X -+ V, fix an integer k ~ r and denote by 4>(U) the space of Ck-solutions of f1l over U for all open subsets U c V. The collection of the spaces 4>(U) comes with an additional structure given by the restriction maps, called 4>(1): 4>(U) -+ 4>(U') for all open subsets U' in U, where 1= I(U', U) stands for the inclusion, I: U' c U, and where 4>(I)(cp) = cpl U' for all cp E 4>(U). The assignment {U H 4>(U), I H 4>(I)} (which is a contravariant functor from the category of open subsets in V to the category of topological spaces) is called the sheaf of Ck-solutions of f1l over V (compare 1.4.2).

Recall that an abstract sheaf 4> over a topological space V (see Godement 1958), by definition, assigns a set 4>(U) to each open subset U c V and a map 4>(1): 4>(U) -+ 4>(U') to each inclusion I: U' c U, such that the following three axioms are satisfied.

(1) Functoriality. If 1': U" c U' and I: U' c U, then the value of 4> at the inclusion 10 1': U" c U abides 4>(1 01') = 4>(1') 0 4>(1).


One also agrees tP(Id: U c U) = Id for all U c V and tP(0) = 0. One calls elements (fJ E tP sections of tP over U and writes (fJ I U' instead of tP(I).

(2) Locality (Uniqueness). If two sections (fJ1 and (fJ2 of tP over U are locally equal, then they are equal, where the local equality means there exists a neighborhood U' c U of every point u E U, such that (fJ11 u' = (fJ21 U'.

(2') Locality (Existence). Let open subsets Ull c U, Jl E M, cover U and let sections (fJ1l E tP(UIl ) satisfy (fJ1l1 Ull nUll' = (fJ1l,1 Ull nUll' for all Jl and Jl' in M. Then there exists a section (fJ E tP(U) [which is unique by (2)J, such that (fJ1 Ull = (fJ1l for all JlEM.

The axioms (2) and (2') show every sheaf tP to be uniquely defined by tP(U.) for any base of open subsets U. c V.

Next, one extends tP to non-open subsets C c V by tP( C) = tP( l!7ftC) which denotes the inductive (direct) limit of tP(U) over all neighborhoods U c V of C (compare 1.4.1). In particular, one defines the stalk tP(v) = tP(l!7ftv) for all VE V and one writes (fJ(V)EtP(V) for (fJ1l!7ftv. Then one can restrict tP to a sheaf over C, called tPlC and defined by (tPlC)(D) = tP(l!7ftD) for all open subsets DeC and for l!7ftD c V. Thus, the sheaf tPl C has the same stalks over the points c E Cas tP.

A sheaf tP is called continuous (or quasi-topological) if every set tP(U), U c V, is endowed with a quasi-topology (see 1.4.1), such that the map tP(I) is continuous for all inclusions I: U' c U. In this case the space tP(C) is equipped with the inductive limit quasi-topology for all subsets C c V.

A homomorphism between continuous sheaves over V, say IX: tP --+ 'P, is a collection of continuous maps lXu: tP(U) --+ 'P(U), for all open U c V which commute which the restrictions of sections, that is lXu,o tP(I) = 'P(I) 0 lXu for all I: U' c U. Finally, one defines a subsheaf tP' c tP by giving a subspace tP'(U) c tP(U) for all U c V, such that tP' satisfies (2) and (2').

2.2.1 Flexibility and the h-Principle for Continuous Sheaves

Consider a continuous sheaf tP over V and define the parametric sheaf tPP over V x P for an arbitrary topological space P by first claiming that its sections are just the continuous families of sections of tP parametrized by P. To complete the definition we only need to specify tPP(U x R) for open sets U c V and ReP. Set tPP(U x R) equal to (tP(U))R, the space of continuous maps R --+ tP(U) with the following quasi-topology. A map Q --+ (tP(U))R is continuous iff the corresponding map R x Q --+ tP(U) is continuous.

Next, we apply this construction to P = V, and then restrict the parametric sheaf tPv over V x V to the diagonal A c V x V. The resulting sheaf over A = V is denoted tP*. Intuitively, sections in tP* are continuous families of germs (fJv c tP(v), v E V. For example, if tP the sheaf of locally constant functions V --+ ~, then tP* is canonically isomorphic to the sheaf of continuous functions on V.

Every section of tP corresponds to a unique constant family of sections with the parameter space V. Thus, we obtain a natural injective homomorphism A: tP --+ tP* which makes tP a subsheaf A(tP) = tP in tP*.


(A) Definitions. A sheaf tP satisfies the (sheaf theoretic) h-principle, if every section q> E tP*(U) can be homotoped to tP(U) c: tP*(U) for all open subsets U c: V. The parametric h-principle for tP claims the homomorphism A to be a weak homotopy equivalence. That is Au: tP(U) -+ tP*(U) is a w.h. equivalence for all open U c: V.

(A') Remark. If tP is the sheaf of solutions of a relation &t c: x(r) and if 'l'is the sheaf of continuous sections V -+ &t, then there is a natural (and obvious) homomorphism J: tP* -+ 'l', such that (J 0 A)f = J,: U -+ &t for all f E tP(U) and all U c: V. If this J is a w.h. equivalence, then the sheaf theoretic h-principle obviously implies the h-principle for &t.

(A") Examples. (a) If &t is an open subset in x(r), then J is clearly a w.h. equivalence (compare 1.4.1).

(b) If &t is given by (*) in (B2) of 1.4.1, then J is a w.h. equivalence, which is proven in 2.3.2. In particular, J is a w.h. equivalence for the sheaf of free isometric Coo-immersions (V, g) -+ (W,h).

(B) Flexibility and Microflexibility. A sheaf tP is called flexible (micro-j1exible) if the restriction map tP( C) -+ tP( C') is a fibration (micro-fibration) for all pairs of compact subsets C and C' c: C in V (compare 1.4.2).

The main result of this section is the following

Theorem. If V is a locally compact countable polyhedron (e.g. a manifold), then every flexible sheaf over V satisfies the parametric h-principle (compare 1.4.2).

Proof Start with the following

(B 1) Definition. A homomorphism IX: tP -+ 'l'is a local w.h. equivalence if IXv: tP(v)-+ 'l'(v) is a w.h. equivalence for v E V.

(B2 ) Local Lemmas. Let tP be any (possibly non-j1exible) sheaf over a locally finite polyhedron V. Then (a) the sheaf tP* is flexible; (b) the inclusion tP = A(tP) c: tP* is a local w.h. equivalence.

Proof The first claim is obvious by the proof of (A') in 1.4.2. To prove (b), take a point v E V and consider a fundamental system of neighborhoods Ui c: V of V,

Ui+l c: Ui' i = 1, 2, .... The space tP(v) is the inductive limit of the sequence tP(Ud -+ tP(U2 ) ••• and the space tP*(v) is the inductive limit of the sequence [tP(U1)]U, -+ [tP(U2 )]U2 -+"', where [tP(Ui)]U, denotes the space of maps Ui -+ tP(Ui). Since V is a polyhedron we can choose all Ui to be contractible, to make the inclusions tP(Ui) -+ [tP(Ui)]U, w.h. equivalences. By passing to the inductive limit we conclude that the map tP(v) -+ tP*(v) is also a w.h. equivalence.

Thus the main theorem reduces to the following


(B) Homomorphism Theorem. Let cP and 'P be flexible sheaves over a locally compact countably compact finite dimensional space V. Then every local w.h. equivalence oc: cP -+ 'P is a weak homotopy equivalence.

Proof We start with the following functorial construction, which transforms an arbitrary continuous map a: X -+ Y into a fibration a: X -+ Y. Start with the space P(Y) of all continuous maps p: [0,1] -+ Y and define X as the subset in X x P(Y) of the pairs (x,p), such that a(x) = p(o). We define the map a by setting a(x, p) = p(1). There is a canonical homotopy equivalence between X and X and the map a: X -+ Y is a Serre fibration. Clearly, the map a is a w.h. equivalence iff a is a w.h. equivalence. Now, an arbitrary Serre fibration is a w.h. equivalence iff the fibers are weakly contractible. (A space Z is called weakly contractible if for an arbitrary compact polyhedron K the space of continuous maps K -+ Z is path connected. For such a Z, the space of maps K -+ Z, in fact, is weakly contractible.)

Let us apply our fibration construction to the homomorphism ex: cP -+ 'P. We _ _ r--J

obtain a new sheaf cP, cP(V) = cP(V), and a homomorphism a, such that the maps au: (j)(V) -+ 'P(V) are Serre fibrations for all open sets V c V. In order to get "a fiber" of a we take a section 'P E 'P(V) and define the fiber sheaf Q = Q", over V by setting Q(V') = a[/(t{!) for all open subsets V' c V. The flexibility of cP and 'P implies (easy to see) the flexibility of Q, which reduces (B) to the following

(B') Contractibility Theorem. Let Q be a flexible sheaf over V. If Q is locally contractible [that is the space Q(v) = Q(CDji(v) is weakly contractible for all VE V] then the space Q(V) is weakly contractible.

Proof We proceed in four steps.

Step 1. If V is a compact subset in [R then the space Q(V) is path connected.

Proof To join sections W l and W 2 in Q(V) by a path wtEQ(V), tE [0,1], we cover V c: ~ by two closed subsets V' and V", such that each of them lies in a disjoint union of small e-intervals in [R and such that the intersection V' n V" is a finite set. Since the space Q(v) = Q(CDjiv) is path connected for all VE V; there is a path w; E Q(V') between wllV" and wZIV" as well as a path w;' E Q(V") between wllV" and w 2 IV", provided the above e > ° is small enough. Since the space Q(V'n V") is simply connected (being the product of the simply connected spaces Q(v), VE V' n V"), there is a homotopy of the path w;1 V' n V" to w;1 V' n V", which extends, by the flexibility, to a homotopy of w; to another path w; E Q(V') between wll V' and w21 V", (where the homotopies by definition are fixed at the ends of the paths). Now the paths w; and w; agrees over CDji(V' n V") and, hence, they define the required path W t over V.

Step 2. The above Q(V) is weakly contractible.

Proof Since Q is flexible, the sheaf V H (Q(V))K over V is also flexible and the previous step applies.

Step 3. If V is a compact subset in [RN then Q(V) is weakly contractible.

Proof Let p: V --+ ~ be an orthogonal projection and let Q. denote the push10rward sheaf over p(V) c ~ defined by Q.(U.) = Q(p-i(U.» for all U. c p(V). Since Q is flexible, the sheaf Q. is also flexible. Furthermore, Q.(v) = Q(p-i(V» for VEp(V), which implies by induction in N the weak contractibility of Q.(v) for all v E p(V), as p-i(V) lies in a hyperplane in ~N. Hence, the space Q(V) = Q.(p(V» is weakly contractible by Step 2.

Step 4. Exhaust an arbitrary V by compact subsets Vi c V2 c ... c Vi c ... c V. Since dim V < 00, every Vi embeds into ~N and so Q(Vi) is weakly contractible

for all i = 1, 2, .... The space Q(V) is the inverse (projective) limit of the sequence of the Serre fibrations, Q(Vi) +-- Q(V2) +-- ••• , and hence, it is weakly contractible as well. Q.E.D.

Exercise. Assume Q(v), VE V, to be (n + k)-connected for n = dim V and a given k = 0,1, .... Prove Q(V) to be k-connected for an arbitrary flexible sheaf Q over V.

2.2.2 Flexibility and Micro-flexibility of Equivariant Sheaves

We have already seen in 1.4.2 that many interesting sheaves over V are acted upon by the group of diffeomorphisms Diff(V). In fact, it is customary to use the pseudogroup of diffeomorphisms, that is the set of all pairs (U,f), where U is an open set and f is a diffeormorphism of U onto another open set U' = f(U)

Examples.Let tP be the sheaf of immersions V --+ W which relates to each open set U c V the space tP(U) of immersions U --+ W. A diffeomorphism f: U --+ U' sends the space 4>(U' ) to 4>(U) by qJ --+ qJ 0 f for each immersion qJ: U ' --+ W

Further examples of such Diff(V)-invariant sheaves are provided by k-mersions (see 1.3.1), by free maps (see 1.1.4) and by maps transversal to a foliation or to a general subbundle as in 2.1.4. On the other hand the sheaf of isometric immersions V --+ W is not Diff(V)-invariant. It is invariant under the (pseudo )-group of isometries of the underlying Riemmanian manifold V. Another important (pseudo) group is attached to a smooth map n: V --+ Yo. This (pseudo) group consists of diffeomorphisms f commuting with n, that is n(f(v» = n(v), VE V. It is called the (pseudo) group Diff(V, n) of fiber preserving diffeomorphisms.

Main Flexibility Theorem. Let V = Vo x ~,and let n: V --+ Vo denote the projection on the first factor. Let tP be a micro flexible sheaf over V, invariant under Diff(V, n). Then the restriction tPl Vo = Vo x 0 is a flexible sheaf over Vo = Vo x O.

We prove this theorem in the next section, but first give some applications.

Flexibility for Diff-invariant Sheaves. Let 4> be a micro flexible Diff(V)-invariant sheaf over a manifold V. Then the restriction to an arbitrary piecewise smooth polyhedron K c V of positive codimension, tPIK, is a flexible sheaf over K.


Proof Start with an arbitrary sheaf IJ' over a locally finite polyhedron K and observe (using the induction by skeletons) the following simple fact

(*) If the restriction of IJ'to every simplex in K is a j1exible (microj1exible) sheaf, then lJ'is also j1exible (microj1exible) over all of K.

By applying this to sufficiently fine subdivisions of K we conclude to the following

Localization Lemma. If for each point k E K the restriction of IJ'to some neighborhood of k is aj1exible (microj1exible) sheaf then IJ' itself is a j1exible (microj1exible) sheaf

Exercise. Prove the localization lemma for an arbitrary locally compact space K.

Let us return to our Diff(V)-invariant sheaf rp. For each simplex L1 in K there is a neighborhood U c V which splits into the product U = Vo x IR such that L1 c Vo = Vo x 0 c U. The restricted sheaf rpl U over U is Diff(U)-invariant and, in particular, it is Diff(U, n)-invariant for the projection n: U ~ Yo. The main flexibility theorem· implies that the sheaf rpl Vo is flexible, so rp 1 L1 is also a flexible sheaf, and Lemma (*) applies.

The h-Principle for Open Manifolds. An arbitrary microj1exible Diff(V)-invariant sheaf rp over an open manifold V satisfies the parametric h-principle.

Proof Take a codimension one polyhedron K c V such that V is isotopic to an arbitrarily small neighborhood of K (see 1.4.1). Since that sheaf rplK is flexible it satisfies the h-principle (see 2.2.1); therefore the sheaf rpl (I) jz(K) = V also satisfies the h-principle. [See Bierstone (1973) for a generalization.]

Remark. As we know (see 1.4.2), the sheaf rp is not flexible in general.

Immersions, Free Maps, etc. All open Diff(V) invariant differential relations over an open manifold V satisfy the parametric h-principle. In particular, maps of rank > k, free maps, hyperbolic immersions (see 3.2.2) and maps V ~ IRq directed by an open set A c Sq-l (see 1.4.4) satisfy this h-principle as long as the underlying manifold V is open.

Proof The openess of a differential condition implies mircoflexibility and it also shows that the sheaf theoretic h-principle is equivalent of the usual one (see 1.4.1).

Isometric Immersions Between Pseudo-Riemannian Manifolds. The sheaf of isometric COO-immersions (V, g) ~ (w, h) is not, in general, microflexible. However, the sub-sheaf rp of free isometric immersions is microflexible (see 2.3.2). Furthermore, if (V, g) = (Vo x IR, go EB 0), then rp is Diff(V, n)-invariant for the projection n: V = Vo x IR ~ Yo. In this case rp is a flexible sheaf, which, moreover, satisfies the h-principle. With this, one obtains isometric immersions V ~ W under suitable dimension assumptions (see 1.4.2, 3.3).


2.2.3 The Proof of the Main Flexibility Theorem

Let cP be an arbitrary sheaf over V. We treat a map t/J: Q --+ cP(A), A c V, as a function t/J = t/J(v,q), vEA, qEQ, where t/J(V,q)EcP(V) denotes the image of the section t/J(q)EcP(A) under the restriction map cP(A) --+ cP(v) = cP(l!?jZ(v». In short, t/J(v,q) is the restriction of t/J(q) to l!?jZ(v).

When Q is split into the product of a compact polyhedron and a closed interval, Q = P x [x,y], then the maps t/J: P x [x,y] --+ cP(A) are called deformations over A. When the set A is compact then according to our definitions (see 1.4.1), each deformation t/J over A is actually defined over an open set U = U(t/J) ::> A, and we denote this extended deformation P x [x, y] --+ U also by t/J. We say that a deformation t/J over A is fixed at a point VE U = U(t/J) if t/J(v;p,t) = t/J(v;p,x) for all pairs (p, t)E P x [x,y]. We call the set of non-fixed points oft/J the support of the deformation, supp t/J c U.

(A) Compressibility. A deformation t/J over A is called compressible if for an arbitrarily small neighbourhood U of A there exists a deformation if: P x [x,y] --+

cP(U), U = U(t/J), with the following three properties:

(i) iflA = t/JIA, that is if(a;p, t) = t/J(a;p, t) for all points aEA, pE P, tE [x,y]. (ii) iflP x x = t/JIP x x, that is if(v;p,x) = t/J(v;p,x) for all VE U and pEP.

(iii) supp if c U.

Observe, that this compressibility property of a deformation t/J, depends only on the behaviour of t/J near A, that is on l!?fi(A) c U = U(t/J), and so the particular choice of the neighborhood U does not affect the content of the notion of compressibility. However, one can not express compressibility in terms of the restricted sheaf cP I A. In fact, if a deformation t/J is given over the whole space where a sheaf is defined, then this t/J is tautologically compressible. Let us show that compressibility is equivalent to flexibility. More precisely, a sheaf cP over a locally compact space V is flexible if for every compact set A c V all deformations over A are compressible.

Proof Let cP be a flexible sheaf. The condition (i) prescribes if on A and the condition (iii) prescribes if on U\ U by requiring it to be fixed there. When the neighbourhood U c U is sufficiently small and its closure in U, CI(U), is compact, then the restriction map cP(A U (CI(U)\ U» --+ cP(CI(U» is a fibration, and we can extend this deformation from AU (CI( U)\ U) to CI( U). Such an extension gives us if on CI(U) and so on U.

Now, let t/J: P x [0,1] --+ cP(A) be a compressible deformation over A and let its restriction to P x ° c P x [0,1], t/JIP x 0, be extended to a larger compact set Be V, B ::> A. The properties (i) and (ii) show that in order to extend t/J, it is sufficient to extend some compression if of t/J. When the neighbourhood U is chosen sufficiently small, the condition (iii) allows us to extend if to B just by making it fixed on B\ U.

Microcompressibility. A deformation t/J: P x [x,y] --+ cP(A) is called microcompressible if there is a positive e, e E (0, X - y], such that the restricted deformation


"'IP X [x,x + e] = "'.: P X [x,x + e] -+ IP(A) is compressible. This property is reminiscent of microflexibility but, in fact, is much stronger. If IP is a microflexible sheaf, then, for a given neighborhood U of a compact set A, we can find a positive e, such that the deformation "'. can be compressed to If/. with supp "'. c U, but this e here depends on U. On the other hand, the microcompressibility property provides a universal e for all (arbitrarily small) neighborhoods U. This difference is crucial as the following lemma shows.

Microcompressibility Lemma. A sheaf IP over a locally compact space is flexible iff all deformations over compact sets are microcompressible.

Proof We must show that microcompressibility implies compressibility. Take a deformation "': P X [x, y] -+ IP(A) and let us apply the microcompressibility property to the following auxiliary deformation,.,: Q x [0,1] -+ IP(A), where Q = P x [x,y] and,., is defined by ,.,(p, t, 1:) = ",(p, min(y, t + 1:», PEP, t E [x, y], 1: E [0,1]. Since ,., is microcompressible with some e > 0, the restriction of", to an arbitrary interval [Xl, Yl] c [x, y] of length::; e, that is "'IP x [Xl' Yl] = '" 1 : P x [Xl' Yl] -+ IP(A), is a compressible deformation. Now, we subdivide the interval [x, y] into some finite number of intervals of length ::; e such that the restriction of", to these intervals are compressible. Now an obvious induction reduces the whole matter to the following.

If for a point ZE[X,y] the restricted deformations "'0 = "'IP x [x,z] and "'1 = "'IP x [z,y] are compressible then", is also compressible.

Indeed, for a given U ~ A, we first compress "'0 to If/ 0 with supp If/ 0 c U. Next, we take a smaller neighbourhood U1 of A such that "'0IU1 ="'0IU1 •

Such a U1 exists by the property (i) above. Finally, we compress "'1 to If/l with supp If/l c U1 and we define If/ by "glueing" If/ 0 and If/l together:

If/(P t) = {~o(P,t) for tE[X,Z], , "'l(P,t) for tE [z,y].

The property (ii) shows that this iii is a correctly defined compression.

(B) Actions of Diffeotopies on IP. Take open subsets U' c V and U c U' and move U in U' by a diffeotopy c5t : U -+ U', tE[O, 1], for c50 = Id: U -+ U cU'. Let 1P' be a subset in IP(U' ) and let c5t act on 1P' by assigning to each section cP E 1P' a homotopy of sections in IP(U}, called c5:cp E IP(U), such that c5~cp = cpl U and such that the following four conditions are satisfied.

(i) If two sections in 1P' are equal at some point in U', say CPl (u~) = CP2(U~), and if c5to(uo) = u~ for some Uo E U and to E [0,1], then (c5t~cpl)(uo) = (c5t~cp2)(uo). This allows us to write cp(c5t(u» for (c5t*cp)(u), UE U.

(ii) Let Uo cUbe the maximal open subset where c5t is constant in t, that is c5t(u) = c5o(u), U E Uo. Then the homotopy c5t* cP (whenever defined) is constant in t over Uo.

(iii) If the diffeotopy £\ is constant in t for t ~ to over all of U, then the homotopy c5t* cP also is constant in t for t ~ to.


(iv) If qJp E t/J', PEP, is a continuous family of sections, then the family (),* qJp is jointly continuous in p and t.

(B') Examples. (a) If the sheaf t/J is acted upon by the group Diff V, then all diffeotopies obviously act on all of t/J. For instance, the diffeotopies in V act on the sections of the trivial fibration X = V x W -+ V. Now, let Y be an open subset in X and consider the sheaf t/J of sections V -+ Y. Since Y is not assumed invariant under Diff(V), the diffeotopies of V do not preserve t/J. However, for an arbitrary compact family of sections qJ E t/J(U'), the action {)t* qJp is defined in t/J for the diffeotopies {)t which are sufficiently CO-close to the identity.

(b) Let a diffeotopy {)t be smooth in t and let (); denote the vector field d{)tldt on Ut = ()t(U) c U'. Take a continuous function (X on U x [0,1] which vanishes at every point (u, t) E U x [0,1], where the vector ();(u) E T,AU'), u' = ()t(u), vanishes. Then the assignment qJ ~ qJ({)t(u)) + J~ (X(u, t) dt defines an action of {)t on (the sheaf of) continuous functions qJ on V.

(C) Sharp DifJeotopies. Let Vo be a closed subset in the above U' c V. A diffeotopy of Vo in U' is by definition a diffeotopy {)t: U -+ U' for an arbitrarily small neighborhood U = (?jt Vo c U'. Fix some metric in V and call some set .91 of diffeotopies Vo -+ U' strictly moving a given subset S c Yo, if dist({)t(S), Yo) ~ Jl > ° for t ~ ! and for all {)t E d.

Call .91 sharp at S if for every v > ° there exists a diffeotopy ()t E .91 such that

(i) ()tl{?jt(v) = ()ol{?jt(v), tE [0,1], for all points VE VO which have dist(v, S) ~ v, where (?jt(v) c Vis an (arbitrarily) small neighborhood of v.

(ii) {)t = {)1/2 for t ~ !. Call a set of diffeotopies sharply moving Vo at S, if it contains a subset .91 which

strictly moves S and is sharp at S. Finally, for a given sheaf t/J on V and for given actions of diffeotopies ()t on

subsets t/J' = t/J~, c t/J(U'), we say that acting difJeotopies sharply move Vo at S, if for every compact family of sections qJp E t/J(U') there exists the above subset .91, such that qJp E t/J~t for all ()t E d.

(C') Let Vo be a submanifold in V, let B c Vo be a codimension zero submanifold bounded by a closed hypersurface S = oB c Vo and let t/I: P x [0,1] -+ «P(B) be a deformation for a given sheaf t/J on V. Denote by U' = U'(t/I) c V the neighborhood of Be V where t/I is actually defined and let V~ = Vo n U'. Take an arbitrary compact subset A c B which does not intersect S = oB and let Uo c V~ be an arbitrary neighborhood of B.

Main Lemma. If the sheaf t/J is micro flexible and if acting difJeotopies ()t: V~ -+ U' sharply move V~ at S, then the deformation t/lIA admits a microcompression in the sheaf t/Jo = «PI VO to a deformation with the support in Uo.

Proof Since t/J is microflexible, there exist e = e(Jl) > ° for all Jl > 0, and a deformation t/I!l: P x [0, e] -+ t/J(U'), which equals '" I P x [0, e] near B and such that the

support supp "'p. lies in the Jl-neighborhood Up. c: U' of B (for a given metric on V). Moreover, there exists 8 = 8(Uo) > 0, such that the above homotopy can be made additionally constant in t E [0,8] on Vo \ Uo, that is

supp("'p.IP x [0,8])n V~ c: Uo.

Next, take a diffeotopy t5t : V~ -+ U', tE [0,1], which does act on "'p. and such that

(i) t5t is constant in t away from a small v-neighborhood Uv c: Uo of S c: Uo for some v > 0, such that

v < min(dist(S, A), dist(S, 0 Uo)).

(ii) 1\ is constant in t for t ~ t. (iii) 151 sends S outside Up.,

15 1 (S) c: U'\ Up."

Let (\ = 15M for A. = 8-1 where (\ for t ~ 8 is defined by (\ = 15 1• Now compress "'p.I V~ to a deformation Ij/ in rpo by putting

Ij/(v,p,t) = {"'p.(~(V),p,t). for_vEB and tE[O,B]; '" p.(t5 t (v), p, mm(t, B)) for v E Vo \B.

These formulae agree along S for t E [6, B], as (\ for t ~ 8 sends S outside the support of "'p., and hence, Ij/ is correctly defined. Since the support of "'p.IV~ lies in Uo and since t5t is constant in t outside Uo, the deformation Ij/I V~ is supported in Uo,

while Ij/IA = "'p.IA = "'IA. Q.E.D.

(C") Say that acting difJeotopies sharply move a submanifold Vo c: V if each point v E Vo admits a neighborhood U' c: V of v, such that acting diffeotopies t5t : V~ = Vo n U' -+ U' sharply move V~ at any given closed hypersurface S c: V~.

Theorem. Let rp be a microj1exible sheaf over V and let a submanifold Vo c: V be sharply movable by acting difJeotopies. Then the sheaf rpo = rplXo is flexible and, hence, it satisfies the h-principle.

Proof An arbitrary deformation over A c: V~ can be microcompressed by applying the main lemma to some hypersurface S around A. Hence [see (A)], rpl Vo is locally flexible and therefore (see the localiiation lemma in 2.2.2) a flexible sheaf.

Corollaries. (a) The main flexibility theorem (see 2.2.2) follows immediately since diffeotopies preserving the fibration V = Vo x IR -+ Vo (obviously) sharply move Vo=VoxOc:v.

(b) Let rp' be an open Diff(V)-invariant subsheaf ofthe sheaf of exterior differential k-forms on V and let rp c: rp' be the subsheaf which consists of the closed forms in rp'. Notice that the sheaf rp (unlike rpl) is not microflexible. Consider the subspace Go c: rp(V) offorms g in rp(V) co homologous to a given closed k-form go on V. Then we "parametrize" Go with the sheaf 'Po of the (k - 1)-form '" on V for which


go + dt/l E cPo The sheaf 'Po is clearly microflexible and the map t/I H go + dt/l is a homotopy equivalence of 'Po(V) onto Go.

lf go =1= 0, then the sheaf 'Po is not Diff(V)-invariant. However, each smooth diffeotopy bt: U - U' defines an action on 'Po as follows. Take the interior product (see 3.4.1) of the field b; = dbt/dt on Ut = bt(U) c: U' with go and then pull-back this product to U by bt. Thus we obtain a (k - I)-form, say gt* = bt*(b;· go) on U for all tE [0, 1].

Assign t/I H bt*(t/I) + J~ gt* dt for all t/I E 'Po. One easily checks this to be an action [compare (b) in (B)] of bt on 'Po which yields the parametric h-principle for 'Po and, hence for cP, provided V is open.

(C III ) Example. Let cP be the sheaf of symplectic (see 3.4.2) forms on an open manifold V and let forms go and gl in CP(V) represent a given class 1X0 EH2(V; ~). lf there exists a homotopy of non-singular (possibly non-closed) 2-forms gt between go and gl' then the h-principle for the sheaf 'Po yields a homotopy of I-forms t/ltE 'Po(V), tE [0, 1], such that t/lo = 0 and dt/ll = gl - go·

Exercises. Let L be a completely non-integrable k-plane field on a Coo-manifold W, that is a k-dimensional subbundle L c: T(W), such that successive Poisson brackets ofthe Coo-sections W - L (that are vector fields on W tangent to L) span the tangent bundle T(W).

(a) Show the sheaf of Coo-immersions ~ - W which are everywhere tangent to L to be microflexible. Hint. This can be done with a straightforward (but lengthy) geometric argument or (much faster) with the analytic techniques of 2.3.8.

(b) Let V be an arbitrary smooth manifold and let~' be the sheaf of Coo-maps F: V x IR - W, such that

(i) the restriction of F to each line v x IR c: V x IR, v E V, is an immersion IR = v x IR - W everywhere tangent to L.

(ii) the map F is transversal to L. Here it means the fiberwise surjectivity of the homomorphism T(V x ~) - T(W)/L obtained by composing the differential DF : T(V x ~) - T(W) with the quotient homomorphism T(W) - T(W)/L.

Prove the sheaf ~' to be microflexible. Then show the sheaf ~ = 'PI V x 0 to be a flexible sheaf on V = V x o.

(b') Prove the sheaf cP of Coo-maps V - W transversal (in the above sense) to L satisfies the CO-dense parametric h-principle.

2.2.4 Equivariant Microextensions

Consider two sheaves cP and iP over V and a homomorphism IX: iP - cP over V. We say that IX is surjective if for each point v the corresponding map lX(lVp(v»: ~(v) - cP(v) is onto. Notice that for a surjective homomorphism IX the map IX(V): iP(V) - cP(V) is not, in general, surjective.

Examples. Let V = V x 0 c V x ~,let tP denote the sheaf (of germs) of immersions V -+ Wand let if> be the restriction to V of the sheaf immersions V x ~ -+ W. The natural restriction homomorphism if> -+ tP is surjective iff dim V < dim W, but the corresponding map if>(V) -+ tP(V) may not be surjective for dim W < 2(dim V) - 1. For instance, if V is the real projective plane and W = ~3, then there is an immersion V -+ ~3 but the product V x ~ has no immersions into ~3. However, for a contractible manifold V, each immersion V -+ Wextends, for dim W > dim V, to an immersion V x ~ -+ W. This allows one to apply the main flexibility theorem to V x ~ and to conclude the flexibility of immersions of a contractible manifold V into W for dim W > dim V. Since any V is covered by balls, one can use the localization lemma (see 2.2.2) and obtain flexibility, and thus the h-principle, for extra-dimensional immersions of non-open manifolds V.

Unfortunately, for more general maps this approach does not work. For example, in general, a map V x M of rank ~ k, even for V = ~n, cannot be extended to a map V x ~ -+ W of rank ~ k + 1, though such extensions may exist at every point VE V.

Let us describe a property of the homomorphism IX: if> -+ tP that "descend flexibility" from if> to tP. Take two sets A and B c A in V and call two sections qJEtP(A) and jpEif>(B) coherent if the restriction qJIBEtP(B) equals to lX(jp)EtP(B). The space of all coherent pairs (qJ, jp) E tP(A) x if>(B) is denoted by Q = Q(A, B) and the natural map if>(A) -+ Q is denoted by " = " (A, B).

Definition. A homomorphism IX is called microextension if it is surjective and if for every pair of compact subsets A and B in V the map ,,: if> -+ Q is a microfibration.

Example. Let tP be the sheaf of immersions V -+ Wand let if> be the restriction to V c V x ~ of the sheaf of immersions V x ~ -+ W. For a set A c V, a section in if>(A) is an immersion of (9p(A) c V x ~ to W, and the space Q(A, B) for B c A consists of the maps of the union (9p(B) U (V n (9p(A», (9p(B) c V x ~, to W, such that the restrictions of these maps to (9p(B) and to V n (9p(A) are immersions: (9p(B) -+ Wand V n (9p(A) -+ W respectively. Here the map ,,: if>(A) -+ Q amounts to the restriction of immersions from (9p(A) to the union of (9p(B) and V n (9p(A).

Since the immersion condition is open, we immediately conclude that " is a microfibration (compare 1.4.2). The same conclusion holds for the restriction map of (k + 1)-mersions V x ~ -+ W to k-mersions V -+ W, and in general, for all open relations whenever the homomorphism" makes sense.

Microextension Theorem. If if> is a flexible sheaf and IX: if> -+ tP is a microextension then the sheaf tP is also flexible. In other words, if tP admits a flexible microextension then tP itself is a flexible sheaf

The proof is given in the next section. This theorem provides the h-principle for immersions and k-mersions in the extra-dimensional case. Now try free maps V = vn -+ ~q. The only suitable extension comes from free maps V x ~ -+ ~q. These maps only exist for q > (n + 1)(n + 4)/2 = [n(n + 3)/2J + n + 2 and only for such


a q the corresponding homomorphism ;p -+ cP is surjective. Thus, we need n + 2 extra dimensions to get the h-principle for the free maps of closed manifolds into IRq. The method of removal of singulatities serves much better for free maps: it only requires one extra dimension.

Invariant Extensions. Let X -+ V be an arbitrary smooth fibration and let y -+ V' :::h V x IR be the fibration induced by the projection n: V x IR -+ V. Denote by II(r): y(r) -+ x(r) the map that assigns to an r-jet represented by a germ g: V' -+ y at v' = (v, t) E V', the jet ofthe restriction of g to V = V x t c V x IR = V'.

For a differential relation fJi c x(r) we call a relation fJi' c y(r) an extension of fJi if the map II(r) sends fJi' onto fJi.

Now, we invoke the group ~ = ~(V', n) ofthe fiber preserving diffeomorphisms for n: V' -+ V and we observe that this ~ naturally acts on Y and on y(r). An extension fJi' c y(r) of fJi c x(r) is called invariant if it is invariant under this action of~.

Examples. For the immersions relation fJi c X(1), X = V x W -+ V, there is a natural non-invariant extension whose solutions are the maps V x IR -+ W such that their restrictions to the manifolds V x t c V x IR are immersions for all t E IR. Notice that this extensions is an open condition in y(1) for Y = V' x W -+ V'. There is also an invariant but not open extension of the immersion relation fJi c X(1). The solutions of this last extension are the maps V x IR -+ W) which are immersions on all manifolds V x t c V x IR and which are constant on all1ines v x IR c V x IR, v E V. These maps are not microflexible.

The immersion relation fJi c X(l) has an extension fJi' c y(l) which is both open and invariant only for dim W> dim V, that is the immersion condition for V' -+ W. (If dim V = dim W this fJi' c y(r) is an empty set and so the projection II(l): fJi' -+ fJi is by no means onto.)

The Open Extension Theorem. If a relation fJi c x(r) admits an open invariant extension fJi' c y(r) then fJi abides by the h-principle and solutions of fJi are flexible.

Proof Denote by ;P' the sheaf of solutions of fJi'. Since fJi' is open the sheaf ;P' is microflexible and by the main flexibility theorem its restriction to V = V x 0 c V x IR = V'is a flexible sheaf ;p over V. There is a natural restriction homomorphism of;P to the sheaf cP of solutions of fJi. Since fJi' c y(r) is open, this homomorphism is a microextension; therefore cP is flexible and it satisfies the h-principle as well (see 2.2.1).

Let us give a canonical method for locating an invariant open extension of fJi c x(r). Take the pullback fJi* = (IIrrl(fJi) c y(r). This fJi* is an extension of fJi which is not invariant. Next, we take the maximal ~-invariant subset in fJi*, namely fJi' = n ~ ~(fJi*), ~here ~ runs over ~. The condition fJi' is an invariant extension of fJi, but it may be not open. Finally, we take the interior of fJi' and project this interior, Int(fJi' ), by II(r) to x(r). The image of this projection is an open set ~ c x(r) which

is contained in fJll and the condition Int(fJll') c y(r) serves as the maximal open invariant extension of ~. This immediately leads to the following conclusion.

A condition fJll c x(r) admits an open invariant extension iff f1 = fJll.

Examples. For the k-mersion condition, fJll c X(1), X = V x W, the condition ~ c X(l) is empty for k > dim W, but for k < dim W we have ~ = fJll. This gives Feit's k-mersion theorem in the extra dimensional case.

Take an arbitrary fibration X --+ V and a subbundle r in the tangent bundle T(X).

Transversality Theorem. If codim(r) > dim(V), then section V --+ X transversal to r satisfy the h-principle.

Proof A straightforward check up shows that the inequality co dim (r) > dim(V) implies f1 = fJll.

Here our condition fJll is not Diff(V)-invariant and even for an open manifold V we can not drop the dimension restriction. Notice also that for the trivial fibration V x W --+ V this transversality theorem reduces to Hirsch's theorem if r equals the kernel of the differential of the projection V x W --+ W

2.2.5 Local Compressibility and the Proof of the Microextension Theorem

We start with an abstract version of the "induction by skeletons" procedure. Consider a compact space A covered by compact subsets Ai' i = 1, ... , k, U7=1 Ai = A. For a subset I c {1, ... ,k} we denote by AI the intersection nieIAi and by VAl the union Ujo(AI n Aj ). Consider some unions of Ai' say B = UjEJAj for J c {1, ... , k} and B' = U j eJ' Aj for J' c J, and observe that the extension of any homotopy from B' to B reduces to extensions from VAl to AI for all I c J. Here, the flexibility (microflexibility) of a sheaf cP over A on the pairs (AI> VAl) implies flexibility (micro flexibility) on (B, B'), where the flexibility (microflexibility) of cP on a pair (C, C) means the restriction map cP( C) --+ cP( C) is a fibration (microfibration).

Lemma. If A admits an arbitrarily fine finite cover by compact subsets Ai such that cP is flexible (micro flexible) on all pairs (AI' VAl), then the sheaf cP is flexible (mirco-flexible).

Proof For every pair (C, C) of compact subsets in A and for all neighborhoods U :::::J C and U':::::J C in A there exists a pair (B, B') built from Ai' such that C c B c U and C c B' c U', provided the cover by Ai is sufficiently fine. Hence, the extension of deformations from (!)/tC to (!)/tC reduces to that from B' to B for sufficiently fine covers A = U Ai. Q.E.D.


Flexibility of Parametric Sheaves. Let rp be a sheaf over A and let Q be a finite polyhedron.

If rp is flexible (micro flexible) then the parametric sheaf rpQ over A x Q (see 2.2.1) is also flexible (micro flexible).

Proof. Cover A by small compact sets Ai' i = 1, ... , k, and subdivide Q into some small simplices Sj' j = 1, ... , 1. The above argument reduces our problem to the following

Sublemma. For every simplex SeQ and for every compact subset B c A the sheaf rps is flexible (micro flexible) on the pair (A x S, B) for B = (B x S) U (A x as).

Proof. Let cp: P ~ rpS(A x S) be an arbitrary map and let 1jI: P x [0, 1] ~ rpS(B) be a deformation such that IjIIP x ° = cpIB. By the definition of rps (see 2.2.1), the map cp is given by a map

cp': S x P ~ rp(A)

and IjI is given by two maps

1jI': (9jt(aS) x P x [0, 1] ~ rp(A), (9jt(aS) c s,

and

1jI": S x P x [0, 1] ~ rp(B).

The maps cp' and 1jI' agree on the intersection (S x P) n ((9jt(aS) x P x [0,1]) =

({9jt(aS)) x PeS x P x [0,1], P = P x ° c P x [0,1], and so they define a map (cp', 1jI'): P' ~ rp(A) for P' = (S x P) U ((9jt(aS) x P x [0,1]) c S x P x [0,1]. Now for the restriction map p: rp(A) ~ rp(B), we have

ljI"IP' = po(cp',IjI'),

and in order to extend the deformation IjI to A x S we must extend the lift (cp', 1jI') of 1jI" from P' to S x P x [0,1]. If we use for (9jt(aS) a small standard collar of the boundary as in S, we get a compact polyhedral pair (P', P), such that P' is a deformation retract of P. When rp is flexible, then p is a fibration and this lift extends to the entire S x P x [0,1]. If the sheaf rp is microflexible, then we only have an extension to (9jt(P') c S x P x [0,1] but this is all we need in this case.

Microextensions of Parametric Sheaves. Let rp and iP be sheaves over A, let a: iP ~ rp be a homomorphism and let Q be a finite polyhedron.

If a is a microextension then the corresponding homomorphism of the parametric sheaves, aQ: iPQ ~ rpQ, is also a microextension.

Proof. The microextension property in particular implies that each map a(a) = a({9jt(a)): iP(a) ~ rp(a), aEA, is a surjective micro fibration. It follows that the maps aQ(a, q): d)Q(a, q) ~ rpQ(a, q) are surjective for all points (a, q) E A x Q.

Now we must show that for an arbitrary pair of compact sets in A x Q, the corresponding map 1JQ is a microfibration (see 2.2.4). We use as earlier some

covers {AJ of A and {SJ of Q which allow us to assume that Q = S and that the pair in question is (A x S, B) for B = (B x S) U (A x as) and B c A. A map ijJ: P --+ «P(A x S) amounts to a map

ijJ': S x P --+ «P(A).

A coherent pair (1/1, ~): P x [0,1] --+ QS, for 1/1: P x [0,1] --+ <pS(A) and ~: P x [0,1] --+ «PS(B), which deforms the projection Yfs 0 ijJ: P --+ QS = Q(A x S, B) (that is YfsoijJ = (I/I,~)IP, P = P x ° c P x [0,1]), is given by the following three maps

1/1': S x P x [0,1] --+ <P(A),

~': (l)fz(aS) x P x [0,1] --+ «P(A), and

~": S x P x [0,1] --+ «P(B),

where the pair (t/J', ~") is coherent, that is po t/J' = IX 0 ~". Here p: <P(A) --+ <P(B) is the restriction map and IY. stands for IY.(B): «P(B) --+ <P(B). Furthermore, the maps ijJ' and~' agree on their common domain of definition, (l)fz(aS) x PeS x P x [0,1] thus providing a map

(ijJ', ~'): P' --+ «P(A),

for P' = (S x P) U ((I)fi(aS) x P x [0,1]) c S x P x [0,1]. The coherent pair (t/J', ~") defines a map (t/J', ~"): S x P x [0,1] --+ Q(A, B) and the pair (ijJ, ~') lifts the restriction (t/J', ~")I P' to «P(A) for the map Yf: «P(A) --+ Q(A, B). Since Yf is a microfibration, this lift extends to a lift (l)fi(P') --+ «P(A), for (l)fz(P') c S x P x [0,1]. Thus, for a small e > 0, we obtain a lift S x P x [0, e] --+ «P(A), which gives us the required lift P x [0, e] --+ «PS(A) of (t/J, ~).

Diagonal Products. Consider a sheaf <P over V and fix a compact polyhedron P. A double-deformation over a compact subset A c V is by definition a continuous map t/J: P x [O,e] x [O,e] --+ <P(A). By restricting t/J to P x [O,e] and to P x J, where J c [0, e] x [0, e] is the diagonal, we obtain two deformations P x [0, e] --+ <P(A), denoted by t/J0 = t/J(p, t, 0) and t/J* = t/J(p, t, t) correspondingly. Furthermore, t/J tautologically defines a deformation t/J': P' x [0, e] --+ <P(A) for P' = P x [0, e]. Namely t/J'(P', r) = t/J(p, t, r) for p' = (p, t). We say t/J is compressible if t/J' is a compressible deformation (see 2.2.3). Call t/J microcompressible if the restriction t/JIP x [0,<5] x [0, <5] is a compressible double-deformation for some positive <5 E [0, e].

Lemma. If t/J and t/J0 are compressible (microcompressible) then the deformation t/J* is also compressible (microcompressible).

Proof First (micro) compress t/J0 to ltI° and then t/J' to ltI', such that the support of ltI' is contained in a small neighbourhood U' :::J A for which ltI°1 U' = 1/1°1 U'. Then define the required (micro)compression ltI* of t/J* by

-* {ltI'(U,p,t,t) for UEU' t/J (U,p,t) = ltI0(u,p,t) for UEU\U',

where U = U(I/I*) is the actual domain of definition of 1/1* (compare 2.2.3). Q.E.D.


Call double-deformations t/Jo, ... , t/Jk composable if t/JiO = t/Ji~1 for i = 1, ... , k. We define for such t/Ji the (diagonal) product by setting o~~o t/Ji = t/J:. The above lemma yields the following

Compression of Products. If the deformation t/J8 and the double-deformations t/Jo, ... , t/Jk are compressible (microcompressible), then the product o}~o t/Ji is a compressible (microcompressible) deformation.

Nonlinear Partition of Vnity. Call a point (a,p)EA x P fixed under a doubledeformation t/J if t/J' (a, p, t) = t/J' (a, p, 0) for all t E [0, el Define supp t/J c A x P to be the closure of the non fixed points of t/J. Say a deformation qJ: P x [0, e] --+ tP(A) admits a partition of unity relative to a given cover of the space A x P by open subsets Vi c A x P, i = 0, ... , k, if qJ = o}~o t/Ji for some composable doubledeformations t/Ji such that supp t/Ji C Vi for i = 1, ... , k, and supp t/J8 c Vo.

Proposition. Let A x P be covered by open subsets Vi' i = 0, ... , k, and assume the restriction tP I A is a j1exible sheaf over A. Then every deformation qJ admits a partition of unity relative to the cover {V;}. Similarly, the microj1exibility of tPlA insures the partition of unity for the restriction qJ I P x [0,0], where 0 is some positive number :s; e.

Proof Let P consist of a single point, assume k = 1, denote S = supp qJ and prove the following

Lemma. If tP is j1exible, then there exists a double-deformation t/J: [0, e] x [0, e] --+ tP(A) such that t/J* = qJ and

(i) supp t/J 0 c Vo n (9ft(S) (ii) supp t/J' C VI n (()ft(S).

Similarly, if tP is microj1exible, the above t/J exists on [0,0] x [0,0] for some positive o :s; e.

Proof. The condition t/J* = qJ defines t/J on the diagonal 11 C [0, e] x [0, el Furthermore, (i) defines the restriction t/JIA\(Vo n (()ftS) on the segment [0, e] x ° C

[0, e] x [0, e] while (ii) defines the restriction t/JIA\(VI n (()ftS) on the entire square [0, e] x [0, e l Since the union 11 U ([0, e] x 0) is a homotopy retract in the square [0, e] x [0, e], this partially defined t/J extends to the required double-deformation [0, e] x [0, e] --+ tP(A) if tP is flexible. If tP is micro-flexible, then the extension is possible over [0,0] x [0,0] for some 0 > 0. Q.E.D.

Now, still assuming P = {p}, we construct by induction on i = 0, 1, ... , k, double-deformations t/Jk' t/Jk-I' ... , t/Jk-i, such that t/J: = qJ and

(i)

(ii)

k-i-I supp t/Jf-i = U ~:

j~O

supp t/J~-j C Vk- j for j = 0, ... , i.

This is done with the above lemma applied to the covering of A by Vk - i and A \ (9ft(A \ Vk -;) at every induction step.

Finally, if P contains more than one point, we apply the above to the parametric sheaf f/JP over A x P.

Local Compressions. A deformation t/J: P x [0,1:] --+ f/J(A) is called S-microcompressible if for an arbitrary neighborhood a c V = V(t/J) (see 2.2) of the support supp t/J c U there exists a smaller positive 6 E [0,1:],6 = 6(0), such that the restricted deformation t/J I P x [0,8] can be compressed to a deformation VI: P x [0,8] --+ f/J(A), such that, in addition to properties (i)-(iii) in 2.2 this compression VI satisfies

suppVl ca. Recall that the compression procedure involves a neighbourhood [J of A and the support supp VI must be contained in this 0. Now, supp t/J must be contained in the intersection On [J, and 6 may depend on 0, but not on [J (compare with the definition of microcompressibility in 2.2.3). It is clear that S-microcompressibility implies microcompressibility. On the other hand, if the sheaf f/J is flexible, one can construct S-compressions with 6 = I: for any a in the same way we did it for the usual compressions in 2.2.3.

There is one important difference between compressions and S-compressions. Namely, the S-microcompressibility of a deformation t/J: P x [0,1:] --+ f/J(A) only depends on the behavior of t/J near the intersection A n supp t/J (but does not on what happens in the interior of A\supp t/J). In other words, if a subset A' in A contains the intersection An (9ft(supp t/J), and if the restricted deformation t/J I A': P x [0, 1:] --+ f/J(A') is S-microcompressible, then the deformation t/J itself is also S-microcompressible.

Now, take a double-deformation t/J: P x [0,1:] X [0,1:] --+ f/J(A) and consider the corresponding deformation t/JP: It x [0,1:] --+ f/JP(A x P), where Ie is the first interval [0,1:]. We say that the double-deformation t/J is S-microcompressible if the restriction t/JPIIb x [0,15]: Ib X [0,15] --+ f/JP(A x P) is an S-microcompressible deformation for some positive 15 E (0, 1:] where I~ = [0,15]. Clearly, S-microcompressibility implies the usual microcompressibility for double-deformations. Again, the property of S-microflexibility of a double-deformation t/J only depends on the behaviour of t/J

r--.....J near supp t/J c A x P.

If f/J is a flexible sheaf, then all double-deformations are S-microcompressible since the flexibility of f/J implies the flexibility of the parametric sheaf f/JP and hence the above remark about deformations applies.

Partial Deformation. Take a map <Po: P --+ f/J(A) and a product Ao x Po c A x P, where Ao is a compact set in A and Po is a subpolyhedron in some subdivision of P. We introduce a partial deformation of <Po as a map t/J: Po x [0,1:] X [0,1:] --+ f/J(Ao) such that t/J(ao, Po, 0, 0) = <po(ao, Po) for all (ao,po)EAo x Po. We say that the map <Po is locally stable if for each point (ao, Po) E A x P there is a product Ao x Po (as above) which contains a neighbourhood of (ao, Po) in A x P, and such that every partial double-deformation Po x [0, 1:] x [0, 1:] --+ f/J(Ao) of <Po is S-microcompressible.


Local Criterion for Microcompressibility. Let t/J be a sheaf over a locally compact space V and let A be a compact subset in V. If the restricted sheaf cP I A is micro flexible and if all maps P -+ cP(A) are locally stable, then all deformations cP: P x [0,1] -+ t/J(A) are microcompressible.

Proof Since cP I A is a microflexible sheaf, the deformation IP I P x [0, e], for some e E [0, 1], can be decomposed into the product of double-deformations with arbitrarily small supports, IPIP x [O,e] = O~=Ot/li' t/li: P x [O,e] x [O,e] -+ cP(A). Now, if the support of the double-deformation !/Ii lies inside a "small product", that

is (!)ft(~ !/Ii) C Ai x Pi' for Ai c A and 1'; c P, then the restricted doubledeformation !/IilAi x Pi: 1'; x [0, e] x [0, e] -+ cP(Ai) is S-microcompressible. Therefore, each double-deformation !/Ii is microcompressible and the product of their microcompressions o~=o t/li' gives us the required microcompression iii of IP.

Proof of the Microextension Theorem. We shall prove this theorem for sheaves over a manifold V, since this is all we need for the further applications. The general case of an arbitrary locally compact space V is left as an exercise.

Now we have a microextension IX: iP -+ cP, where iP is a flexible sheaf and we have to show that cP is also flexible. We may assume, by induction on dim V that for each hypersurface H c V the restriction cP I H is already a flexible sheaf. Actually, we shall only use the microflexibility of cPIH. (In most examples the microflexibility of cP is not a problem anyway).

In order to prove flexibility of cP we only need to establish microcompressibility of deformations IP over a given compact set A in V (see 2.2.3). The compression of a deformation IP to iii with supp iii c (J is equivalent to the compression of the restriction of IP to a hypersurface H c (J, which separates A and the boundary of (J in V. (See the main lemma in 2.2.3). So we can work on H, where cP is microflexible. To save the notations we assume that the sheaf t/JIA is microflexible and continue to work on A.

Let us enumerate the relevant properties of the microextension IX: iP -+ cPo

(1) For every section lPoEcP(A) and for each point aEA there is a compact subset Ao in A such that Ao :::J An (!)fi(a), and such that the section lPolAo lifts (by surjectivity of IX) to a section iiio E iP(Ao). That is lXo(lPo) = lPolAo, where lXo is an abbreviation for IXIAo. (2) For every deformation IP: [0,1] -+ cP(A), IP(O) = lPo there exist a positive e E (0,1] and a partial lift ip: [0, e] -+ iP(Ao)' That is ip = ipo and lXo(ip(t» = lP(t)IAo for t E [0, e]. In fact, for every compact subset Bo in Ao the map ,.,: cP(Ao) -+ Q(Ao, Bo) is a microfibration. If Bo is empty, then Q(Ao, Bo) = cP(Ao) and the map '1 reduces to lXo = IXIAo. Therefore the map lXil: cP(Ao) -+ cP(Ao) is a microfibration and the initial deformations can be lifted. (3) This property refines (2) by giving us a lift ip which has almost the same support as IP. Namely, for an arbitrary neighbourhood 0 c V of Ao n supp IP, there is a positive 6 E (0, e] for which one can find a lift ip: [0,6] -+ cl>(Ao) such that supp ip c O. We construct this rp by first interpreting the restriction rpolAo \0 as the constant deformation ip0: [0,1] -+ iP(Ao \ 0) rpO(a, t) = rpo(a) for a E Ao \ 0 and for alIt E [0,1].

Then, for the pair of sets (Ao, Ao \ ao), we have the coherent pair of deformations, (cp I Ao, cpO). Finally, we lift the initial phase of this pair of deformations to (Ao) for the micro fibration 1]: (Ao) ~ Q(Ao, Ao Vo)· (4) The properties (1)-(3) hold if we pass to the parametric sheaves and to the corresponding homomorphism aQ: Q ~ r[JQ, as it was shown in the beginning of this section for all finite polihedra Q.

Now, we prove the microextension theorem by applying the previous local criterion. For a map CPo: P ~ r[J(A) and a point (a,p)EA x P, we find a product Ao x Po c A x P, such that Ao x Po contains (!)jt(a,p) and such that the restriction CPolAo x Po lifts to a map CPo: Po ~ (Ao). Next, for a partial double-deformation 1/1: Po x [0,1] x [0, 1] ~ r[J(Ao), we lift its restriction 1/I1P0 x [0,1:] x ° to (Ao), and then, with a given neighbourhood a in Ao x Po of supp 1/1 c Ao x Po, we find an extension of the last lift to a map Po x [0,1:] x [OJ] ~ r[J(Ao) for some positive l E (0, 1:], such that the restriction of this map to Po x [OJ] x [OJ], called

~: Po x [0, l] x [0, l] ~ r[J(Ao) is a double deformation with ~(~) c a. Since the sheaf is flexible, ~ can be S-microcompressed (even without making l smaller).

Finally, by projecting the S-microcompressed ~ back to r[J(Ao) by ao we obtain a S-microcompression of 1/1.

2.2.6 An Application: Inducing Euclidean Connections

Recall, that a connection in a COO-smooth vector bundle E ~ V is a rule that assignes to each COO-smooth vector field L on Va first order linear differential operator in the space Coo(E) of smooth sections V ~ E. This operator, called the covariant derivation in the direction L and denoted by VL : COO (E) ~ Coo(E), must satisfy the formal properties of a derivative. (See Kobayashi-Nomizu 1969.)

If u1 , ••• , Un are some local coordinates in V, then each connection in E is uniquely determined over this coordinate chart by the n operators of covariant derivations in the directions a/aui, i = 1, ... , n, denoted by Vi: C'J(X) ~ Coo(X).

Examples. Each trivial bundle V x [Rm ~ V carries the standard flat connection whose covariant derivatives Vi of a section V ~ V x [Rm are just the usual derivatives a/oui of the map f: V ~ [Rm, which corresponds to the section. For a field L = 2:7=1 ai(%uJ the derivatives VL in this case amounts to the Lie derivative LJ = 2:7=1 ai(of/ou;).

Take a bundle E with a connection V and a Coo-subbundle E' c E. Each linear COO-projection P: E ~ E', p 2 = Id, gives rise to a connection V' in E'. Namely, V' = po V, that is V~(X) = po JdX) for all fields L on V and for all sections X E Coo(E') c Coo(E).

Exercise. Let F be the trivial bundle V x [Rq ~ V with the standard flat connection, denoted by VF and let E c F be a sub bundle of dimension m. Suppose that q ~ m(n + 1) + n, n = dim V, and that E c F is a generic Coo-subbundle. Show that for an arbitrary connection V in E there exists a projection P: F ~ E such that poVF = V.


Induced Connections. Let E ~ V be a bundle with a connection V and let 9 = V' ~ V be a Coo-map of a manifold V' into V. Denote by E' ~ V' the induced bundle g*(E) over V' and let G: E' ~ E denote the corresponding fiberwise isomorphic map between the bundles E' and E. The bundle E carries a connection V', called the induced connection g*(I7), which is uniquely characterized by the following property.

If two sections X: V ~ E and X': V' ~ E are related by the equality Go X' = X 0 g, and if two fields L: V ~ T(V) and L': V' ~ T(V') are related at a point v' E V' by Dg(L'(v'» = L(g(v'», then G«(I7'LX')(V'» = (l7LX)(g(V'».

For example, if 9 is constant then the induced bundle E' is trivial and V' = g*(I7) is the (standard) flat connection in E'.

Euclidean Connection. We now assume that the bundle E ~ V is given a Euclidean structure that is a field of Euclidean metrics in the fibers Ev c E for v running over V. A connection V in such a bundle is called Euclidean if the Lie derivative of the scalar product abides by the Leibniz rule:

L(X, Y) = (VLX, Y) + (X, VL Y),

for all tangent fields on V and for all section X and Y in Coo (E).

Examples. The standard flat connection in a trivial bundle is Euclidean. Let E' c E be a subbundle with the induced Euclidean structure and let

P: E ~ E' be the orthogonal projection. Then for a Euclidean connection V in E the connection po V in E' is also Euclidean.

Let us apply this construction to the canonical m-dimensional bundle over the Grassmann manifold ofm-planes in ~q. This bundle, H ~ Gr = Grm(~q), is realized as a subbundle of the trivial bundle F = Gr x ~q ~ Gr. The standard flat connection VF in F and the orthogonal projection P: F ~ H yield the connection po VF

in H, called the canonical connection. Let W be a Riemannian Coo-manifold. The tangent bundle T(W) ~ W has a

distinguished Euclidean connection, called the Riemannian connection VW. If W is isometrically Coo-immersed into ~q, then the tangential Gauss map g: W ~ Grm(~q), m = dim W, induces VW from the canonical connection in the bundle H ~ Grm(~q).

Now suppose the Riemannian manifold W has dimension m + n. Let an ndimensional submanifold in Wand let E ~ V be the normal bundle of V in W. By restricting the connection VW to T(W)I V and by applying the orthogonal projection T(W)I V ~ E we obtain the so-called normal connection in E.

Exercises. (a) Show that for an arbitrary connection V in an abstract bundle E ~ V, there exists a Riemannian metric in the total space E, such that the normal connection of the zero section V c. E equals V.

(b) Recall the existence of an isometric Coo-immersion of every (m + n)dimensional manifold into ~qO, for qo = (m + n + 2)(m + n + 3)/2 (see 3.1.7). Show that an arbitrary Euclidean connection in an m-dimensional bundle over an n-dimensional manifold V can be induced from the canonical connection in H ~ Grm(~qo).

The problem of inducing Euclidean connections was first considered by Narasimhan and Ramanan (1961), who also studied non-Euclidean connections. Our aim is to obtain a connection inducing map V -+ Grm(lRq) for a relatively small q.

Our major tool is the h-principle for a class of such maps. In fact, we study the following more general problem. Let E and F be two bundles over V with arbitrary Euclidean connections, V in F and V' in E. We seek connection homomorphisms X: E -+ F. Namely X must be an isometric isomorphism of E onto a subbundle E', such that V' = po V for the orthogonal projection P: F = E' = X(E) ~ E.

We work for a while over a fixed system of local coordinates u, ... , Un in V and we also fix, over this local chart, an orthonormal frame of sections ek : V -+ E, k = 1, ... , m, where m is the dimension of the bundle E. With these fixed data, we can express the connection V' in terms of Christoffel's coefficients F;kl = <ek , Viel ),

1 ~ i ~ n, 1 ~ k, 1 ~ m. Since the vectors ek are orthonormal, that is < ek , el ) = bkl ,

we have -::,0 < ek , el ) = 0 for all i, k and l, and so F;kl = - F;lk. UUi

An isometric homomorphism X: E -+ F (over our local chart) now amounts to a system of orthonormal sections X k : V -+ F, k = 1, ... , m,

k, 1 = 1, ... , m,

and the condition V' = po V is expressed by the following system of partial differential equations in unknowns Xl, ... , X m,

If the bundle F has dimension q, then each section X k is given by q real functions and the conditions (*) and (**) represent [m(m + 1)/2] + [nm(m - 1)/2] equations in mq unknown functions on V We must solve these equations with arbitrary given nm(m - 1)/2 functions F;kl = - F;lk on V The system of these equations is underdetermined for q > [(m + 1)/2] + [n(m - 1)/2], and so one expects it to be solvable for q ~ mn/2. But our method provides solvability only for q ~ mn. [See D'Ambra (1985) for the case q ~ mn/2.] This restriction is due to the fact that we can only handle some special homomorphisms X: E -+ F, called regular, that are distinguished by the condition of linear independence of the section X k and ViXk ,

i = 1, ... , n, k = 1, ... , m, in each fiber Fv c F, v E V Observe, that the span of the vectors X k and ViXk only depends on the homomorphism X and not on the choice of the frame {ed or of the coordinate system {u;}. So, the definition of regularity is correct and it is also clear that the connection V' in E plays no role in this definition. It is equally clear, that regular maps may exist only for q 2 m(n + 1).

Example. Let E be the trivial line bundle. Then isometric homomorphisms E -+ F are unitary fields V -+ F that are sections of the unit sphere bundle associated to F. If F is the trivial flat bundle, F = V x IRq -+ V, then regular unitary fields V -+ F correspond to immersions V -+ Sq-l.

(A) Theorem. The regular connection Coo-homomorphisms X: E -+ F, satisfy the h

principle for q 2 m(n + 2), where n = dim V and m and q are the dimensions of the bundles E and F respectively.

Corollary. If q > qo = max(m(n + 2), m(n + 1) + n), then there always exists a regular connection Coo-homomorphism E -+ F. In particular, every connection in E can be induced by a map g: V -+ Grm(lRqo) from the canonical connection.

Proof of the Corollary. It will become clear in the proof of the main lemma below that the differential condition~, that governs our homomorphisms X: E -+ F, fibers over V with fibers ~v, v E V, which are homotopy equivalent to the Stiefel manifold Sts(lRq) for s = m(n + 1). Therefore, we have a section V -+ ~ for q > m(n + 1) + n. Then the theorem applies for q :?: m(n + 2).

Proof of the theorem is based on the following

Main Lemma. The sheaf of germs of regular connection COO-homomorphisms X: E -+ F, is micro flexible.

Proof Since microflexibility is a local property (see 2.2.3) we only need to establish microflexibility for regular solutions ofthe systems (*) and (**). We proceed inductively for k = 1, ... , m, and thus reduce the solution of the systems (*) and (**) to the solution of a sequence of m open differential conditions, ~(k) imposed on X k ,

k = 1, ... , m, where the vectors Xl' ... , Xz, ... , X k - l are fixed. So we work with the following system of conditions imposed on Xl> ... , X k :

(a) the frame (Xl' ... , X k ) is regular, that is the vectors XI and ViXI are linearly independent for I = 1, ... , k, and i = 1, ... , n.

(b) 1 (XII.,Xp) = fJII.P'

(XII.ViXp) = F;II.P,

ex, p = 1, ... , k, i = 1, ... , n.

We must show, that for any given solution (Xf, ... , X~-d of the systems ~k-l the conditions imposed by the system ~k on X k can be reduced to an open condition. The only differential equations in ~k which involve X k are

ex = 1, ... , k - 1, i = 1, ... , n.

These equations can be written equivalently as

(ViX2,Xk) = _F;lI.k,

and so all equations in (b) imposed on X k become algebraic. Furthermore, all the equations in (b) are linear in Xk, with the only exception of

(Xk,Xk) = fJkk = 1.

Since the vectors XII. and ViXII., at < k, are linearly independent, the linear equations, namely

(x2,xk ) = fJak = 0 and } (Vi X2,Xk ) = ~rll.'k, at = 1, ... , k - 1, i = 1, ... , n, ~o(k)

define an affine subbundle A in F of codimension (n + 1)(k - 1), such that sections V -+ A correspond to solutions X k of these equations. The non-linear equation

(Xk' X k) = 1 prescribes a sphere Sv in each fiber Av c A, v E V. Observe that the set Sv c Av may be, apriory, empty, or it may consist of one single point. But we could assume from the beginning the existence of at least one solution Xf of ~o(k), such that the frame (X?, ... , Xf-l' Xf) is regular and, in particular, xf is not contained in the span of X~ and ViX~ for rt. < k; otherwise, there is no solution to speak about. This independence of the vectors X~, ViX~ and some vectors xf implies that our sets Sv c Av are actual spheres and that they form a sphere bundle S --+ V, such that sections V --+ S are exactly solutions X k satisfying the system (b) in ~k with fixed Xa = X~, rt. < k. In order to satisfy the entire system (a) we must add the regularity condition for the frames (X?, ... , Xf-l, Xd, but this is an open differential condition in X k • In fact, there is a field of n-planes in the total space S, which are transversal to all fibers Sv c S, and such that the sections V --+ X transversal to this field are the solutions of ~k relative to X k • Observe that the I-jets ofthese transversal sections V --+ S at each point v E V form a manifold of the homotopy type of the Stiefel manifold Stn+1 (W), where p denotes the dimension of fibers Av c A, v E V. This shows that our original condition ,q{ --+ V is fibered in the way we asserted in the proof of the previous corollary,

Notice finally, that by reducing ~ to a sequence of open relations, we have not only obtained microflexibility, but also the rest of the nice "open properties". In particular, the h-principle for ~ is equivalent to the sheaf theoretic h-principle and we also have the following

Microextension Lemma. Let Vo be a submanifold in V and let Eo and Fo denote the restrictions EI VO and FI VO respectively with induced connections V~ and Vo. Let (/>

denote the sheaf of connection homomorphisms E --+ F and let (/>0 be the corresponding sheaf of germs of connection homomorphisms Fo --+ Eo. If q ~ m(n + 1), that is if ~ is not the empty set, then the restriction homomorphism (/> 1 Vo --+ (/>0 is a microextension.

Now, we introduce the bundles E and F over V x IR which are induced, (together with their connections,) from E and F respecitively by the projection n: V x IR --+ V. The regular connection homomorphisms E --+ F are naturally acted upon by Diff(V x IR, n) and the theory of sheaves (see 2.2.3) yields the h-principle for connection homomorphisms E --+ F without any restriction on the dimension. Finally, the above lemma brings this h-principle back to V for q ~ m(n + 2).

Exercises. (a) Prove for q ~ k(n + 2) that the natural map of the space of solutions of the system ~k (over a fixed coordinate chart) to the space of solutions of the system ~k-l is a Serre fibration [Hint. Use the group Diff(V x IR, n)].

(b) Describe non-open conditions ~ which can be reduced to open conditions. Define an appropriate maximal invariant extension (as in 2.2.4 for the open case) and give a criterion for validity of the h-principle generalizing the theorem on inducing connections. Consider, in particular, the connection inducing problem for pseudo-Euclidean connections.

(B) Semiregular Homomorphisms. Let m = 2 and let (e 1 ,e2 ) be an orthonormal frame in E which is parallel at a given point v E V, that is Tik,I(V) = 0, i = 1, ... , n, k, 1= 1,2. A connection inducing isometric homomorphism X: E --+ F is called semi-


regular at v ifthe n vectors (SVi X1 + CVi X2 )(V)EFv are linearly independent for all pairs of real numbers sand c such that S2 + c2 = 1.

Exercise. Show that no homomorphism is semiregular if q = n + 2 for n even. However, if n is odd, then semiregular homomorphisms do (locally) exist for all q ~ n + 2.

Our analysis of regular homomorphisms equally applies to semiregular ones. In particular, we have the following

Corollary. Let V be a parallelizable manifold and let the bundles E and F be trivial. If q ~ n + 4, or if q is even and ~ n + 3, then there exists a connection inducing isometric Coo-homomorphism E -+ F which is semiregular at all points VE V.

Exercise. Define semiregular homomorphisms for m ~ 3 and prove the pertinent h-principle.

(B') Remark. We shall need later (see 3.1.7) the following version of the above corollary.

Let V split into the product, V = Vo x ~, where Vo is a parallelizable manifold, let. the bundles E and F be trivial and let e: V -+ E be a unitary Coo-section. If q ~ n + 3, then there exists a connection inducing isometric Coo-homomorphism X: E -+ F such that

(i) the section X(e): V -+ F is regular; (ii) the restriction of X to the bundle Et = EI V; over V; = V x t c V x ~ is an

everywhere semiregular homomorphism of Et to Ft = F 1 V; for all t E ~.

Proof. The microflexibility of the pertinent homomorphisms X is established as earlier, which yields the existence of X via the h-principle.

2.2.7 Non-flexible Sheaves

The lack of the (micro) flexibility does not necessarily impairs the h-principle. For example, Can-maps V -+ Ware not microflexible but the h-principle does not suffer due to the following fact which summarizes basic results by Whitney (1934), H. Cartan (1957) and Grauert (1958).

Let X -+ V be a Can-submersion and let fo: V -+ X be a COO-section whose jet JIo: V -+ x(r), for a given r = 0, 1, ... , is real analytic on some analytic (possibly singular) subvariety Vo c V. Then fo admits a fine Coo-approximation by Can-sections f: V -+ X, such that JII VO = JIol Yo.

This can be refined for vector bundles X -+ V with the cohomology of the sheaf C/J of Can-sections V -+ X. Namely, Hi(C/J) = 0 for i ~ 1 (Cartan 1957).

For general (non-linear!) continuous sheaves there is no (?) cohomology groups to measure the degree of non-microflexibility.

However, the deviations from the h-principle is reasonably controled for the non-flexible sheaves in the following examples.

(A) Degenerate Maps. A continuous map f: A --+ C is called k-contractible if there exists a k-dimensional polyhedron B and continuous maps g: A --+ Band h: B --+ C, such that hog is homotopic to f

Exercises. Let f: V --+ W be a C1-map, such that rank Df(v) ::;; k for all VE V. (a) Show the map f to be k-contractible. (b) Assume rank Df(v) == k and let V be a closed manifold. Construct a

k-dimensional manifold B and two maps of constant rank = k, say g: V --+ B and h: B --+ W such thatf = hog.

(b') Assume the manifolds V and Was well as the map f to be real analytic and construct the above B, g and h for open manifolds V. Find counterexamples in the Coo-case.

(A') The h-Principle for Maps of Constant Rank (Gromov 1973; Phillips 1974). Let V be an open manifold and let cP: T(V) --+ T(W) be a homomorphism of constant rank k. Then the following condition (*) is necessary and sufficient for the existence of a COO-map f: V --+ W of constant rank k whose differential is homotopic to cP by a homotopy of homomorphisms of rank k.

(*) There exists a k-dimensional bundle over a k-dimensional polyhedron, say K --+ B, and vector bundle homomorphisms CP1: T(V) --+ K and CP2: K --+ T(W), of constant rank k, such that CP2 a CP1: T(V) --+ T(W) is homotopic (via homomorphisms of constant rank) to cp.

Proof First, let V be obtained from the ball Bn, n = dim V, by attaching I-handles H' = B' X IRn-1 for 1 ::;; I ;s; k < n/2. If a map f of rank k is given on a small neighborhood (!}jzSl-l c H' of the sphere S'-1 = aB x 0 c H' and if the ball B' = B' x 0 c H' is transversal to the subbundle Ker Df c T(V) near s'-1, then one easily extends f to {!}jzB' c H' by applying the h-principle to immersions B' --+ W, for k < dim W. (If k = dim W the h-principle in question amounts to Phillips' submersion theorem.) Furthermore, if B' is not transversal to Ker Df , then there exists a CO-small diffeotopy of the identity map {!}jzS'-1;::> which achieves the transversality. This follows from the h-principle for immersions S'-1 --+ (!}jzS'-1 transversal to Ker Df and from the inequality 2k < n. Thus, we obtain the h-principle for maps f: V --+ W of rank k for our special manifold V = Bn UiHfi with an obvious induction in i = 1, 2, ....

Now, for any V, we construct with (*) an n'-dimensional manifold V' =

Bn'UiHfi,n' = 2k + 3,ahomomorphismcp~: T(V')--+ T(W) of rank k and a homomorphism CPt: T(V) --+ T(V') such that the composition cP~ a CPt is a homomorphism T(V) --+ T(W) of rank k homotopic to cp. Then we have, as earlier, a Coo-map f': V' --+ W of rank k. Furthermore, the h-principle for maps V --+ V' transversal to Ker Df , yields such a map, say 1": V --+ V', for which the composition f' a I" is the required map V --+ W of rank k.

(A") Exercises. (a) Fill in the detail for the above proof. (b) (Chen 1971). Take two loops Y1 and Y2 at some point in a connected manifold

V and let W 1 and W 2 be closed C1-smooth I-forms on V such that w1 /\ w2 == 0 and such that the period matrix (L, wj ), i,j = 1,2, has rank 2. Assume dimH1(V;~) = 2 and show Abel's period map

V --. T2 = H1(V; ~)/H1(V; Z)

to have rank :::; 1. Then construct a homomorphism of 7t1 (V) into the free group F2 = Z * Z, such that the images of[y;] E 7t1 (V), i = 1,2, are freely independent in F2.

(b') Let dimH1(V;~) ~ 3 and apply (b) to the covering V --. V for which the subgroup 7t1(V) c 7t1(V) is generated by [Y1] and [Y2]. Thus show the homotopy classes [y;] E 7t1 (V) to be freely independent in 7t1 (V) (with no assumptions on dimH1).

(b") Assume dimH1(V;~) = d < 00 and study homomorphisms 7t1(V) --. F2 by means of Abel's map V --. Td.

(c) Degenerate 2-Forms. Take a submanifold Vo c V, let H2(VO;~) ~ H2(v,~) = 0 and consider a closed 2-form Wo on Vo for which Wo /\ Wo == O. Show that Wo does not extend to a closed form W on V for which W /\ W == 0 on V, unless Ie ao /\ Wo = 0 for every 3-cycle c in Vo, homologous to zero in V and for every I-form ao on Vo, such that dao = Woo Apply this to Vo = S3 C B4 = V and to the pullback Wo ofthe area form on S2 under the Hopfmap S3 --. S2.

(d) Locally Flat Immersions V --. ~q. Call an immersion f: V --. ~q locally k-flat if a small neighborhood of each point v E V is sent by f into an affine k-dimensional subspace in ~q. C1-Approximate an arbitrary C1-immersion f: V --. ~q by k-flat Coo-immersions for k :::; ko = ko(dim V), where ko(1) = 2 and ko(2) :::; 7. Hint. Start with a piecewise linear approximation of f for a suitable triangulation of V.

(d') Show the normal Pontryagin classes Pi of a k-flat immersion to vanish for i > 2(k - n), n = dim V, and thus obtain a lower bound for ko(n).

(B) Integrable Subbundles of T(V). Let K c T(V) be a Coo-smooth subbundle of codimension k and let a: T(V) --. N = T(V)/K be the quotient homomorphism. There (obviously) exists a unique vector bundle homomorphism A: K ® K --. N such that A(a1 ® 132) = a([a1,a2]) for every pair of sections 131 and 132: V --. K, where [ , ] stands for the Poisson bracket of vector fields on V. The subbundle K is called integrable if A == O.

Theorem (Bott 1968). If K is integrable, then the Pontryagin classes Pi E H4i(v, ~) of N can be represented by closed Coo-smooth 4i-forms Wi on V such that every form W on V which is a polynomial in Wi with constant coefficients identically vanishes on V for degw > 2k.

Proof Take a small neighborhood U c V, fix a frame of independent sections Xj: V --. K, j = 1, ... , n - k, and let Y1' ... , fie be transversal to K tangent fields on U such that the sections VI = a(Yj): U --. N, I = 1, ... , k are linearly independent. Then there (obviously) exists a unique (affine) connection V in the bundle NI U, for


which I, m = 1, ... , k

j = 1, ... , n - k, m = 1, ... , k.

Furthermore, the identity A == 0 shows the derivative V x v is independent of the choices of Xi and Yi for all fields X: U ~ K and all sections v: U ~ N. Moreover, the curvature.Qx.y: N ~ N,defined by .Qx,y(v) = VxVyv - VyVxv - V[X,yjvvanishes on the pairs of vectors X and Y in K c T(V), as a straightforward computation shows. By patching these connections on open subsets with a partition of unity on V we arrive at a (Bott) connection i7 on V whose curvature ti also vanishes on K c T(V). Now, there is a unique (Chern-Weill polynomial of degree 2i in ti (viewed as a matrix valued 2-form on V) which gives us a closed (scalar valued) 4i-form OJi on V, such that [OJ;] = Pi' Since ti vanishes on K and codim K = k, every polynomial of degree > k in .Q is zero on V. Q.E.D.

Exercises. (a) Show, by producing examples and using Bott's theorem, that the integrability relation violates the h-principle for 2 ::::; k ::::; n - 2. (One does not know if there are obstructions to this h-principle besides Bott's theorem.)

(b) (Shulman 1972). Show the Massey products (as well as ordinary cupproducts) of Pi vanish beyond the dimension 2k for all integrable subbundles K c T( V) of codimension k.

(b') Let CM: V ~ GrM!(IRM'), M' ~ M! + dim V denote the classifying map for the bundle (M!)N = fV EB NEB'" EB l'f. Assume V is compact and K is integrable.

Al! Show the map CM to be 2k-contractible for all sufficiently large integers M.

Hint. Replace the Grassmann manifold by the space G' of operators P: IRM' ~ IRM', such that p 2 = P and rankP = M!. Take a classifying map C': V ~ G' (for a large enough M') which induces a Bott connection i7 on V from the (obvious) canonical connection in the natural M!-dimensional bundle over G'. Study C' by means of Sullivan's minimal model.

(c) Secondary Classes (Godbillon-Vey 1971). Observe that every affine connection in the trivial line bundle N ~ V is given by a I-form, say p on the associated principle bundle which is fiberwise diffeomorphic to V x IR ~ V. Assume N = T(V)/K for an integrable subbundle K c T(V) of codimension one, let p correspond to a Bott connection and take a section s: V ~ V x IR. Show that the 3-form s*(P A dP) on V is closed and the cohomology class h = [s*(P A dP)] E H 3 (V; IR) depends only on K.

(c') Let r be a cocompact discrete subgroup in SL2 1R and let T be the subgroup of upper triangular matrices. Let Ko c T(Vo), for Vo = SL2 1R/ r, consist of the vectors tangent to the T-orbits in Yo. Show Ko to be integrable with the GodbillonVey class h i= O. Imbed Yo into IR" for n ~ 6 and show that there is no integrable subbundle K c T(IR") for which K n T(Vo) = Ko. (Thus integrable co dimension one subbundles violate the h-principle for extensions.)

(B') Foliations. Let (jJ denote the sheaf of Coo-submersions V ~ IRk and let 'I' be the sheaf of subbundles K c T(V) of codimension k. The correspondence


fl--+ Ker Df C T(V), f E t/>, defines a homomorphism of sheaves t/> ~ 'P whose image, say ~ c 'P, consists, by Frobenius theorem, of (exactly and only) integrable subbundles. Thus, for every integrable subbundle K c T(V), there exist submersions Ji: Ui ~ IRk for some open cover U i Ui = V, such that Ker Dfl = KI Ui for all i = 1, .... This defines a unique partition, called a foliation of V into connected (n - k)-dimensional submanifolds (which may be non-closed subsets in V) called leaves !l' c V, which are everywhere tangent to K, and such that each intersection !l' n Ui is a countable union of connection components of Ji-pullbacks of some points in IRk.

Example. Let the structure group of a k-dimensional bundle N ~ V with the fibers Nv ~ IRk, VE V, reduces to a discrete subgroup in DifflRk. Then. there exists a Galois covering V ~ V, such that the lift N ~ V is a trivial fibration over V. Moreover, there is a splitting N = V x IRk, where each submanifold Vx = V x x c V x IRk, x E IRk is the graph of a section V ~ N, such that the subbundle K = U xe iJ;l' TCp':) c T(N) is invariant under the action ofthe Galois group ron N. Thus we obtain an integrable subbundle on N = Nlr, say K c T(N), which is transversal to the fibers Nv c N, v E V, and whose leaves are covered by the manifolds v".

Corollary (Phillips 1969). If the structure group of the quotient bundle N of a subbundle K c T(V) reduces to a discrete subgroup, then K is homotopic to an integrable subbundle K' c T(V), provided V is an open manifold. [In fact, this is also true, according to Thurston (1974, 1976), for closed manifolds v.]

Proof. The h-principle for maps V ~ N transversal to K gives us a transversal map f: V ~ N, for which K' = Djl(K) c T(V) is homotopic to K.Q.E.D.

Exercise. Apply this corollary to subbundles K c T(V) of codimension one.

See Lawson (1974), Fuks (1981), and Reinhart (1983) for further information and references on foliations.

(C) Complex Structures. An almost complex structure on V is an automorphism J: T(V)p such that J2(-r) = --r for all -rE T,,(v) and VE V. This is equivalent to a reduction of the structure group GLnlR of the bundle T(V) to GLmC, where n = 2m = dim V. An almost complex structure is called complex (or integrable) if it is locally isomorphic to the standard (integrable!) almost complex structure on cm. This can be expressed with the sheaf 'P of immersions V ~ cm and with the obvious homomorphism h of 'P to the sheaf t/> of almost complex structures on V by defining the sheaf of complex structures on V as the subsheaf t/>' = 1m h c t/>.

Every almost complex structure J on V defines a (unique up to a homotopy) classifying map C] of V to the complex Grassmann manifold GrmCN, for a given N> 2n, such that the complex vector bundle (T(V),J) [where (a + J=ib)-r d~ a + bJ(-r)] is induced by C] from the canonical bundle over GrmCN •


Proposition (Gromov 1973; Landweber 1974). If the manifold V is open and if the map CJ is (m + 1)-contractible, then the almost complex structure J is homotopic to a complex one. [Compare Adachi (1979).]

Proof Proceed as in the proof of (A') with the total reality condition [see (C) in 2.4.5] for embeddings Sli-1 -4 (!)jzS1i- 1 c Ht instead of the transversality. The details are left to the reader.

Corollary. If V is an open manifold of dimension n = 2m ::; 6, then the sheaf if> of complex structures on V satisfies the h-principle.

Indeed, any map V -4 CrmlRN is (m + 1)-contractible in this case.

(D) Classifying Space. Consider a topological space P and a continous sheaf if> over an arbitrary manifold V. Denote by if>~ the subsheaf of the sheaf if>P over V x P (see 1.5.5) for which if>~(U x R) consists of the locally constant maps R -4 if>(U) for all open subsets U x ReV x P. Next, for a continuous map f: P -4 V we define the pull-back sheaf f*(if» over P as the restriction of if>~ to the graph rJ = P c V x P. The contravariant functor!F from topological spaces to sets, which assigns to each P the set of pairs (f, cp), where f: P -4 V is a continuous map and cp Ef*(if>)(P), satisfies:

(*) The restriction of!F to the category of open subsets in P is a sheaf over P. (**) The sheaf axioms are also satisfied for finite coverings of P by closed subsets Pi c P, i = 1, ... , k. Namely, if some elements tfJi E !F(P;) agree (in the obvious sense) on the intersections Pi n~, i,j = 1, ... , k, then there is a unique tfJ E !F(P) such that

tfJ I Pi = tfJi'

If X is an arbitrary topological space then the functor

PH {continuous maps P -4 X}

obviously satisfies (*) and (**). With this in mind, we define a "space" as an arbitrary contravariant functor from topological spaces to sets which abides (*) and (**).

Example. The functor which assigns to each P all n-dimensional vector bundles over P is a "space".

Warning. All bundles over P do not form a set with the usual meaning of "all". Here and below, we restrict our "aIls" to objects to a fixed sufficiently large set theoretic universe. The reader may entertain himself by putting the logic straight.

One obviously extends to "spaces" the usual topological notions, like continuous maps, Serre fibrations, homotopy groups etc. Furthermore, one can assign (in many ways) to each "space" !F an ordinary space, say [!F] and a weak homotopy equivalence [!F] -4 !F. For example, let ,100 be the infinite dimensional simplex with countably many vertices. Then one constructs a cell complex [!F], such that the set


of k-cells in [ff] is the union U"ff(CT) where CT runs over all k-faces in Aoo , and where a (k - I)-cell 11.' Eff(CT') is attached to II.Eff(CT) as a face iff CT' is a face of CT and 11.1 CT' = 11.'. This [ff] comes with a natural continuous "map" [ff] ~ ff which clearly is a weak homotopy equivalence. (For the above vector bundle functor this is the ordinary construction of the classifying space).

Exercises. (a) Define sheaves over V with values in "spaces" and extend the results in 2.2.1-2.2.4 to these sheaves.

(b) Let tP be a micro flexible sheave over an arbitrary manifold V and let qJ E tP*(V). Construct a continuous map f: V ~ V, which lies in a given CO-fine neighborhood of the identity map, and a section t/I Ef*(tP)(V) which is (in an obvious sense) "homotopic" to qJ. Derive the h-principle for Diff(V)-invariant microflexible sheaves over open manifolds V from the CO-dense h-principle for maps @jt Vo ~ V x V whose projections on the second factor are immersions @jt Vo ~ V, where @jtVo c V is an (arbitrarily) small neighborhood of a given subpolyhedron Vo c V of positive codimension.

(D') Fix a Diff-invariant sheaf tP over ~n and consider a fibration T ~ P with the fiber ~n and the structure group Diff~n. Since this group operates on tP(~n), we can difine the associated fibration, say Tcp ~ P with the fiber tP(~n).

Definitions. (a) A tP-bundle is a fibration T ~ P with a given section called a tP-structure, P ~ Tcp, that is a family qJpE tP(I;, ~ ~n) continuous in p. (b) Aflat connection in T is a subsheaf F in the sheaf of sections P ~ T such that (i) for each point t E Tp C T, PEP, for each pair of small neighborhoods ScI;, of

t and U c P of p, and for each point s E S there exists a unique section f = f. E F(U) for which f(p) = s;

(ii) the map gq: S ~ 1'q defined by s f--+ f.(q) for all q E U, is a diffeomorphism of S onto an open subset in the fiber 1'q which COO-continuously depends on q E U.

(c) A flat tP-bundle is a pair consisting of a tP-bundle given by sections qJp E tP(Tp ~ ~n), PEP, and of a flat connection in T, such that the diffeomorphism gq: S ~ 1'q sends (for the Diff-action in tP) qJplS to qJq 1 gq(S) for all tE Tp, and pE P and all q E U. [Compare Segal (1978).]

(c') Examples. The trivial bundle T = P x ~n ~ P carries the canonical flat connection given by the sheaf of (the graphs of) locally constant maps P ~ ~n. Each section qJ E tP(~n) extends to a unique tP-structure on this T which is flat for this connection and which is called the constant tP-structure.

Let us project the product V x V to the diagonal A c V x V by (VI' V2) f--+

(VI' vd. Then there is an arbitrarily small neighborhood T c V x Vof A for which the restricted projection T ~ A is fiberwise diffeomorphic to the tangent bundle T(V) ~ V = A. A natural connection in this T is given by the sheaf of those sections s: A ~ T c V x V whose projections on the second V-factor are locally constant.

Let If' be a Diff-invariant sheaf on V which is locally isomorphic to tP. Then each section t/I E IJ'*(V) defines (by the definition of t/I*, see 1.5.5) a tP-structure on


small neighborhoods T ~ T(V) of L1 = V. This structure is flat for the above connection if and only if'" is contained in IJT(V) c IJ'*(V).

(d) Consider the "space" which relates to each P the set of all 4>-bundles over P and let [4>]* be the corresponding classifying space. Similarly define the (HaefligerMilnor) classifying space [4>]0 for the "space" of flat 4>-bundles. Furthermore, consider a natural (forgetful) map between these spaces, say H: [4>]0 --+ [4>]*.

Exercises. (1) Show the map H to be a weak homotopy equivalence for micro flexible sheaves 4> on IRn.

(2) Let IJT be a Diff-invariant sheaf on a smooth manifold V which is locally isomorphic to 4>. Observe a correspondence between 4>-structures on the tangent bundle T(V) --+ V and sections in IJ'*(V). Let V be an open manifold. Show that the existence of a flat 4>-structure in T(V) implies the existence of a section in IJT(V). Moreover, a section", E IJ'*(V) :::l IJT(V) can be homotoped to IJT(V) if and only if the classifying map Clp: V --+ [4>]* can be "lifted" to a continuous map Co: V --+ [4>]0, such that HoC is homotopic to Clp'

Remark. If [$'] is a classifying space for a "space" $' then each '" E $'(P), for all cell complexes P, defines a (unique up to a homotopy) classifying map Clp: P --+ [$'] which is consistent with the weak homotopy equivalence [$'] --+ $'.

(3) Reformulate the above (A'), (B) and (C) in terms of the respective maps H: [4>]0 --+ [4>]*.

(4) Take a subset of tangent n-planes of a manifold W, say A c Grn W, and let 4> be the sheaf of immersions f: IRn --+ W, such that D,CT,,(lRn» E A, for all v E IRn (compare 2.4.4). Observe the canonical map [4>]* --+ A and prove it to be a weak homotopy equivalence for open subsets A.

(5) Take a closed k-form g on a manifold Wand let 4> be the sheaf of g-isotropic immersions f: IRn --+ W that is f*(g) == O. Show the homotopy fiber of the map H: [~]O --+ [~]* to be the Eilenberg-MacLane space K(IR,k - 1). Prove the map H turned into a fibration to be induced from the canonical fibration over K(IR, k) [with the fiber K(IR,k - 1)] by the composition of the obvious map [4>]0 --+ W with the (classifying) map W --+ K(IR, k) which represents the cohomology class [g] E Hk(W; IR).

(6) Define the classifying sheaf for a given sheaf 4> on V with no (DifT- V)-action. Reduce this sheaf to a single space if 4> is acted upon by a transitive (pseudo) group of diffeomorphisms on V. [Compare Brown (1962).]

(D") The Weak h-Principle. Fix a Diff-invariant sheaf 4> over IRn and consider a locally isomorphic (to 4» sheaf IJT (which is also DifT-invariant) over an ndimensional manifold V. Say that IJT satisfies the weak h-principle on V if for an arbitrary flat 4>-structure CPo on T(V) --+ V there exists a section", 1 E IJT(V) whose associated flat 4>-structure CP1 on T(V) [see the above (c')] is "homotopic" to CPo. This means the existence of a flat 4>-bundle (T, cp) over V x [0,1] such that (T, cp)1 V x 0 = (T(V), CPo) and (T, cp) V x 1 = (T(v)cpd·

Exercises. Prove the weak h-principle for an arbitrary Diff-invariant sheaf IfF on an open manifold V.

Show the weak h-principle to be equivalent to the ordinary h-principle for all micro flexible Diff-invariant sheaves IfF on all manifolds V.

Relate flat cP-bundles over V to pairs (f, t/J) where f: V -+ V is a continuous map and t/J E f*( IfF) (V). Then generalize the above to arbitrary (non-Diff-invariant) sheaves IfF over V.

(E) Foliations on Closed Manifolds. Thurston (1974, 1976) has proved the following

Theorem. The sheaf of Ci-foliations, i = 1,2, ... , 00, on an arbitrary (possibly closed!) manifold V satisfies the weak h-principle. Moreover, 2-dimensional foliations satisfy the ordinary h-principle.

Thurston's proof is based on a far-reaching generalization of Reeb's construction [see (c) in (E')]. Another approach (related to the surgery of singularities) is due to Misharchev and Eliashberg (1977).

Exercise. Derive the following corollaries from Thurston's theorem (a) If a subbundle K c T(V) has a trivial normal bundle N = T(V)jK, then K

is homotopic to an integrable Coo-subbundle in T(V). (b) Every co dimension one subbundle in T(V) is homotopic to an integrable

Coo -subbundle. (c) Every 2-dimensional subbundle is homotopic to a Coo-integrable one. (d) The sphere S7 admits a k-dimensional Coo-foliation of codimension k for all

k = 1, ... ,6. (No simple construction is known for these foliations.)

(E') can-Foliations. The weak h-principle fails to be true for can-foliations of codimension one on closed manifolds due to the following

Theorem (Haefliger 1962). No closed simply connected manifold V admits a real analytic foliation of codimension one.

Haefliger proves this by applying the classical Poincare-Bendixson theorem to the intersection of a given C<X)-foliation on V with an appropriately immersed disc D2 -+ V. Thus he finds an immersed cylinder Sl x [0, IJ -+ V transversal to the foliation, such that the closed leaves of the induced foliation on the cylinder are exactly the circles Sl x t for t E [0, Ij2J.

Examples of Foliations. (a) Let the fundamental group r of a closed manifold Vo act by C-diffeomorphisms on a closed manifold F. Then the group r diagonally acts on Vo x F, where Vo is the universal covering of V, and the closed manifold V = Vo x Fjr carries a natural no-dimensional Ci-foliation for no = dim Yo' For instance, the unit tangent bundle of an no-dimensional manifold Vo of constant negative curvature carries an no-dimensional Can-foliation transversal to the fibers of the bundle.

Remark. It is very hard, in general, to find a F-action on a given manifold F.

(b) Let F be a discrete subgroup in a Lie group G. Then there is an (obvious) one-to-one correspondence between G-invariant foliations on G/F and connected subgroups in G.

(c) The action xH2 i x, iEll., XE~n\{o}, preserves the foliation of Rn\{o} into parallel k-planes. Thus one obtains a can-foliation of dimension k, for any k = 1, ... , n - 1, on the Hopf manifold (~n\ {O})lll. ~ sn-l X Sl. Furthermore, one obtains a Can-foliation of codimension one on (~"t. \O)lll. ~ Dn- 1 x Sl, such that the boundary o(Dn- 1 x Sl) ~ sn-2 X Sl is a leaf of this foliation. In particular, one gets such a foliation on the solid torus D2 x Sl. By gluing two solid tori over the boundary one obtains Reeb's foliation on S3 which is COO-smooth (but not can!) and whose only closed leaf is the (Clifford) torus T2 c S3.

(F) Compact Complex Manifolds. Complex structures on closed 4-dimensional manifolds violate the weak h-principle as seen in the following

ExampJe(Yau 1976). The(parallelizable)manifold(T3 # p3~) x Sl admits no complex structure.

Proof Every compact complex surface V whose first Betti number is even, dimHl(V; [R) = 2m, admits (see Kodaira 1964) m linearly independent closed holomorphic I-forms. These forms define the (Abel) holomorphic (!) period map A: V -+ T Zm = e m/!!! for the period lattice!!! ~ Hl(V;£:) ~ ll.Zm in em, such that A*: Hl(T2m; ll.) -+ Hl(V; ll.) is an isomorphism. The above manifold V ~ (T3 # p3[R) X Sl admits a basis hi E Hl(V; £:) ~ £:4, i = 1, ... ,4, such that the cup product hl U hz U h3 U h4 is a generator in H4(V; £:) ~ ll.. Hence, any map A: V -+ T4 isomorphic on Hl must have the topological degree ± 1. The desired contradiction now follows from the (obvious) fact that every proper holomorphic map of degree one between equidimensional complex manifolds induces an isomorphism of the respective fundamental groups.

(G) <1>-Cycles. Let <1> be a Diff-invariant sheaf over [Rn, let [<1>Jo be the classifying space for the flat <1>-bundles and denote by ff -+ [<1>Jo the canonical fibration (whose flat <1>-structure we forget for the moment) regarded here as an n-dimensional vector bundle over [<1>Jo. (The vector bundle structure is unique up to an isomorphism.) Then we consider sheaves 'Plocally isomorphic to <1> over closed oriented manifolds V, dim = n. Recall that every section !/J E 'P(V) defines a flat <1>-bundle over V and, hence, a classifying map C'I': V -+ [<1>Jo. The maps which arise this way are called <1>-cycles in [<1>Jo. Each <1>-cycle C in [<1>Jo defines an integral homology class [CJ E Hn([ <1>JO) that is the image of the fundamental class of the underlying manifold V. The following condition (*) is obviously necessary for the representation of a class hEHn([<1>JO) by a <1>-cycle,

(*) There exists a closed oriented manifold V and a continuous map B: V -+ [<1>J o, such that the induced bundle B*(ff) over V is isomorphic to T(V) -+ V.

Furthermore, if the sheaves 'P over all closed manifolds V satisfy the weak h-principle, then (*) is clearly sufficient for a representation of h by a <P-cycle.

Remark. It would be far more interesting to prove (rather than to use) the h-principle by representing the homology of [<P]O by <P-cycles. But little is known in this direction [compare (H) and (H')].

Let us relax (*) by allowing a stable isomorphism between the bundles B*(ff) and T(V), that is an isomorphism B*(ff) EEl I ~ T(V) EEl I, for the trivial bundle l=VxlR-+V.

Lemma. If n ~ 3 then the stable condition (*) implies (*).

Indeed, for an arbitrary n-dimensional bundle T -+ V which is stably isomorphic to T(V) there exists a stably parallelizable manifold (n ~ 3) Yo, such that the tangent bundle of the connected sum V # Vo is isomorphic to the induced bundle p*(T) over V # Vo for the obvious (pinching) map p: V # Vo -+ V. Q.E.D.

Let us reduce the stable condition (*) to a pure homotopy condition. To do this we fix a countable (n + l)-dimnesional subcomplex K c [<P]O, such that the group Hn(K) injects into Hn([<P]O) and such that a given class hEH"([<P]O) comes from some class hi E Hn(K). Then there (obviously) exist a (unique up to a diffeomorphism) N-dimensional manifold X for a given N ~ 2n + 2 and a homotopy equivalence y: X -+ K, such that the Whitney sum T(X) EEl y*(ffl) is a trivial bundle over X for the bundle ff' = fflK over K c [<P]o.

Now, the class hE Hn([<P]O) can be represented by a map B: V -+ [<P]O satisfying the stable condition (*) if and only if the class y;l(h')E Hn(X) can be represented by a submanifold in X with the trivial normal bundle. Such submanifolds form a (cobordism) group which is isomorphic (via Pontryagin-Thorn construction) to the (cohomotopy) group of homotopy classes of maps X -+ sN-n for the one-point compactification X of X.

Corollary. If n ~ 3 and if the bundles 'P locally isomorphic to n and take an open subset of oriented tangent n-planes in W; say A c Grn(W). Assume smooth immersions f: V -+ W directed by A [which means Dj(T,,(V))EA, VE V, compare 2.4.4] satisfy the hprinciple for all n-dimensional manifolds V. Take a class hE Hn(V) and let hi be a class in Hn(A) which goes to h under the projection A -+ V. Then in the following two cases the class (m!)h, for all sufficiently large m, can be represented by an immersion directed by A of some closed oriented manifolds V into W;

(1) n ~ 3, (2) there exists a continuous map IX: sn -+ Aw = An Grn(Tw(W)) for all WE W; such

that the induced n-dimensional bundle over sn is isomorphic to T(sn).

Indeed, the case (1) is covered by the above corollary and (2) allows one to derive the non-stable condition (*) from the stable one. [See Eliashberg (1984) for a comprehensive study of cobordisms of differential relations.]

Exercises. Develop cobordism theories for (a) (Wells 1966) immersions of positive codimensions; (b) Lagrange and Legendre (see 3.4.2, 3.4.3) immersions into symplectic (respec

tively contact) manifolds. [See Arnold (1980) for an elementary treatment of low dimensional examples and Audin (1984) for a general approach.]

(c) Free isotropic (see 3.3.4) immersions into pseudo-Riemannian manifolds; (d) Immersions transversal to a given subbundle in T(W) of co dimension

~n + 1. Show, for instance, that a non-zero integer multiple of each class in Hn(W)

can be represented by an immersion of parallelizable manifold into W for 3'::;n<dimW

Remark. The tangent bundle T(G) of the Grassmann manifold G = Grn(W) -+ W contains a canonical subbundle, say K c T( G) of codimension q - n for q = dim W, such that the natural lift of every submanifold V" c W to G is everywhere tangent to K. Next, for a submanifold A c G, we put K' c T(A) n K and we view [following Thorn (1959)] maps V -+ A tangent to K' as generalized maps V -+ W directed by A. For example, if q = n + 1 and if A is an open subset in G, then these are just Legendre maps V -+ A for the contact structure KIA. Legendre maps unlike the maps V -+ W directed by A, always satisfy the h-principle (see 3.4.3) which makes the cobordism theory of the generalized A-directed cycles quite easy (see the above (b)). In fact, Thorn (1959), developed a homology theory for such cycles for all q - n without use of the h-principle.

(H) Curvature Relations. Consider Riemannian metrics g on a manifold V whose sectional curvatures do not vanish that is either K(g) > ° or K(g) < ° everywhere on V. If V is a closed connected surface then the parametric h-principle refined by the Gauss-Bonnet theorem holds true:

If the Euler characteristic X(V) is positive, then the metrics g on V with K(g) > ° constitute a non-empty contractible space. If x(V) < 0, then the metrics with K(g) < ° form such a space.

This is an easy application of the uniformization theory (compare 3.2.4). However, there is no simple refinement of the h-principle for extensions.

Example. Let V be a compact surface bounded by a simple closed curve S ~ av. Then, by the Gauss-Bonnet formula, the geodesic curvature /((s), s E S, satisfies fs /((s) ds < 2nx(V) if K(g) > ° and fs /((s) ds > 2nX(V) for K(g) < 0. Furthermore, let S+ c S be a (connected!) segment oflength (T +, such that fs+ /((s) ds ~ /(+ > 2n, and let every open (possibly disconnected) subset S_ c S oftotallength .::; (T + satisfy fs /((s) ds ~ L. Then,for K(g) > 0, the numbers /(+ and L satisfy /(+ + /(- < 2n. If K(g) < 0, then /(+ + L < 2n(1 - X(V)) + fs /((s) ds. This is proved by deforming

S+ to the shortest curve y in V with oy = OS+ and by applying the above inequalities to the complement V\y. [Compare Gromoll-Klingenberg-Meyer (1968).]

Thus, the extension of metrics with non-vanishing curvature from 0 V to V meets an obstruction which is not are accounted for by the h-principle nor by the GaussBonnet theorem. [Compare Gromoll-Meyer (1969).]

If dim V ~ 3, then the relations K(g) > 0 and K(g) < 0 completely deviate from the h-principle. For instance, closed manifolds (V,g) with K(g) > 0, have finite fundamental group, while the inequality K(g) < 0 makes 11:1 (V) irifinite. This totally desagrees with the parametric h-principle which predicts the space of metrics 9 satisfying K(g) > 0 (or, as well K(g) < 0) to be a (non-empty!) contractible space. In fact, this space may be disconnected. See Hitchin (1974) for an example of two metrics of constant positive curvature on S8 which can not be joined by any homotopy ofmetrics 9 with K(g) > O. In fact, Hitchin proves that no homotopy of metrics with non-negative scalar curvature exists between these metrics. (No such example is known for K(g) < 0.)

The classifying space for the sheaf lP + of metrics on IRn with K > 0 is clearly weakly homotopy equivalent to the Grassmann manifold GrnlRoo, n = dim V, and the same is true for the relations K ~ 0, K < 0 and K ~ O. It is unknown whether a non-zero integer multiple of every class hEHn(GrnIROO) can be represented by a lP + -cycle. In plain words we ask for a (possibly disconnected) oriented manifold (V, g) with K(g) > 0 and with given sufficiently divisible characteristic numbers. If V is a closed connected manifold with K ~ 0, then the Euler characteristic and the signature are bounded by Ixl + lui ~ constn, (Gromov 1981), which strongly restricts connected cycles. However, disjoint unions of spheres and of products of complex projective spaces (which carry obvious metrics with K ~ 0) may have arbitrarily large (vectors of) characteristic numbers. Similarly, compact hyperbolic manifolds and products of complex hyperbolic manifolds (which have K ~ 0) generate a subgroup of finite index in Hn(GrnIR OO ). On the other hand, the only known restriction on characteristic numbers of a closed manifold (V, g) with K(g) ~ 0 is X(V) ~ 0 for dim V = 4, with the strict inequality X(V) > 0 for K>O.

(H') The Scalar Curvature. The failure of the h-principle for metrics 9 with the scalar curvature S(g) > 0 less drastic than for K > O. For example, these metrics are amenable to a surgery (Shoen-Yau 1979; Gromov-Lawson 1979) which allows one to produce many closed manifolds with S > 0 starting from standard examples (like the above lP + -cycles). In particular, every simply connected manifold V whose second Stiefel class W2 does not vanish admits a metric 9 with S(g) > 0, provided dim V ~ 5 (Gromov-Lawson 1979).

On the other hand if W2 = 0, and if V is homeomorphic to a product Vi x V2

where the manifold Vi admits a metric gl with K(gd ~ 0 (like Vi ~ Tm) and V2 has non-zero A-genus (e.g. dim V2 = 0 or dim V2 = 4 and u(V) f= 0) then Vadmits no metric 9 with S(g) > o. [See Gromov-Lawson (1983) and Shoen (1984) for further information and references.]

Exercises. (a) Let f7t c x(r) --+ X --+ V be an open differential relation, let Vo c V be an arbitrary submanifold and let fo: V --+ X be a C-solution of f7t [i.e. Jio(V) c f7t]. Let F denote the space of C-solutions f: V --+ X of f7t, such that Ji- 1

1 Vo = Ji~ll Vo, and let Fo be the space of jets cp: Vo --+ f7t of such solutions near Vo. That is cp E Fo if and only if there exists a solution 1': (!)jz Vo --+ X of f7t such that Ji;-ll Vo = Ji~ll Vo and for which Ji' I Vo = cp. Prove the following

Weak Flexibility Lemma. The map ff--»JiVo is a Serre fibration F --+ Fo.

Hint. Use the induction in dim Vand codim Vo, starting with dim V = 1, dim Vo = 0.

(b) Apply (a) to the differential relations K(g) > 0, K(g) < 0, S(g) > 0, and to a closed geodesic Vo c (V, go)· Thus deform a given Riemannian metric go which satisfies one of the above inequalities to a metric g whose sectional curvature is constant near Vo, while satisfying the same curvature inequality as go everywhere on V.

(c) Construct a metric of positive scalar curvature on the connected sum of n-dimensional manifolds, n ~ 3, with constant sectional curvature> 0. Then, using (b), make this work for all manifolds with (now non-constant) positive scalar curvature.

(d) (Yau). Construct metrics of negative Ricci curvature on the connected sums of manifolds of negative Ricci curvature.

Scalar Curvature S < 0. Riemannian metrics g with S(g) < ° on n-dimensional manifolds are likely to satisfy the parametric h-principle for n ~ 3. In fact, the non-parametric h-principle is established in the following stronger form by KazdanWarner (1975).

Let cp be a COO-function on a connected manifold V, dim V ~ 3, which is somewhere negative at some point in V. Then there is a COO-metric g on V, such that S(g) == cp.

A similar existence theorem (with an easy proof left to the reader) holds for the equation S(g) dg == w, where dg is the Riemannian volume form and W is a given n-form on V which is negative at some point. In fact, one expects the solvability of the system dg = wo, S(g) = cp for the above cp. Furthermore, it does not seem to be hard to solve the equation P(g) = W for a top-dimensional Pontryagin-Chern-Weil fOfm P(g) and for a given n-form w, for which [w] = [P(g)] E Hn(v) and which is assumed to change sign [compare (H")]. Moreover, one hopes to solve systems of equations Pi(g) = Wi' i = 1, ... , k, for k small compared to n. A similar but easier problem consists in finding connections in a given bundle with given Pontryagin forms.

(HI!) Exercises. (a) (Gromov-Eliashberg 1973). Consider n-dimensional manifolds V and W with given n-forms Wo on Wand w on V, where Wo nowhere vanishes on W, and prove the following h-principle for maps f: V --+ W, such that f*(wo) = w.

Let fo: V --+ W be a continuous map which lifts to a homomorphism of the tangent bundles T(V) --+ T(W) of rank ~ n - 1 on every fibre T.,(v), v E V, and let


fo*[wo] = [w] for the cohomology classes [wo] E Hn(w; IR) and [w] E Hn(v; IR). Then there exists, in the following two cases, a COO -map f: V -+ W of rank :? n - 1, such that f*(wo) = w. (1) The manifold V is open. (2) V is connected and the form w changes the sign somewhere on V (i.e. neither w :? °

nor w ::::;; ° relative to a fixed volume form).

Show, in particular for an arbitrary COO-function cp on a connected stably parallelizable Riemannian manifold (V, g), such that Svcpdg = 0, there exists a Coo_ map f: V -+ IRn, n = dim V, whose Jacobian satisfies Jj == cp.

(a') Let W be the Grassmann manifold GrmlRN and let Wo be a SO(N)-invariant closed non-zero form on W. Homotope a given immersion fo: V -+ W to a map f: V -+ W, such that f*(wo) = w, provided fo*[wo] = [w] and one of the above (1) and (2) is satisfied. Then construct a connection in an m-dimensional bundle over V whose given Pontryagin form P equals w, provided [P] = [w]. Replace (1) and (2) by a suitable condition on Wo by using the techniques in 3.4.2.

Remark. The action of Diff V on (scalar curvature) functions has a non-empty (!) open orbit which is equally true for the action on n-forms. This suggests a unified approach to equations like S(g) = cp and f*(wo) = w.

(G) Folded Maps and Sections. We have seen in (H) how the h-principle may fail for microflexible sheaves 'I' over a closed manifold V. However, one can save the h-principle by slightly modifying the sheaf '1'.

The Singular h-Principle. Let '1" be a microj7exible sheaf over V' which is acted upon by the (pseudo) group DiffV and let f: V -+ V', for dim V = dim V' = n:? 2, be a COO -map whose Jacobian changes sign on every connected component of V (i.e. there is a pair of points, say v+ and v_ in each component of V, such that the local degree of fat v+ equals + 1 and degvJ = -1, where one defines these degrees on the oriented double coverings of V and V' in case the manifolds are non-orientable). Then the pull-back sheaf 'I' = f*('1") over V satisfies the h-principle. [Compare 6.2.5 in Gromov (1972) and 3.1 in Gromov (1971).]

Proof First, the formalism in (D), reduces the problem to the sheaf 'l'h of continuous maps V' -+ Q c IRn x P whose projections to IRn are COO-immersions, where P is a polyhedron and Q is an open subset in IRn x P. If the only singularity of f is a folding along a hypersurface Vo c V, then the proof is concluded with the equidimensional folding theorem of Eliashberg (see 2.1.3). In general, assuming V is compact, one takes a small open neighborhood V' c V' of the set of the critical values of f, such that the (topological) boundary av' is a Coo-hypersurface in V'. Then one considers the submanifold VI = V\f-I(V) c V whose boundary aVI = f- 1(aV') is immersed by f onto av' c V'. If there are at most two points in the pull-back f- 1(u) c aVI for all u E au', then, by identifying these points in pairs, one obtains another manifold out of VI' say V2 , which is continuously mapped by f into V' with a topological folding (like x 1-+ Ixl) over those points u E V where f-I(u) contains


exactly two points. In fact, the folding locus is non-empty due to the sign change condition imposed on f This folding is good enough to apply Eliashberg's theorem to V2, and then to extend maps V2 -+ Q to all of V. If f-l(U) contains many points, then they can be locally organized into pairs to give local folded maps to which the extension version of Eliashberg's theorem applies. Finally, if V is an open manifold, then a modification (in fact, a simplification) of the above argument establishes the h-principle for 'I' = f*( '1") with no assumptions on the map f what-so-ever. Filling n the detail in this "proof" is left to the reader.

Remarks. (a) The most interesting (and the easiest) case of the singular h-principle concerns maps f: V -+ V' whose only singularity is a folding along some Vo c V, as the sheaf f*('1") for such an f is the nearest to '1".

(b) Saving the h-principle with a folding is also possible for some nonmicroflexible and non-DifT-invariant sheaves. An instance of that is the case (2) of the equidimensional equation f*(wo) = w [see (H")], where the sign change condition on w insures "foldings" of maps f: V -+ W

Exercises. (a) State and prove the weak h-principle for the sheaf 'I' = f*('1") without assuming '1" microflexible [compare (0")].

(b) Let 'l'be a Diff-invariant microflexible sheaf over an n-dimensional manifold V, n ~ 2, with a non empty boundary, and let a finite group r act freely on an open tubular neighborhood Vo of the boundary. Show that sections t/J in 'I'(V) which are r-equivariant on Vo (i.e. y(t/Jo) = t/Jo for t/Jo = t/JiVo, YEF) satisfy the h-principle. Generalize this to infinite groups r whose action on Vo is discrete. Prove the weak version of this h-principle for non-microjlexible sheaves '1'.

Locally Split Metrics. A Riemannian Coo-manifold V = (V, g) is called locally split, if a small neighborhood V c V of each point v E V isometrically splits, (V, gl V) =

(Vo x (0, s), go EB dt 2 ).

(c) Show that the Euler and Pontryagin numbers of every locally split manifold vanish.

(c') Construct a locally split Coo-metric on every manifold V which admits a free Sl-action. (Simply connected manifolds carry no real analytic locally split metrics.)

(d) Consider the sheaf if> of locally split Coo-metrics on [Rn and prove every n-dimensional bundle over an (n - 1)-dimensional polyhedron to carry a flat if>structure. Thus show every open n-dimensional manifold to admit a locally split COO-metric. [Compare Pasternak (1975).J

(d') Let V be closed oriented manifold and let C: V -+ G = Grn=l[RN be the clasifying map for the stabilized tangent bundle, T(V) EB 1-+ V for I = V x [R -t V, where G is the Grassmann manifold of oriented (n + I)-planes in [RN for N > 2n = 2 dim V. Let 'I' be the sheaf oflocally split COO-metrics on V and assume the map Cis (n - I)-contractible. Construct a Coo-map f: V -+ V homotopic to the identity whose only singularity is a folding along some closed (possibly disconnected) Coo-hypersurface Vo c V, such that the induced sheaf f*('I') admits a section t/J* E f*( '1').


(d") Let the Pontryagin numbers of a connected oriented manifold V' vanish. Show the classifying map of the manifold V = 2 V' = V' # V' into G = Grn+1 ~N to be (n - I)-contractible. Construct with the above ifJ* a COO-metric g on V which admits an orientation reversing isometric involution I: (U,g)+=J for an open subset U c V with an (n - I)-dimensional complement. Prove with (C") in 2.1.3 the vanishing of the signature u(V) = 2U(V/) = 0, and conclude to the following

Thom-Hirzebruch Signature Theorem. There exists a rational cohomology class Ln c Hn(G; 0) (which is a certain polynomial in the Pontryagin classes PiE H4i(G; Z) with rational coefficients) such that every closed orientable manifold V, dim V = n = 4m, has u(V) = LiV) for Ln(V) d:,[ C*(Ln) where C is the classifying map V -+ G. [The polynomial Ln can be explicitly determined by substituting in the equality u(V) = Ln(V) the products of complex projective spaces for v.]

2.3 Inversion of Differential Operators

In the previous section we have obtained the h-principle for some P.D.E. systems which are locally solvable by a purely algebraic procedure. Now, we turn to more general non-linear systems that become algebraically solvable only after they have been linearized. In order to come back from (solutions of) the linearized sytems to (solutions of) the non-linear systems themselves one needs an appropriate infinite dimensional implicit function theorem. Such a theorem was discovered by Nash (1956) in the course of his solution of the isometric imbedding problem. In the following sections we develop Nash's theory in the context of differential operators and related sheaves of solutions of non-linear P.D.E. systems.

2.3.1 Linearization and the Linear Inversion

Let X denote, as usual, a COO-fibration over an n-dimensional manifold V and let G -+ V be a COO-smooth vector bundle. We denote by f1£rt and f§rt respectively the spaces of Crt-sections ofthe fibrations X and G for alIa = 0, 1, ... , 00. Let~: f1£' -+ f§o be a differential operator of order r. The expression "differential of order r" means that for each C'-section x: V -+ X the value ofthe section ~(x): V -+ G at any given point VE V depends only on the r-jet J~(V)EX('). In other words, the operator ~ is given by a map L\: X(') -+ G, namely ~(x) = A 0 J~. We say that ~ is a Crt-operator, a = 0, 1, ... , oo,ifthemapA is Crt-smooth. We assume belowthat~isa Coo-operator and so we have continuous maps ~: f1£rt+, -+ f§rt for alIa = 0, 1, ... , 00. The word "continuous" equally applies here to the usual and to the fine topologies in our spaces of sections.

Example. Let G denote the symmetric square of the cotangent bundle of v, let W be a manifold with a quadratic differential form h, and let X = W x V -+ V. We get

2.3 Inversion of Differential Operators 115

an operator ~ by relating to C1-maps x: V -+ W the induced quadratic forms, ~(x) = x*(h). This operator ~ = ~h has first order and, for a Coo-form h, this is a Coo-operator. In the case when (w, h) = (IRq, L:1 dxf) we write (compare 1.1.5).

~(x) = g = {gij} = {(;;/ ;~)}, i,j = 1, ... , n.

Linearization. Denote by Yvert(X) c T(X) the space of those vectors in T(X) which are tangent to the fibers of the fibration X -+ V, and for a section x: V -+ X we denote by y" -+ v the induced vector bundle, x*(Yvert(X)). If the section x is CCl-smooth, then y" -+ X is a CCl-smooth bundle and for each P :::;; a. we denote by qy! the vector space of CP-sections V -+ Y". The space qy: can be also defined as the infinite dimensional tangent space T,,(~Cl). If X -+ V is a vector bundle, then each bundle y" is canonically isomorphic to X and each space qy! is canonically isomorphic to ~p.

Now, we assume that the fibers of the fibration X -+ V have no boundaries and we observe that for each pair (x, y) of C' -sections x: V -+ X and y: V -+ y" there exists a cr -smooth family of sections Xt: V -+ X, t E [0, 1], such that Xo = x and iJxt/iJt = y for t = O. We define the linearization of ~ at x, called Lx: qy; -+ f§0, by the formula

iJ LAy) = L(x,y) = iJt~(xt)lt = O.

This definition does not depend on a specific choice of the family X t that represents y E qy; and one can interpret

Lx: Tx(~r) = qy; -+ T(f§°) = f§0

as the differential of our operator ~: ~r -+ f§0 at x E ~r. It is also clear, that Lx is a linear differential operator of order r in y. Furthermore, if x is of class CCl+r, then Lx is a CCl-operator. Moreover, L(x, y) is a differential operator of order r in both variables x and y. This "global" operator L acts on the space ,1r of Cr-sections V -+ Yvert(X), and since ~ is a Coo-operator, L is also a Coo-operator, L: ,1Cl+r -+ f§Cl. If we interpret ,1Cl+r as the tangent bundle, ,1Cl+r = T(~Cl+r), then L = L(~) amounts to the (global) differential of ~.

If X -+ V is a vector bundle we have Lx: ~r = qy; -+ f§0 and L: ~r x ~r -+ f§0, namely

L(x,y) = limt-l[~(x + ty) - ~(x)]. t .... O

Exercise. Extend the definition of L to the case when both, X and G over V, are general (non-vector) fibration.

Irifinitesimal Inversions of~. We say that the operator ~ is infinitesimally invertible over a subset d in the space of sections x: V -+ X if there exists a family of linear differential operators of a certain order s, namely Mx: f§s -+ qy~, for xEd, such that the following three properties are satisfied.

(1) There is an integer d ~ r, called the defect of the irifinitesimal inversion M, such that d is contained in ~d, and furthermore, d = dd consists (exactly and only)


of Cd-solutions of an open differential relation A c X(d). In particular, the sets dlt+d = .91 n !:rlt+dare open in !:rlt+d in the respective fine CIt+d-topologies for all 01: = 0,1, ... ,00.

(2) The operator Mx(g) = M(x,g) is a (non-linear) differential operator in x of order d. Moreover, the "global" operator

M: dd x ~S _ ,10 = T(!:r0 )

is a differential operator, that is given by a Coo-map A EB G(s) - 4ert(X). (3) Lx 0 Mx = Id, that is

L(x,M(x,g)) = g for all xEdd+r and gE~r+s.

Examples. If f!} is a zero order operator and if the corresponding map J: X - G is a submersion on an open set A c X, then, obviously, f!} is infinitesimally invertible over the corresponding .91 = d° with order and defect zero.

Our next example is more exciting.

Theorem (Nash 1956). Let f!} be the operator relating to C1-maps x: V - IRq the

induced forms f!}(x) = g = {g;J = {(;~;' ;~)}. Then, over the space of free maps

V - IRq, this operator f!lJ admits an irifinitesimal inversion M of defect d = 2 and of order s = o.

Proof We clearly have

{ lox Oy) lox Oy)} L(x,y) = \ou;' OUj + \ou/ou; , i,j= 1, ... ,n=dimV.

(Accidentally, this L is linear in x as well as in y but this is irrelevant to the subject). In order to construct M we must resolve relative to y the following P.D.E. system

(L) ( ox Oy) (ox Oy) -,- + -,- = gu' oU; oUj oUj oU;

Following Nash we add to (L) the equations

(N) (;~;,y) = 0, i = 1, ... , n.

Then we differentiate (N) and get

(N/) I ox Oy) + I ~ y) = 0 \ou;' OUj \ou;ou/ .

Finally we alternate i and j and conclude that under condition (N) the system (L) is equivalent to the following system,

(N*)

In particular, every solution y of the algebraic (in y) joint system (N) + (N*) also satisfies (L).

. ax ax As x IS a free map, the vectors -;-(v), and -:1--(V)E IRq are linearly independent

UU i uuiouj

for all v E V and so the system (N) + (N*) is solvable at every point v E V In order to get a solution y over the whole manifold V we must specify a canonical procedure for picking up one solution y(v) at each point v. [This problem does not appear for q = n(n + 3)/2 when the solution y(v) is unique, but only for q > n(n + 3)/2.] We make our choice by taking at each point v E V the solution y that minimizes the norm Ilyll = <y,y)1/2, and with this minimal y = y(x, g) we put M(x, g) = y(x,g).

Generalization (Green 1970). For an arbitrary pseudo-Riemannian manifold (w, h) the form inducing operator on free maps 8t\(x) = x*(h), admits an iriflnitesimal inversion of orders s = 0 and of defect d = 2 over free maps x: V -+ W

Indeed, the calculation above goes through with the covariant derivatives V; and Vb in place of Ojou i and 02/0UiOUj, and the minimal y may be taken with any auxiliary Riemannian metric.

See 2.3.8 for further examples oflinear P.D.E. systems that are solvable by purely algebraic manipulations.

2.3.2 Basic Properties of Infinitesimally Invertible Operators

In the theorem below we always refer to the respective fine topologies in the spaces frll, <§P and frll x <§p. For a subset d c fro we denote by d ll c frll, rt.. = 0, 1, ... , 00,

the intersection d n frll again with the induced fine CIl-topology. We apply the same rule to subsets in <§o and in fro x <§o, that is for {fI c fro x <§o we put {fI1l,P = {fI n (frll x <§P) and we deal with the fine CIl x CP-topology in {fI1l,P.

Let ~: frll+r -+ <§Il be a differential COO -operator of order r and let ~ admit, over an open set d = dd C frd, an infinitesimal inversion M of order s and of defect d. Let us fix an integer (J ° which satisfies the following inequality.

(JO > S = max(d,2r + s). Finally, we fix an arbitrary Riemannian metric in the underlying manifold V

Main Theorem. There exists a family of sets {fix c <§Go+S for all x E d Go +r+s, and a family of operators ~;1: {fix -+ d with the following five properties. (1) Neighborhood Property: Each set {fix contains a neighborhood of zero in the space

<§(1o+s. Furthermore, the union {fI = {x} x {fix where x runs over d Go+r+s, is an open subset in the space d(1o+r+s X <§Go+s.

(2) Normalization Property: ~;1(0) = x for all x E d Go +r+s•

(3) Inversion Property: ~ 0 ~;l - ~(x) = Id, for all x E d Go +r+s, that is

~(~;l(g)) = ~(x) + g,

for all pairs (x, g) E {fl.

(4) Regularity and Continuity: If the section xEd is C", +r+'-smooth and if g E!!4x is CU'+'-smooth for 0"0 ~ 0"1 ~ '11> then the section ~;l(g) is CU-smooth for all 0" < 0"1' Moreover the operator ~-1: !!4",+r+.,u,+. --+ d U, ~-l(X,g) = ~;l(g), is jointly continuous in the variables x and g. Furthermore, for '11 > 0"1' the section ~-l(X,g) is CU'-smooth and the map ~-1: !!4",+r+.,u,+. --+ d U' is continuous.

(5) Locality: The value of the section ~;1 (g): V --+ X at any given point v E V does not depend on the behavior of x and g outside the unit ball Bv(l) in V with center v, and so the equality (x,g)IBv(l) = (x',g')IBv(1) implies (~;l(g»(V) = (~;:l(g'»(V).

The proof is given in 2.3.3-2.3.6, where we also consider Holder spaces f'§U and !!l"" for all real 0" and '1.

Corollaries. (A) Implicit Function Theorem. For every xoEdoo there exists a fine C·+·+1-neighborhood of zero in the space f'§s+.+1, S = max(d,2r + s), say !!4o c f'§'+s+1, such that for each CU+'-section g E !!40 , 0" ~ S + 1, the equation ~(x) = ~(xo) + g has a CU-solution.

In the case of isometric immersions this is the famous theorem of Nash. If a metric go on V can be realized by afree COO-immersion xo: V --+ ~q then the

CU-metric go + g for 0" ~ 3 can be realized by CU-immersions for all g that are C3-small.

(B) The operator ~: d OO --+ f'§oo is an open map in the respective fine COO-topologies. Indeed, for every Xo E d oo, the operator ~ is invertible near the point go =

~(xo) E f'§OO by the operator go + g f--+ ~;"l(g) that is Coo-continuous in g.

(C) Approximation Theorem. If g E f'§oo, then every solution Xo E d,,+r+. of the equation ~(x) = g admits, for '1 > S, a fine CU-approximation by Coo-solutions for all 0" < '1.

Proof First, we C,,+r+'-approximate Xo by an arbitrary XEdOO . Then the section g' = g - ~(x) is C,,+s-small. Now, by the neighborhood property, the operator ~Ag') is defined for g' and

~(~;l(g'» = ~(x) + g' = g,

so that x' = ~;l(g') satisfies ~(x') = g. Finally, by the regularity and continuity of ~-1 in x and g, the solution x' is Coo-smooth and is CU-close to Xo = ~;"l(O).

Relation r7l = r71(A,~, g) and the Sheaf cP of Its Solution. Let us fix a Coo -section g: v --+ G and call a COO-germ x: (f)jt(v) --+ X, v E V, an infinitesimal solution of order a of the equation ~(x) = g, if at the point v the germ g' = g - ~(x) has zero a-jet, J;.(v) = O. This property of x depends only on the jet represented by x, called j = J~+<X(v), and we denote by r71<X(~, g) c x(r+<x) the set of all jets that are represented by these infinitesimal solutions over all points v E V. This relation r71<X(~, g) has


exactly the same C+a-solutions as the equation f0(x) = g. Now, we recall the open relation A c X(d) defining the set d c q;d and for IY. 2:: d - r we put

gJa = gJAA, f0, g) = Ar+a- d n gJa(f0, g) c x(r+a),

where Ar+a-d denotes the pull-back (p~+arl(A) for pta: x(r+a) -+ X(d). In other words, the relation gJa c x(r+a) corresponds to those infinitesimal solutions of order IY. of the equation f0(x) = g, which also satisfy the relation A c X(d). A C+a-section x: V -+ X satisfies fJlta iff f0(x) = g and xEd, and so all relations fJlta, IY. = d - r, ... , have the same COO-solutions. We set gJ = fJltd- r and denote by cP = cP(fJIt) = cP(A, f0, g) the sheaf of COO-solutions of gJ.

Local Solution of the Equation f0(x) = g. Take a jet j E gJa c x(r+a) over v = pr+a(j)E V and representj by a COO-germ x: (l)jZ(v) -+ X, such thatj = J~+a(v).

(D) If IY. > S + s = max(d + s,2r + 2s), then there exists a continuous family of COO-germs, x(t): (l)jZ(v) -+ X, t E [1,0], such that x(O) = x, x(1) E cP(v) and J~tj(v) E fJlta for all t E [0, 1]. In other words, infinitesimal solutions of gJa can be deformed to local solutions.

Proof Take a small neighborhood Vo c V of v such that the section x is defined over this Vo and there satisfies the relation A c X(d). Since j E gJa c gJa(f0, g), we have J;,(v) = 0 for g' = g - f0(x), and so we can find another COO-section gl: Vo -+ G that is arbitrarily small on Vo in the fine Ca-topology and such that gll{l)jZ(v) = g'I{I)jZ(v). Now, we apply the main theorem with Vo in place of V and obtain our family by setting x(t) = f0- 1 (x,tgd.

Local w.h. Equivalence. Denote by F(gJa) the sheaf of germs of sections V -+ fJlta and consider the homomorphism J: cP -+ F(gJa)' J(q» = J~+a.

(D') If IY. > S + s, then J is a local w.h. equivalence, that is the maps J(v): cP(v)-+ (F(gJa))(v) are weak homotopy equivalences for all v E V.

Proof We combine the deformation procedure Xf-+x(t) above with the following general considerations.

Let F = F OO (X) be the sheaf of COO -sections V -+ X and consider the associated sheaf F*. There is a natural (jet) homomorphism J = Jr+a of F* to the sheaf of section F(x(r+a»). Indeed, sections y E F*(U), U c V, are continuous families of germs Xv E F(v) where v runs over U. One gets q> = J(y): U -+ x(r+a) by taking q>(v) = J~+a(v), VE U. There also exists a homomorphism I: F(x(r+a») -+ F* such that J 0/= Id. In fact, such an I assigns to every jet j E x(r+a) a specific germ x = x(j)EF over (l)jZ(v), v = p(r+a)(j), that represents thisj = J~+a(v). Furthermore, for each x E F(v) the germ (1 0 J)(x) is homotopic to x and the homotopy can be chosen simultaneously for all x E F(v), v E V. In particular, for the pull-back l'(gJa) = J-l(F(gJa)) C F*, the map J: l'(gJa) -+ F(fJlta) is a w.h. equivalence.


Now we tum to the sheaf if> and we also take if>* c F*. This sheaf if>* is contained in l'(Plla) c F*; because sections in if>* are families of local solutions of Plla, while sections in l'(PllJ are families of irifinitesimal solutions. The deformation procedure x f---+ x(t) applies to families of infinitesimal solutions, as long as these solutions are defined over a fixed open set Vo c V. It follows, that (l'(Plla))(v) can be deformed to if>*(v) c (l'(Plla))(v) for all v E V, and so the inclusion i: if>* --+ l'(Plla) is a local eq ui valence.

Finally we invoke the local equivalence L1: if> --+ if>* (see 2.2.1; do not confuse with the map L1: x(r) --+ G where defines .@) and observe that the homomorphism J: if> --+ F(Plla) decomposes into three local equivalences, J = J 0 i 0 L1. Hence J is a local w.h. equivalence.

(D") Remark. The Theorems (D) and (D') also hold for 0( = S + s. To show this we must transform infinitesimal solutions x of order 0( = S + s to such solutions of order 0( + 1.

Suppose that X --+ V is a vector bundle and let M = M(x, g) be an infinitesimal inversion of.@ of order s (in g). Then the section Xl = X + M(x, g'),for g' = g - .@(x), is an infinitesimal solution of order p = 2(0( - r - s) + 1. In particular for 0( ~ s + s ~ 2r + 2s we get p ~ 0( + 1.

Proof By Taylor's formula, f(x + y) = f(x) + f' (x)y + B(x, y)y2, one has .@(x + y) = .@(x) + L(x, y) + B, where B = B(x, y, y, y) is a differential operator of order r, and B is bilinear in the last two arguments. In particular, the identity J~+r(v) = ° implies Jjk+1(V) = 0.

Now, as J;,(v) = 0, we have J;-S(v) = ° for y = M(x, g'). Furthermore, with this y we have L(x,y) = g - .@(x), since LxoMx = Id. It follows that .@(x l ) = .@(x + y) = g + B with J~(v) = 0.

Exercise. Prove Taylor's formula for .@(x + y) by applying the usual formula to the map L1: x(r) --+ G and thus completing the argument above.

Microflexibility of if>. Take a section x E if>(V) and consider a deformation of x over a compact set K c V. This deformation, X t E if> ( (9fi(K)), t E [0, 1], Xo = x I (9fi(K), extends to a COO-continuous deformation Xt: V --+ X, Xo = x. Fix an open set Vo ::::> K in (9fi(K), such that xtlVo = xtlVo, tE [0, 1], and choose a metric in V such that dist(K, V\ Yo) > 1. With this metric we take the operator .@-l and apply it to (Xt' g;), for g; = .@(x) - .@(xt) = g - .@(xt), for t in a small interval [0, e] c [0,1], e > 0. The resulting new deformation x; = .@-l(Xt,g;) satisfies the equation .@(x;) = g and by the locality of .@-l we have x;IK = xtlK, tE[O,e]. Finally we observe that this construction equally applies to the families of sections Xp,t for p running over a polyhedron, and to all open sets U ::::> K in place of V. Therefore all restriction maps if>(U) --+ if>(K) are microfibrations

Example. The sheaf offree isometric COO-immersions (V, g) --+ (w, h) is microflexible, as it was claimed in 1.4.1.


(E) The Transversality Theorem for Solutions of f7ia. Let IX ~ d - r and observe (compare the above) qta to be a (locally closed) Coo-submanifold in x(r+a). Consider a stratified subset E c qta with COO-strata.

(E') The jet J~+a: V -4 qta of a generic COO-solution x: V -4 X of f7ia is transversal to each stratum of E.

Proof The above discussion allows enough deformations of (jets of) solutions of qta is order to apply the argument of l.3.2(D).

Example. A generic free isometric Coo-immersion x: V -4 W is transversal to any given (and fixed) Coo-submanifold Wo c W

2.3.3 The Nash (Newton-Moser) Process

Let us reduce the Main Theorem to the case when X -4 V is a vector bundle. To do that, we realize X as a Coo-subfibration in the trivial bundle, X c X = V x IRm -4 V for m large and then we extend A: x(r) -4 G to a map .1: x(r) -4 G, such that in a normal neighborhood U of X in X we have .1 = A 0 P for the normal fiberwise projection P: U -4 X. Since P is a submersion, the corresponding zero order differential operator F(U) -4!'£ is infinitesimally invertible and so the operator {g: :f -4 ~ associated to .1 is also infinitesimally invertible over (9ft(d) c fld ~ PEd, and with {g-I for {g we put .@-I = po {g-I.

At this point we assume that X is a vector bundle.

Smoothing Operators. We choose and fix a sequence oflinear operators Si: !'£O -4 PEoo, i = 0, 1, ... , and another such sequence in ~ that is also denoted with a stretch of language, Si: ~o -4 ~oo. We call them local smoothing operators if the following two properties hold.

Locality. Every Si' i = 0, 1, ... , does not enlarge supports of sections PE more than by 8i = (2(i + 1))-2, that is, with our fixed metric in V, the value (Si(X»(V) only depends on this x within the ball Bv(e;), for all x in PEo or in ~O) and for all VE V

Convergence. If x is Ca-smooth, IX = 0, 1, ... , then Si(X) --t X as i -4 00. Moreover, Ca-convergence Xi -4 X implies Ca-convergence Si(X;) -4

cx.

Remark. The notion of convergence always refers to the usual (not fine) topologies in the respective spaces.

Exercise. Show that the fine convergence Xi -4 X makes all Xi for i ~ io equal to x outside a fixed compact set in V

Basic Notations. For a finite or infinite sequence x(i) E PE(r), we write x(i) = Si(x(i» and with the linearization L = L(.@) we consider

L *(x(i), y) = L(x(i), y) - II L(x(i) + ty, y) dt.


Then we put y(i) = x(i) - x(i - 1), i = 1, ... , and we take

e(i) = L *(x(i - 1), y(i», ;

E(i) = Le(j) 1

and

E*(i) = e(i) + (S; - S;-1)E(i - 1), for e(i) = S;(e;).

Observe that

iii

LE*(j) = LSj(E(j) - E(j - 1» + L(Sj - Sj_dE(j - 1) = S;(E(i», 1 1 1

where we assume E(O) = O. Now, with a given pair (xo,g)Ed x q;O, de f!{, we define Nash's process

directed by (xo, g) as a finite or infinite sequence x(O) = X o, x(1), x(2), ... , x(i), ... E f!{d with the following two properties

(A) X(i) E d, i = 0, ... , and so the infinitesimal inversion M is defined at X(i). (B) x(i + 1) = x(i) + M(x(i), g'(i», for g'(i) = (Si+1 - S;)g + E*(i), i = 0, 1, ... ,

where we assume E*(O) = O.

Observe that the condition (B) inductively defines x(i) for all i as long as (A) holds.

"Raison d'etre" of Nash's Process. Suppose that the sections x(i) are defined for all i = 0, 1, ... , and let the sequence x(i) C' -converge as i ..... 00, to x = x( (0). In this case we define g)-1 = g)-1(xo,g) by setting g)-1(xo,g) = x(oo).

If the "total error" sequence E(i) CO-converges as i ..... 00, then this operator g)-1 satisfies the normalization, the inversion and the locality properties of the main theorem of 2.3.2. (We now use X o in place of x of 2.3.2.)

Proof If 9 = 0, then (Si+1 - S;)g = 0 for all i = 0, 1, ... , and, by induction, we have all y(i) and E*(i) equal zero, so that x(i) = x(O) = Xo for all i = 0, 1, ... , and x( (0) = Xo as well.

In order to prove the inversion property, g)(x( (0» = g)(xo) + g, we first recall d

that dt g)(x + ty) = L(x + ty, y), and then we have

g)(x(i + 1» - g)(x(i» = f L(x(i) + ty(i + 1), y(i + 1»dt

= L(x(i),y(i + 1» - L*(x(i),y(i + 1» = L(x(i), M(x(i), g'(i» - e(i + 1)

= g'(i) - e(i + 1),

and so we conclude

i

f0(x(i + 1)) - f0(XO) = I,g'(j) - E(i + 1)

° i i

= ~)Sj+1 - Sj)g + I,E*(j) - E(i + 1) ° 1

= Si+1 (g) + Si(E(i)) - E(i + 1),

and for i -+ 00 we get

f0(x( 00)) - f0(xo) = g.

To prove the locality, we observe that x(i + 1) is obtained from X(j),j = 0, 1, ... , i, and g by some differential operators and by the smoothing operators Si-1' Si and Si+1. It follows, that value x(i + 1)(v) is determined by all x(j),j ~ i, and g within the ball Bv(Si-1), Si-1 = (2;t2. Since I,f (2;t2 < t, the values x(i)(v) are determined by (xo, g)IBv(1) for all VE V and for all i = 0, 1, ... , and so this is also true for x(oo) = f0- 1 (xo,g).

Remark. For Si = Id, i = 0, 1, ... , Nash's process reduces to the usual Newton's method of solving non-linear equations. Unfortunately, Newton's process (without smoothing) diverges in most interesting cases. To make it converge one should use appropriate smoothing operators.

2.3.4 Deep Smoothing Operators

For a function f: ~n -+ ~q we denote by Ilfllo its CO-norm, Ilfllo = SUpVElRn Ilfll, and for o(E(O, 1) we denote by Ilfll" its Holder C"-norm,

Ilfll" = max(llfllo,sup(llwll-"llf(v + w) - f(v)II)), V.w

where v runs over ~n and w runs over all non-zero vectors in the unit ball in ~n around the origin. For an arbitrary 0( = j + (),j = 0, 1, ... , ()E [0,1), we put Ilfll" =

II J} 11o, and if the map f is not Cj-smooth we assume II fII" = 00, for all 0( ~ j.

Digression. The existence of operators f 1--+ Sf satisfying the smoothing estimates [see (1)-(3) below] needed for Nash's iteration process depends in a crucial way on Taylor's remainder theorem. A more geometric aspect of this theorem is revealed in

Yomdin's Proof of A.P. Morse' Lemma. Fix a large positive constant c = c(n,r) for n, r = 1,2, ... , and consider a C"-function

f: ~n -+ ~ for 0( E [r, r + 1].

Cover the unit ball B c ~n by N = NE ~ cs-n balls BE of radius s for some small s > ° and let p: BE -+ ~ be the Taylor polynomial of f on BE of degree degp = r.

Let

Ae = {xEBelllgradpll:::;; c'e ll- 1 }

for c' = cllflill and observe with Taylor's theorem that the c' ell-neighborhood of the image p(Ae) c IR contains the set of critical values of flBe. That is the image f(Ie) c IR of the zero set Ie C Be of grad f on Be. Since the set Ae is semialgebraic and p is a polynomial there exists a (semi) algebraic curve Ce c Ae of degree d :::;; (n deg p t < c which meets the level Ae(Y) = Ae n p -1 (y) for all Y E IR. To construct Ce take a generic linear function Lon IRn, consider the critical set Ce(y) of L on (each stratum of the canonical stratification of) Ae(Y) and let C = Uy A.(y). over all Y E IR (compare 1.3.2). The length of Ce is estimated by Crofton formula,

length Ce :::;; cd(Diam Ce) :::;; 2n c2 e.

Since IlgradplCel1 :::;; c'e ll- 1 and p(Ae) = p(Ce), the c'ell-neighborhood of p(Ae) consists of at most cd intervals of length:::;; e" ell for e" = 2n+1 c2 c', where we assume e:::;; 1. Therefore, the set of critical values of fiB can be covered by at most ce-n

intervals of length:::;; e" ell for all e E (0,1). In particular, if ex > n, then the set of critical values has measure zero (compare 1.3.2).

The reader interested in further applications of Yomdin's method is referred to his forthcoming book in Asterisque series and to my Bourbaki talk in June 1986. Here are some samples suggested as exercises.

(a) Let the set of critical values of a Cil-function on a compact n-dimensional manifold contain the subset {I, r P, ••• , C P, ••• } c IR for some p > 0. Then ex :::;; n(p + 1).

(b) Let f be a Cil-smooth map of a compact n-dimensional manifold V into IRq and let Io c V be the zero set of the differential of f Then the image f(Io) c IRq can be covered by at most ce -n balls of radius ell for some c = c(f) and all e E (0, 1).

(b') Let Ii c V be the subset where rankf :::;; i. Then f(Ii) can be covered by at most ce-d balls in IRq of radius e for d = i + ex-1 (n - i) and for all positive e :::;; 1.

(c) Let p be a polynomial map of the unit ball Be IRn into IRq and define A = Ap: B -+ IR+ by A = (det DD*)1/2, where D = Dx: IRn -+ IRq is the differential of p at x E IRn and D* is the adjoint operator IRq -+ IRn. Consider the subset

Ae = {xEBIA(x):::;; e}

and estimate the q-dimensional measure of the image p(Ae) c IRq by Jl[p(Ae)] :::;; ce for some constant c = c(n,q,degp) and all eE[O, 1].

n a (d) Consider a vector field X = L Pi- on IRn where p/s are polynomials and

i=1 OXi

construct an algebraic hypersurface H c IRn of degree:::;; c = c(n, deg Pi) which meets every orbit of X.

(d') Consider a q-codimensional foliation on IRn whose tangent bundle is an algebraic subset in Grn_q(lRq) and find an algebraic subset in IRn of dimension q which meets every leave of the foliation.

Now, for the manifold V, we fix a locally finite cover by relatively compact coordinate neighborhoods, V = U Jl UJl , Jl. = 1, 2, ... , and we take a partition of unity {PJl} inscribed in {UJl}' Then, we fix some trivializations ofthe bundles G and X over each UJland we divide sectionsfintothe sums,f = 'IJlfJlfor jIl = PJlf: UJl = R" -+ IRq. Finally, for a subset K in V, we select all those neighborhoods among UJl , which intersect K, called {U.} c {UJl }, and we define Holder's (semi)norms II IL.(K) as Ilfll .. (K) = suP. 111"11 .. ·

Exercise 1. Let a sequence /;, i = 1,2, ... , satisfy the following inequalities

11/;llj(K) ~ constii-3 + i2U- .. )-1),

for two given successive integer values ofj,j = j1 ~ ° andj = j2 = j1 + 1, for a fixed 0( in the open interval (j 1 ,j2) and for all i = 1,2, .... Show that f = 'Ii/; is a section of class C" on K, [i.e. Ilfll .. (K) < 00], and that

00

II 'I /;II/J(K) -+ 0, as io -+ 00, i=io

for every (3 < 0(.

Hint. Estimate IIJj'(v + w) - Jj'(v)11 for Ilwll = 8 ~ 1, by using the sum D~l /; with i~ ~ 8-1 < (io + If·

We say that a sequence of smoothing operators So, Sl' ... , has (Nash's) depth if, if for every compact set K c V there are some constants C .. , 0( E [0, (0), uniformly bounded on every finite interval [0, a] c [0, (0), such that all sections f satisfy the following inequalities with the norms II II .. = II II..(K) for all 0( E [0,(0) and for all i = 1,2, ....

Smoothing Estimates

(1) for ° ~ {3 ~ 0(.

(2) II(Si - Si-dfll .. ~ C .. (i- 2d- 1 + i-2/J-1)llfll .. +/J' for -0( ~ (3 < 00.

(3) IISi-1(f) - fll .. ~ C .. W2d + i-2/J)llfll"+/J' for {3 ~ 0.

These formulae and the exercise above show that smoothing operators of depth d ~ 1 naturally interpolate the scale cj to Holder's C .. for all real 0( ~ 0. However, in our exposition below and in Sect. 4.3.5 Holder's spaces are not essential for the proof ofthe main theorem for integral 0"0' 0"1 and '71, and the reader who does not like Holder's spaces may assume all 0( and (3 below to be integers.

To construct "deep smoothing" we start with a COO-function S: IR" -+ IR with support in the unit ball Bo(1) c IRn, and for a continuous function f on IRn we write

S * f = (S * f)(v) = r S(w)f(v + w)dw, J~n

V, WE IRn.

Then we take S;.(v) = An S(AV) for A ~ 1, and observe that supp(S;.) c Bo(A -1) and


Therefore,

liS;. * fiL. ~ CIIfIL.. for all ot: ~ 0,

(4) C = C(S) = r I S;.(w) I dw = r IS(w)1 dw

J~n J~n

Lemma. If the function S is orthogonal to the polynomials Q on ~n of degrees 0, 1, ... , d, then

IIS;.*fllo ~ C.A.-Pllfll p,

for all .A. ~ 1 and {3 E [O,d + 1].

Proof. By Taylor's formula f(v + w) = Qv(w) + Rv(w), where Q is a polinomial in w of degree ~ d and where the remainder R satisfies the following estimate for II w II ~ 1

Rv(w) ~ IlwllPllfllp'

since f SQ = 0, we have

I(S;. * f)(v) = It S;'(W)Rv(W)dWI, for B = Bo(.A. -1),

and so

I(S;. * f)(v) I ~ CIIfllpsup IRJw)1 ~ C2-Pllfll p. WEB

Corollary. For ot: E [j,j + 1],j = 0, 1, ... , and for {3 E [ -ot:,d] one has

(5)

Proof. First show that

(5')

this time for {3 E [ - j, d + 1]. To do that we must estimate Ilok(S;. * f)llo for all partial derivatives Ok of order k ~ j. But Ok(S;. * f) = ( - .A.)k(iJk S);. * f, and the partial derivatives cis are orthogonal to all polynomials of degree ~k + d, so we obtain (5') by applying the lemma to OkS.

Now, for a real ot: E [j,j + 1], we interpolate (5') to (5) by means of the following elementary relation

that holds for allll and v such that j ~ Ilot: ~ v ~ j + 1 and with those a and b, for which a + b = 1 and all + bv = ot:.

Remark. For an arbitrary function S, with no orthogonality conditions, we have

(6)

for IXE [j,j + 1],j = 0,1, ... , and for 0::;; p ::;; IX.

Proof For IX = j the inequality (6) follows from (5') with d = -1, and for P = 0 it follows from (4) for all IX. With these two special cases on hand, we get the remaining IX and P by the interpolation inequality (*).

At this point we normalize S by the condition

r S(v)dv = 1. J~n

If S is so normalized, then

(7) liJi(S;. * f - f)(v)1 ::;; c sup I (aif)(v + w) - (aif)(v)l, WEB

for B = Bo(). -1), for all v E IRn and for all partial derivatives ai . In particular the operators fl-+ S;. * f converge, as A -+ 00, in all Ci-topologies to the identity operator. (Compare with the convergence condition of 2.3.3.)

Proof Since ai(s;. * f) = s;. * aif, we have with jj = aif the following identity

ai(s;. * f - f)(v) = L S;.(w)(jj(v + w) - jj(v)) dw,

and we get (7) with the constant C of (4).

Warning. For Ca-functions with a non-integer IX, there may be no Ca-convergence S). * f -+ f for A -+ 00. For example, the Ca-function f(v) = Ilvlla, 0 < IX < 1, admits no Ca-approximation by COO-functions at all. However, liS;. * f - flip -+ 0 for all f3 < IX.

We say that a normalized function has depth d, if it is orthogonal to the homogeneous polynomials of degrees 1, ... , d. One produces such functions out of any given normalized function S by taking linear combinations

Ii snew = L akS;'k'

k=O

The normalization condition for snew amounts to the identity L~ ak = 1, and the depth d condition is expressed by the equations L~=o akA;P = 0 for p = 1, ... , d.

Notice, that the derivative of S). in A, that is S~ = d~ S;., can be written as

128

where

2. Methods to Prove the h-Principle

" as ~ = ~(Vl""'V,,) = nS + IVj~'

1 uVj

and if S has depth if, then this ~ is orthogonal to all polynomials of degrees 0, 1, ... , [(with deg = 0 included this time).

Theorem. If S has depth if, then the operators S;(f) = S},; * f for A; = (i + 1)2 satisfy the smoothing estimates (1), (2), and (3).

Proof The estimate (1) follows from (6). To get (2) we write

f),; f},1 (S; - S;-l)f = (SA * f) dA = A -1 ~}, * fdA,

},;-I },I-I

and by applying (5) to ~ we get

II(S; - S;-l)fll .. ~ C .. i-2/J-lllfll .. +p for - tx ~ P ~ if, that is equivalent to (2) for -tx ~ P < 00.

The estimate (3) for P = 0 follows from (4). If tx is an integer, and P ~ 1, then (3) follows from (7). Now we can write

00

f = I (S; - S;-dJ, i=l

and 00

f - Sj-l (f) = I (Sj - Sj-1)J, j=i

so that (2) implies (3) for all tx and P ~ 1. For the rest of tx and P we interpolate according to (*).

Remark. This construction of smoothing equally applies to vector-functions f: IR" -+ IRq for all q ~ 1.

Smoothing on V. With the partitions of unity {U,.,P,.} and with some constants A,. ~ 1 assigned to each J1. = 1, 2, ... , we obtain smooth sections

f=If", ,. by setting Si(f) = I,. s,.,; * f", for a fixed function S: IR" = U,. -+ IR of depth [ and for S,.,; = S}" where A = (i + 1)2A,..

These operators Si satisfy the smoothing estimates (1), (2), and (3) and also the convergence condition of 2.3.3, and by choosing the constants A,. sufficiently large we satisfy the locality condition of 2.3.3 as well. This is all we need as long as the space '# is concerned, but for the smoothing in :r we choose the parameters A.,. more carefully in order to meet the requirement x(i) E d of Nash's process. Namely, instead of numbers A,. now we take functions A.,. = A,.(Xo) for Xo E d = dd, such that


every AI" Jl = 1, 2, ... , is continuous in Xo relative to the usual Cd-topology in .91 and such that with these AI' we have Si(XO) E .91 for all Xo E .91 and for all i = 0, 1, .... Such a smoothing, or rather a family Sf 0: f!(0 -+ f!(oo, for all Xo E .91, is called d-stable. Now the constants C/Z in the estimates (1), (2), and (3) depend on Xo, but with Aixo) constinuous in Xo, we can choose all C/Z(xo) = sUPO:5:P:5:/Z Cp(xo) also continuous in Xo' In fact, we only need these contants to be locally bounded in Xo.

Smoothing on Manifolds with Boundaries. If V has a boundary, then some of the neighborhoods UI' are. modeled by IR':. = {Vl, ... ,Vn,Vl ~ o} and the smoothing problem on V is reduced to the smoothing of compactly supported functions f on IR':.. To smooth such an J, we first extend it to a function 1 on IRn and then we take Si(f) = Si(l)IIR':.. The extension operator f -+ 1 must be bounded in all norms II II/Z and it is constructed as follows. Writef = f(v, w), where v = Vl' w = (V2'''''Vn), and seek 1 on IR~ in the form Rv, w) = Li~oaJ( -2iv, w), for VE( -00,0]. The conditions Ok 1 = okf for v 1 = ° amount to the following infinite system of equations in the unknowns ai'

00

L ai( _2i)k = 1, k = 0,1, ... , i=O

We satisfy (**) by taking an entire function p(z) = D> akzk such that p(z = 2k) = (_I)k for k = 0, 1, ... , [see Seely (1964) and Ogradska (1967)].

Non-local Smoothing. All our estimates of the "convolution" integrals S;. * f hold for rapidly decreasing functions S, possibly with infinite supports. "Rapidly decreasing" means that the products of all partial derivatives of S by polynomials of all degrees are bounded on IRn, that is IIJ~(v)llllvllj s constj for all vElRn andj = 0, 1, .... In particular, if one takes a rapidly decreasing function $ with zero partial derivatives at the identity, (Oi$)(O) for allj > 0, then the Fourier transform of $has infinite depth and the corresponding smoothing operators have no "i- 2d" term in the estimates (2) and (3) (see Nash 1956). Of course such an infinitely deep smoothing is never local in our sense and so it is less convenient for non-compact manifolds.

Analytic Smoothing. An open set U c en = IRn Ei1lRn J=1 is called obtuse if with

every point u = x + yJ=1 E U, x, yE IRn, it contains the cone which has u as the vertex and the Euclidean ball BAp)ElRn, p = 211yll, as the base. We denote by WU c U U IRn c en the n-dimensional manifold obtained by joining the complement IRn\Bx(p) with the cone from UE U over the boundary sphere oB(p). We restrict the standard holomorphic n-form dU 1 /\ dU2 /\ ... dUn to W U and call it dw. If a function f: IRn -+ IR extends to a (unique) function fU on IRn U U which is holomorphic on U, then the integral fwufU(w)dw does not depend on the point u, and so it is equal to the Euclidean integral f[J;lnf(w)dw.

For a function f as above we denote by Ji(w) either the usual jet at w = v E IRn

or the "holomorphic" j-th orther jet of fU at w = U E U, and set II f IJf = supw Jj(w) for w running over the union IRn U U. If there is no analytic continuation of f to U we assume IlfilY = 00 for allj = 0,1, ....

Now, we consider a linear combination


d S = L akexp( -IIAkvIl 2 ), VE~",

k=O

which is made (by an appropriate choice of Ak and ak) normalized and deep of depth if. Every such function S extends to a holomorphic function S(u), u E C", and S(u - uo) is uniformly rapidly decreasing on the manifold WUo. That is IIJ~(u - uo)llllu - uoll j ~ constj , for all UoEcn and UE WUo and for allj = 0,1, ... , where the constants constj depend on S but not on the point uo.

Theorem (Whitney 1934; Nash 1966). For all bounded C0 1unctions f: ~"- ~ the "convolutions"

are real analytic functions on ~" that extend to holomorphic functions on CR. Furthermore, the operators

for Ai = (i + if, satisfy the estimates (1), (2) and (3) for all integers oc and p and for all norms II 1111 = II II~ with the constants CIl, oc = 0, 1, ... , independent of the obtuse set UeC".

Proof Write

(S;. * f)(u) = r S;.(w - u)f(w) dw, J8ln

for UEC".

These integrals uniformly converge for IIIm(u)1I :$;; const and so S;. * f is holomorphic. Now, we prove (1) (2) and (3) exactly as we did it before with only one new

ingredient: when estimating our integrals, for example, (S;. * f) at u E U, we employ WU in place of ~n,

(S;. * f)(u) = r S;.(u - w)f(w)dw, Jwu

and we use Taylor's expansion of f at u. Since the function S;. rapidly decreases on W U, all the calculations go through.

Exercise 2. Fill in the details for this argument.

Exercise 3. Prove Whiteney's theorem: Every Coo1unction f on ~n admits a fine Coo-approximation by real analytic functions.

Hint. Take an exhaustion of en by relatively compact obtuse sets ... Uk ::::> Uk- 1 •••

and, with a given number j = 0, 1, ... , and with a function e = e(v) > ° construct functions it such that

II (Jjk - Jj)(v) II :$; (1 -:- ~)e(V)' II h 117k < 00,

II ft - h+1 Ilfk :$; 2-1,

and take limt .... oo h as an e-approximation to f

for all v E Uk n !Rn,

Exercise 4. Generalize Whitney's theorem to Can-fibrations X --.. V.

Hint. Use Can-embeddings of X and V to !RN.

2.3.5 The Existence and Convergence of Nash's Process

131

We work with a fixed local smoothing Si in f'§ of depth d and with an d-stable smoothing Si = Sio, Xo E .91, in !!E, also of depth if We fix a compact set K in V and we write II 11,. for II IL.(K).

For a section XEd we denote by R(x) the lower bound of those numbers R > 0 for which the inequality II y II d :$; R -1 implies X + Y E d for all sections y with support in K. Since the smoothing in !!E is d-stable the upper bound SUPO~i<oo R(Sio(xo)) is finite and so for some Ro < 00 the inequality Ilylld:$; R01 implies SUPO~i<oo R(Sio(xo + y)) :$; Ro· The lower bound ofthese numbers Ro is denoted by

Ro = R(xo) = R(xo, SiD).

A section g E f'§ is called localized (in K) if the I-neighborhood of its support is contained in K, that is dist(supp(g), V\K) ~ 1. If x(i), i = 0, 1, ... , io, is Nash's process directed by (xo, g) and if the section g is localized, then the sections y(i) have supports in K (see 2.3.3). Hence, the condition D~1 II y(i)//d :$; Ro1is sufficient for the existence of y(io + 1) = M(x(io), g'(io» (see 2.3.3) and of x(io + 1) = x(io) + y(io + 1). Furthermore, since all differential operations (of M) in this expression for y(io + 1) are preceded by smoothing operators that are continuous maps !!Eo --.. f!(OO

and f'§o --.. f'§oo, we have (by induction as in 2.3.3) continuous dependence of every y(i)E!!Eoo on the directing pair (xo,g)Edd x f'§o. In particular, if /Ig/l0--,,0, then /I y(io)/Ia --.. 0 for every (J E [0,00) (see 2.3.3) and we come to the following conclusion.

Initial Estimate. For every io = 1, ... , there exists a positive number e =

e(io,Ro, /IxO /l d), such that the inequality /Ig/lo:$; e implies the existence of Nash's process x(i) for i = 0, 1, ... , io. Furthermore,

io

L /I y(i)/Ia :$; ba(e), i=1

where ba(e) --.. 0 for every fixed (J ~ O. • .... 0

Now, for a small g, we want to have an infinite Nash process and we proceed with an estimate for i --.. 00 as follows. First we recall the numbers d and s, the orders

of M(x, g) in x and g, and the order r of the operator L. We assume the depth d of the smoothing to be large compared to d, rand s, for example

d ~ 10(s + 1), for s = max(d,2r + s).

Then we take four numbers (xo, 0"0' 110 and v such that

s + 3v :::;:; 0"0 :::;:; (Xo :::;:; 110'

0< v < 1,

and set

Qo = Qo(i,lX) = Qo(i,lX - 0"0' d) = i- 24 + i2(,,-ao)-1.

Next, with an arbitrary constant Co > 0 we consider all localized sections g and all Xo E d such that

IIxo 11"0+r+8 :::;:; Co

IIgllao+':::;:; Co·

Finally, we consider all Nash's processes xCi) = Xo + :D=1 y(i) directed by these pairs (xo, g) and we ask ourselves under what condition the following two inequalities hold.

Lemma. There exists a constant Co = CoCCo, 0"0' 110' v) with the following property. If, for a given number io, the inequality (E) holds for i = io and if (Co) holds for all lX :$ lXo + r + sand i = 1, ... , io, then the sections y(i) for i :$ 10 4- 1 satisfy the inequality (C~) below.

(C~) for all (X :::;:; 110 + r + s.

Remark. The constant Co also depends on our fixed data, such as the "smoothing" constant C" of 2.3.4 and on the operators Land M. But what is really important, Co does not depend on io.

Corollary. There exists a positive number e = e(Ro, Co), such that the conditions (Ao) together with the additionalinequality IIglio < e imply the existence of Nash's process xCi) for all i = 0, 1, ... , and for this process xCi) the sections y(i) = x(i) - x(i - 1) satisfy the inequality (C~) for all i.

Proof Since v > 0, for some io we have CO :::;:; i(;Co, and, if necessary, by choosing io greater, we also make

00

(**) L iVCoQo(i,lX = d):::;:; tRol. i=io+1

Then, by the initial estimate, we have for IIgllo:::;:; e, with sufficiently small e =


8(io) > 0, the inequality

as well as the inequality

(Coio) Ily(i)IL. ~ iVCoQo(i,a) for i ~ io and for a ~ ao + r + s. The inequalities (**) and (Bo) show that (B) holds as long as (Co) does, and since Co ~ iVCo for i ~ io, the inequality (C~) is stronger, for i ~ io, than (Co). Therefore, by the lemma, "(Co) for i" implies "(Co) for i + 1" for all i ~ io and with (Coio) we have (Co) as well as (CO) for all i x 0, 1, ... ,

Proof of the Lemma. For two related positive numbers, or for two sequences of numbers, such as IIy(i)IL. and IIx(i)IL .. we say that the estimate

II y(i)IL. « Q(i, a)la ~ Po,

implies

IIx(i) II .. «Q'(i,a)la ~ P~,

if for every constant C there is another constant C' = C'( C, Q, Q', Po, P~) such that the inequality

II y(i)II .. ~ CQ(i, a), for alIa ~ Po,

implies the inequality

IIx(i)II .. ~ C' Q'(i, a), for alIa ~ P~,

for all sequences y(i) and x(i) in question, regardless of their length.

Example. We write

(0)

(00)

(1)

(1')

IIxo II~o+r+s « 1,

IIgllao+s« 1,

IIy(i)II .. « iVQo(i,a)la ~ ao + r + s,

II y(i) II .. « Qo(i, a)la ~ '10 + r + s, and we now express the lemma by saying that (0), (00), (1) and the inequality (B) imply (1').

We also write a « 1 in place of a ~ Po if a relevant estimate holds for alIa in any given interval [0, Po].

Example. Since the constants C .. of the smoothing estimate (2) of 2.3.4 are bounded on finite intervals by C .. « 1 for a« 1, the estimate (00) implies the following estimate for g(i) = (Si+1 - Si)g,

(00) IIU(i)II .. « i- 2d- 1 + i2( .. -ao-S)-1Irx« 1.

Our next estimate concerns x(i) = Xo + L}=1 y(j). Since v > 0 and d ~ 1, the estimates (0) and (1) imply

(2) Ilx(i)ll .. « 1 + i2 ( .. -ao+v)lrx ~ rxo + r + s

This estimate (2) implies [via (1) of 2.3.4] the following estimate for x(i) = Sj(x(i)),

(2) Ilx(i)ll .. « 1 + i2( .. -ao+v)lrx ~ 1'10 + r + s + d,

(in fact, we could write rx « 1), and with (3) of 2.3.4 we have for ~(i) = x(i) - x(i),

To proceed further we need some general facts concerning the differential operators Land M. First, for two functions in rx ~ 0, say for Q(rx) and Q'(rx), we write

Q(rx) * Q'(rx) = sup Q(f3)Q(rx - f3).

Then we observe the following Leibniz' inequality

(3) IIJJ'II .. « IIJII .. * IIJ'II .. , where J and J' are two vector functions V -+ ~N, and JJ' may denote any given bilinear form in J and J'.

To prove (3) we write rx = j + e, for j = 0, 1, ... , and for 0 < e < 1. Then the ordinary product of two real functions satisfies

IIJJ'lle ~ IIJllelIJ'llo + IIJlloIIJ'lle·

Then we obtain (3) with the inequality

IIJJ'II .. ~ Cj sup II (OkJ)(OIJ' )lle. k+l:S;j

The next fact we need is the

Chain Rule Estimate. Consider a COO-function F on some domain A c ~N, and a map J: V -+ A. We denote by R(J) the CO-norm of the function <5-1 = <5-1(v) for <5(v) = dist(J(v), ~N\A), that is R(J) = 11<5-1 110' Notice, that R = 0 for A = ~".

If IIJllo« 1 and R(J)« 1, then

(4) IIF(J) II .. « 1 + IIJII .. *··· * IIJII .. , \ v..-------'

j+1

where j = ent(rx).

Proof Our assumptions imply that all partial derivatives Fia) = (0" F)(a) satisfy IIF,.(a)llo« 1. Then we expand the partial derivatives ol(F(J)) into sums of products like II, = 011(J) ... Olk(J)F,.(J), for I ~ j and '1 + '" + Ik ~ j. Since the norm IIF(J)II .. for rx = j + e is estimated by II II, lie for all products II" I ~ j, we get (4) by using (3) and the following obvious inequality for 0 ~ e ~ 1,


IJFIl(J)lle;5; IIJlleIIFIl+t(J)llo + IIFIl(J)llo'

Example. If

II J Iia « Q(i, oe) = 1 + i2a- Yloe ;5; Po, then for y ;;:: 0 we have the identical estimate for F(J),

(4*) IIF(J)lla« Q(i,oe)loe;5; Po· Indeed Q(i, oe) * Q(i, oe) « 1 + i2a- y', for y' = min(y,2y).

135

The Operator M. Finally we turn to differential operators [such as M(x,g)] linear in the second argument. With the jets J = J; and J' = J;, we write M(x, g) = F(J)J' for the usual local coordinate expression

M = I.Fk(J)a'kg, k

and since IIJlla = IIxlla+d and 111'lla = Ilglla+s> we have, under the assumptions used for (4) and (4*),

(5) IIM(x, g)lla «(1 + Ilxlla+d) * Ilglla+s· The Operator L'. Let

L'(x,u,y) = :tL(X + tu,y)lt = O.

Then

(-) L(x + u,y) - L(x,y) = It L'(x + tu,u,y)dt.

Observe, that L' is a differential operator of order r and it is bilinear in u and y, and so we write L' = F'(J)Ji, for J = J~, J = J~ and i = J;.

This operator L' is related to the operator L * of 2.3.3 as follows. Put

L = L(x(i - 1),y(i)),

L; = L'(x(i - 1) + tx(i - 1), x(i - 1), y(i)),

and

L;,t = L'(x(i - 1) + try(i), ry(i), y(i)).

Then with x(i - 1) = x(i - 1) - x(i - 1) we have, according to (-) above and to the formulae of 2.3.3,

e(i) = L *(x(i - 1), y(i)) = It J: (L; - L;t) dt dr.

Let u denote tx(i - 1) or try(i) for some t and r in the interval [0,1]. Then by (1) and (2) we have (keeping in mind d;;:: 1 ;;:: v),

IIJ = J~lla« i-d + i2(a-ao+r+v)loe;5; oeo + s,


and for J = J~, y = y(i) we have

II JII .. ~ i-d + i2( .. -ao+r)+v-1 11X ~ 1X0 + s.

These estimates imply

II JJIL. « II JII .. * II JII .. « i- 2d + i2"-YIIX ~ 1X0 + s, where y is the minimum of the following three numbers

d + 2(0"0 - r - v),

d + 2(0"0 - r) -'- v + 1

and

2(20"0 - 2r - v) - v + 1.

Next, with (2), (4*) and with the estimate for J above we come to the following estimates for J = J~(i-l) + J and for F'(1),

IIF'(1)II .. « IIJII .. « 1 + i 2( .. -ao+r+v)11X ~ 1X0 + s,

and since Ile(i)II .. « IIF'(J)JJII .. « II J II .. * II JJII .. , we have

Ile(i)ll .. « i- 2d + i2"-Y'11X ~ 1X0 + s, where y' = min(y, y + 2(0"0 - r - v),2(d + 0"0 - r - v».

Finally, we recall that 0"0 ~ 2r + s + 3v, and that d ~ 2(8" + v + 1) [see (*)], and get y' = y > 2(0"0 + s + v) + 1; hence,

(6)

Estimates for E*(i) and for g'(i). The estimate (6) implies

i-I

IIE(i - 1) = L e(j)ll .. « 1 + i 2( .. -ao-0)-vl lX ~ 1X0 + s, 1

and with (1) and (2) of 2.3.4 we estimate e(i) = 8j (e(i» and £(i - 1) = (8j -

8j _ 1 )E(i - 1) as follows

lIe(i)lI .. « i-2d + i 2( .. -ao-0-v')-111X « 1,

where v' = min(v,lXo - 0"0) ~ 0, and

11£(i - 1)11 .. « i- 2d- 1 + i 2( .. -ao-0)-v'-111X « 1.

Finally we get

(7*) IIE*(i) = e(i) + .E(i - 1)11 .. « i- 2d + i2(a-ao-0)-v'-111X « 1,

where v' = min(v,lXo - 0"0) ~ 0, and with (00) we conclude

(7') II g'(i) = g(i) + E*(i)ll .. « i- 2d + i 2( .. -ao-o)-111X « 1,

Estimate for y(i + 1) = M(x(i), g'(i». We came full circle. The inequality (B) allows us to use (4) and with (5) we derive from (2) and (7'), the required estimate (1'),

Ily(i + 1)lla« Ilx(i)II:+dllg'(i)lla+s« i-iii + i 2a-"lo:::;; 110 + r + s, where a = min(2ao + 1,4ao - 2d - 2v + 1,2(d - d + ao - v)) = 2ao + 1.

137

Proof of the Main Theoren of 2.3.2. Since Nash's process is local (see 2.3.3), its behavior at a given point v E V does not change if we localize g in the ball around v ofradius p ~ 2 by taking Pg in place of g, where P is a COO-function with support in K and PIB1 (v) == 1. Now, we can use the corollary of our lemma to obtain infinite Nash's processes x(i) for all directions (xo, g) provided Xo E d"o+r+s and g is C"o+s_ small. This amounts to the neighborhood property for ~-1 (xo, g) = x( (0), since the limit x( (0) does exist according to the estimate (1'). Furthermore, according to (6) the "total error" sequence E(i) also converges, and so the inversion property holds as well (see 2.3.3).

Regularity and Continuity of ~-1 = x( (0). In addition, let us assume that Xo is C~' +r+s-smooth and g is C'" +s-smooth for a 0 ::;; a 1 ::;; 11 l' Then, starting with the estimate (1'), namely,

Ily(i)lla« Qo(i,d,o: - ao)lo:::;; 110 + r + s, and by applying the lemma several times with a 1 and 11 1 instead of a 0 and 110' (i.e. with the estimates Ilxo II~, +r+s « 1, and Ilgll", +s « 1 in place of (0) and (00) we come up with the following stronger estimate for Ily(i)lla,

(1 1) lIy(i)lIa« Qo(O: - ad = i-iii + i 2(a-",)-1Io:::;; 111 + r + s.

With this (1 1 ) we get II x(i) - x( (0) II fl -+ 0;_00, for all f3 < a 1, and for a non-integer a 1 we also conclude that x( (0) is CU'-smooth, (see Exercise 1 in 2.3.4). This settles the regularity of ~-1 for these a l' Notice also, that the continuity of ~-1: !JI~' +r+s,'" +s -+

f!{fl, f3 < a 1, is immediate from the continuity of every x(i) in Xo and g (see the initial continuity discussion at the beginning of this section), and from the uniformity of our estimates for lIy(i)lIa, and so for Ilx(i) - x(oo)lla as i -+ 00.

Now, let us assume a 1 to be an integer and let us prove C"'-convergence x(i) -+ x( (0) for 11 1 > a l' To do that, we rewrite y(i + 1) = x(i + 1) - x(i) as

y(i + 1) = M(x(i),g(i) + E*(i)) = M(x(i),g(i)) + M(x(i),E*(i)),

and observe that (1 1 ) implies the following estimate (7n in place of (7*) above,

(7n IIE*(i)lla« i- 2d + i 2(a-",-s)-1-v'lo:« 1,

where v' is a positive number due to the inequality 11 1 > a l' Furthermore, (11) implies

Ilx(i)lla« 1 + i 2(a-",+v)lo: ::;; '11 + r + s,

Ilx(i)lla« 1 + i 2(a-",+v)lo:::;; 111 + r + s + d,

Ilx(i)lla« i- 2d + i 2(a-",+v)lo:::;; '11 + r + s,

for x = x - x, and from (1d, and (2 1) we derive

Ilx(i) - x( 00 )11,. « i-2d+1 + i 2(a-al +V)la :::;;; '71 + r + s.

Observe, that these estimates hold for an arbitrary fixed v > O. Since IIM(x(i), E*(i))lla « llx(i)lla+d * IIE*(i)lla+.' we conclude as before that

IIM(x(i),E*(i))lla« i-2d + i 2(a-a!l-1-v'la:::;;; '71 + r + s,

and with v:_> 0 the sum Lj;1 l\.1(x(i), E*(i)) Cal-converges. Thus, the Cal-convergence xCi) I::! x( (0) is reduced to Cal-convergence of the sum Lj;1 M(x(i), g(i)), where g(i) = (Si+1 - Si)g and, since Ilgllal+'« 1, we have

Let us show that the estimates (2d, (2d and ( ,..., ) imply the Cal-convergence of the sum Lj;1 M (x(i), g(i)). We do this by proving CO -convergence of the derivatives Li oal M as follows. We set J(i) = J~(i) and P(i) = Jzt:. Then M(x(i), g(i)) = F(J(i))]O(i) and the derivatives oal M expand into the sums ofthe following products Ilk,l(i) = okF(J(i))P(i), k + 1= 0'1' Since dis large, we have

II Ilk, I(i) II ° « II xCi) II k+d II g(i) 11.+1 « i-2d-1 + i2(I-al)-1 + i2(d+v-d)-1 + i2(d-al +v)-1

and with v < 1, we get the CO-convergence of all sums Lj;1 Ilk,l(i) for k ~ 1 and I < 0'1' Let us concentrate on the remaining sum Lj;o Ilo,al (i) = Lj;1 F(J(i))]al(i). Set J(oo) = J~(oo) and J(i) = J(i) - J(oo). Then we have,

- r1 d ~ F(J(i)) = F(J(oo)) + Jo dt F(J(oo) + tJ(i))dt.

The last derivative (compare with L') can be written as F'(J(oo) + tJ(i))J(i), and since IIx(oo)IIO'I-v« 1, we derive from (21) the following estimate for our integral

I1 = I1 :tF = I1 F'J(i),

II fJlo « i- 2d+1 + i2(d-a l+v),

and so the sum Lj;1 II(g)]al(i)llo converges. Finally we are left with the sum

00 00

L F(J(oo))·]al(i) = F(J(oo)) L ]al(i), i=O i=O

for ]al(i) = Ji(t:', g(i) = (Si+1 - Si)g. Since 9 is Cal+'-smooth the sums L~=1 g(i) = Sk+1 (g) cal +'-converge to 9 as k -+ 00 [see (7) of 2.3.4] and so the sums L~=1 ]al(i) converge in CO-topology. Q.E.D.

With this proof of the regularity we automatically have the required continuity of the inversion

for all 0'1 = S + 1, s + 2, ... , and for all real '71 > 0'1'

Exercise. Prove CUI-continuity (but not the CUI-convergence) of .@-1 for all real

0"1 < 1'11'

Hint. Apply the difference operators Ow of the form (owf)(v) = f(v + w) - f(v), to the derivatives oj M for j = ent(O" 1)'

2.3.6 The Modified Nash Process and Special Inversions of the Operator .@

We work in this section with a fixed smoothing in fl£, rather than with an d-stable family as in 2.3.5. We modify Nash's process of 2.3.3 by replacing x(i) in L * and in M by Xo + z(i), for z(i) = L~=1 y(j) and z(i) = Si(z(i)). This modification does not effect the basic properties of x(i) = Xo + z(i) and the limit x( (0), still serves as .@-1(xo,g). Furthermore, the condition Xo + z(i) E d guarantying the existence of our new

y(i + 1) = M(xo + z(i), g'(i)),

can be numerically expressed with R(x) (see the beginning of 2.3.5) by the inequality Ilz(i)lId < R-1 (xO)' and so the modified Nash process exists as long as this inequality holds. Moreover, now we are able to give an explicit bound for Ilgllul+s needed for the existence of the modified process x(i). This estimate depends on the data of the following two types.

(1) Fixed Data. First, we fix a partition of unity on V and a compact K c V, so that we have specific norms II Iia in the spaces fl£ and f§. Then, we have the smoothing operators Si in fl£ and in f§ of depth d, where

d 2:: 10(8 + 1), 8 = max(d,2r + s),

and we denote by C(S) the upper bound of the smoothing constants Ca , for o :s; rJ. :s; d, of the inequalities (1), (2) and (3) of 2.3.4, that is

C(S) = sup Ca' Os;aS;d

Finally, we fix a number 0"0 > S; we additionally assume 0"0 :s; 8 + 1 and we take v = 1(0"0 - 8).

(2) Variable Data. We characterize sections XEd by the following quantity

N(x) = Ilxllpo + 2R(x) + 1, for Po = 0"0 + r + s + d.

Then we need specific numerical bounds for the operators L~ = F'(J~)JJ and Mx = F(J~)J' (see 2.3.5). In order to avoid cumbersome notations, we assume that V = IRn and that the bundles X and G are trivial. In this case our F' and F are actual Euclidean functions. Indeed, we have x(r) = IRk for some k and F' maps this IRk = x(r) into another Euclidean space, namely to the total space of the bundle Hom(X(r) EB x(r), G) ~ V, and F maps the open set A c X(d) = IRb to the space Hom(G(S),X) = IRe.

Now, the norms of the partial derivatives of F and F', in the estimates of 2.3.5, have specific values.

We denote by IIFj(x)llo the CO-norm of the function IIJJ(J~(v))II, VE V, for the j-th order jet J~, and we define IIFj'(x)lIo as the CO-norm of IIJ~,(J~(v))ll. Then

with a given section xed we consider all sections ye~d such that Ilylld:$; min(1,!R-1(x» and set IIMllix) = supyllFj(x + y)llo. We also define IIL'lI j (x) = SUPllyllr:S: 1 IIFj'(x + y)llo·

Let us give an explicit estimate of y(i + 1) in terms of y(j) for j :$; i, and of the "norms" N(xo), IlL' II (xo), IIMII (xo) and Ilglla+s. We agree to write "const" for positive constants that depend only on our fixed data.

First, we recall Qo = i-id + i 2( .. -ao)-1, and we take a small positive number o > 0, such that

co

2C(S)<5 L Qo(i,a = d):$; min(1,!R-1 (xo»' i=1

This amounts to an inequality

o :$; const(R(xo) + 1)-1.

Then we consider the modified Nash process x(i) = Xo + z(i) = Xo + L}=1 y(j) directed by (xo, g) with a localized section g, and let

for i = 1, 2, ... , io.

We estimate z(i) and z(i) = z(i) - z(i) as in 2.3.5 and now we denote by u(i) either tz(i - 1) or t'ry(i) for O:$; t, 't :$; 1. Since r :$; d, we have, according to (*) above, Ilz(i - 1)lIr + Ilu(i)llr :$; 1 and so

IIFp(x(i - 1) + u(i) = Xo + z(i - 1) + u(i»llo :$; IIL'IIj/(xo)'

We also have [compare with (2) of 2.3.5]

Ilu(i)II .. :$; consto~(i,Q()IQ(:$; 0"0 + r + s, for ~ = i-2d + i2( .. -ao+v).

With the notation II II~) = II II .. * ... * II II .. (see 2.3.5) and with 1= s + r + 1 we "---y----J ,

have for Q( :$; 0"0 + r the following relation

11L'(x(i - 1) + u(i), u(i), y(i)ll ..

:$; const(IIL'II .. (xo»llx(i - 1) + u(i) + u(i)II~t * Ilu(i)II .. +r * Ily(i)II .. +r>

and so for I = s + r + 1 we obtain

Ile(i) = L'II .. :$; consto2(IIL'II,(xo»N'(xo)Q'(i,a)la:$; 0"0 + s, where Q' = i-2;; + i2( .. -ao-s-v)-1 [compare with (6) of2.3.5J. Then, as in (7*) of2.3.5, we get

IIE*(i)II .. :$; consto2 Cdxo)Q(i,Q()la:$; if,

where CL(xo) = IIL'II,(xo)N'(xo) for I = s + r + 1, and Q = i-2;; + i2:l!r-ao-s)-1.

Finally, under assumption Ilgllao+s :$; e, we obtain

Ilg'II .. :$; const(e + o2C(xo»Q(i,Q()IQ(:$; d,

and for y(i + 1) = M(xo + z(i), g'(i» we obtain the following estimate

(1') Ily(i + 1)11 .. :$; constCM(xO)(e + 02CL(XO»Qo(i, a)la :$; O"g + r + s,


where Qo is the old expression i- 2d + i2("- l1ol-1, and

for m = s + r + s + 1.

Observe, .that here we do need the section Xo to be of class C/o+r+s+d, rather than only C/o+r+s-smooth as in 2.3.5. Indeed the section Xo is not acted upon by smoothing operators anymore, and so estimates for Ily(i + 1)11" require Ilxo 11,,+d' However, now we have an advantage of a better control over the relation Ilgll l1o+s ::;; e(xo) needed for the existence of Nash's process. Namely, there is a positive number So > 0, that depends only on our fixed data, such that the inequalities

(0)

00

(0*) 2C(S)Sl/2 L Qo(i,d)::;; min(1,tR-1(xo», ;=1

imply the existence of the modified Nash's process x(i) such that

(1) Ily(i)li" ::;; !5Q(i, IX) I IX ::;; (10 + r + s, for!5 = Sl/2.

Indeed, under the condition (0), the inequality (1') becomes Ily(i + 1)11" ::;; eo!5 const Qo(i, IX), and for So ::;; conse1 we get our assertion by induction, as (0*) implies (*) above.

The inequalities (0) and (0*) determine a neighborhood [J4 = [J4Po, l1o+s c f!.(Po

X t:§l1o+s, for Po = «(10 + r + s) + d and the new operator ~-1 maps [J4 to d"°.

Furthermore, the argument of 2.3.5 applies to the new ~-1 and shows this ~-1 to be a continuous map

for Pi - (r + s + d) > (11 ~ (10'

So, our new ~-1 is somewhat less regular than the old one, but it behaves better in other respects and is easier to construct as well. Below we give three applications ofthe modified Nash process and ofthe new operator ~-1.

(A) Operators of Polynomial Growth. We say that a Coo-map F: A -+ ~c for A c ~b, has polynomial growth if all jets Ji, j = 0, 1, ... satisfy the following inequalities

II Ji(a) II ::;; Cj(llall + (dist(a, ~b\An-l + l)ki, aEA,

for some positive numbers Cj and kj • If A = ~b, these inequalities become

IIJHa)11 ::;; Cj (1 + Ilall)ki .

Now, for a manifold V with a partition of unity {UI" PI'} as in 2.3.4, differential operators N between vector bundles are given over each UI' = ~n, by maps of the corresponding jet spaces (who are Euclidean spaces) or of open subsets A in these spaces to Euclidean spaces. Then we put NI' = PI'N and we say that N has polynomial growth if the maps corresponding to NI' for all Jl. = 1, ... , have polynomial growth. Clearly, this definition does not depend on a particular choice of {UI" PI'} and on trivializations of vector bundles over UI"


Theorem. If the operator q; and its irifznitesimal inversion have polynomial growth, then for every compact set K c V there is a number k = 1, 2, ... , such that the inequality

Ilglluo+s::::; (2 + Ilxoll po + R(XOWk,

for /30 = (10 + r + s + d and for a given (10 > S; implies the existence of the modified Nash process yielding the (new) operator q;-1 (xo, g) over K.

Proof The polynomial gorwth of q; implies such a growth for the operator L' and the proposition above applies.

Observe, that many interesting operators have polynomial growth. For example, the metric inducing operator on maps x: V -+ IRq, q;(x) = f*(h), for h = L1 dXf, and its infinitesimal inversion (constructed in 2.3.2) have polynomial growth.

(B) The Initial Value (Cauchy) Problem. We call q;-1 a k-consistent inversion of q; along a closed subset Vo in V if the equality J:+s- 11 Vo = 0 implies J!-ll Vo = J!;ll Vo, for x = q;-1 (xo, g).

Let Vo c V be a codimension one COO -submanifold without boundary that divides V, let (10 > s and let k ~ 2r + s be an integer.

Theorem. There exists a k-consistent inversion q;-1: fJI -+ sI for

/30 = (10 + r + s + d,

that continuously maps every space

fJlPl ,u l +k+2s to slUl for /31 - (r + s + d) > (11 ~ (10'

and this q;-1 also enjoys all other properties [i.e. (2), (3) and (5)] of q;-1 of 2.3.2.

Proof Fix a COO -function cp on V which vanishes exactly on Vo and whose differential dcp does not vanish on Vo, and introduce, with the operators Si' new smoothing operators, called Sr, as follows,

for y E!!l;

Sr(g) = cpk+S Si( cp -k-s g) for g E~,

Next, we introduce new norms, Ilyll~ = Ilcp-kyll" in !!l and Ilgll~ = Ilcp-k-Sgll a in ~ for II II" = II Ila(K), and we observe that relative to these new norms the operators Sr satisfy the smoothing estimates (1), (2) and (3) of 2.3.4 as well as the locality and convergence properties (see 2.3.3). Furthermore, the operators M and L' satisfy, relative to the new norms, the same estimates as before. Indeed,

IIM(x, cpk+Sg)ll~ = IIM'P(x, g)lla

where M'P(x,g) = cp-kM(x,cpk-sg) is again a differential Coo-operator of the same order as M, and for k ~ 2r + s the corresponding operator (L')'P(x, u, y) = cp -k-s L' (X,cpk U, cpk y) is also a differential CQ -operator. Finally we denote by ~'P C ~

the space of sections of the form cpk+s g for all g E~, and we observe that a section h E ~k+s is contained in ~'P iff J~+s-ll Vo = O. Since the operator (~'P)"+k+s -+ ~", given

by g H cp -k-sg, is continuous for alIa, the estimate Ilg Ilao+k+2s « 1 for g E '§'" implies Ilgll~o+s « 1. Now we can see that the modified Nash process constructed with the new operators Sr in place of Si and directed by (xo, g) for Xo E !![Po and gE('§"'to+k+2s, satisfies the same estimate relative to the norms II II: we had before for the norms II Ila. Therefore for Xo Ef![Pl and g E('§"'t ' +k+2s we get Ilxo - x( (0) = D~l Yi II~, < 00, and in particular, J~;ll Vo = J~(~)I Vo·

Remark. The non-modified Nash process does not work with the operators Sr since the non-modified u contains xo, which unlike z(i) does not v2.nish on Vo and so the corresponding (non-modified) operator L' is discontinuous in the norms II II"'.

Corollary. Let Xo E '§Po and g E ,§po-r be sections such that J;olVo = J!lVo for go = £C(xo) and for some 1 ~ Po - r. If Po> s + r + s + max(d,2r + 2s) and if 1 ~ s + 2r + 3s, then, for a non-integer Po, there is a section x~ E sro for a 0 = Po - r - s - max(d,2r + 2s) such that £C(x~)llDjl(Vo) = gllDjl(Vo) and J;&,+s-llVo = J;~+s-ll Vo. If Po is an integer, then such an x~ exists in all spaces sr for a < ao.

Proof The condition J!ol Vo = J!I V implies that the section g - go is C1-small on lDjl(Vo) C Vo and under our assumption there is a k-consistent operator £C-1(xo,g - go) for k = 2r + s.

Remark. If we drop the last requirement, J;,:+s-ll Vo = J;~+s-ll Vo, then with our old non-modified £C- 1 we get x~ under less restrictive conditions, namely for 1 > s + s and for Po > s + r + s. This x~ itself is of class Po - r - s, or anything less than that, if Po is an integer, [compare with (D) of 2.3.2].

Example [compare Jacobowitz (1974)]. For the isometric immersion problem we have r = 1, s = 0, d = s = 2, so that the corollary applies to free maps Xo of class CPo, Po > 5, which are infinitesimally isometric of order 4 along Vo. That is, the original metric g in V and the induced metric agree along Vo with the derivatives of order :0;; 4. The corollary allows us to construct a new map, x~ of class 130 - 3, isometric on lDjl(Vo) such that J';ol Vo = J';ol Vo·

Question. Is it possible for metrices g of class COO to extend Xo from Vo to an isometric immersion lDjl(Vo) --+ ~q without this unpleasant loss of three (and 3 + B for an integer Po) degrees of smoothness?

Exercise 1. Prove the theorem and the corollary for an arbitrary Coo-submanifold Vo c V without boundary, and also analyze the case, when Vo has a boundary, for example co dim Vo = O.

(C) Global Analytic Inversions. Let us describe the "variable data", discussed earlier, in slightly different term. Let Po = s + r + s + d + 2 and let us lift the set A c X(d)

to X(Po) by taking the pullback Ao = (p~or1(A) C X(Po). The "functions" F and F' are also lifted to Ao and denoted by F and F~ respectively. (In fact F~ is defined everywhere on X(Po).) Now, for the same reason as before, we assume V = ~n, so that

Fo and Fe are actual functions on the domain Ao in the Euclidean space IRm = X<Po).

We put R-1(a) = dist(a, IRm\Ao), aeAo, and let J and J' denote the jets of order Po of the functions Fo and Fe respectively. Then, with a point aeAo, we consider all those points beAo for which dist(a, b) = Iia - bll =:; p = min(1,R-1(a)/2) and set

IIMII(a) = sup IIJ(b)ll, b

IlL' II (a) = sup IIJ'(b)ll, b

Iia - bll =:; p,

lIa - bll =:; p.

Finally, we denote by N(a) the sum Iiall + 2R(a) + 1, and set

Co(a) = (N(a) + Wllo(IIMII(a) + 2)2(IIL'II(a) + 2)2.

This huge "constant" is meant to dominate the constants CL ' and CM of the modified Nash process.

Now for the "fixed data" we take the analytic smoothing operators in V = IR" of 2.3.4 of the depth d ~ 10(s + 1) and we take 0"0 = S + 1 and k = 2r + s + 2.

Theorem. Let q>: Ao -+ IR be a positive real analytic function, such that q>(a)Co(a) =:; 80

where 80 is a sufficiently small positive number which depends only on the fixed data above. Then there exists an inversion .@-l(xo,g)definedforall those sections xo e .JiIllo and go e ~tto+', for which

IIJ;o+'(v)11 =:; (q>(J!B(v»)k+'+l, v e V,

and the section .@-l(xo,g) is Ctto-smooth. Furthermore, if the operators .@ and Mare real analytic (that is the maps L1: fI' -+ G and F on A c Xl') are real analytic), then the section .@-l(xo,g) is real analytic for real analytic pairs (xo, g). Finally, this operator .@-1 enjoys the normalization and inversion (properties) and also the following

Continuity. If

and

for Jo(v) = Jf~(v), then

IIJ;o(v) - J;p(v)11 =:; <>q>o(Jo(v» for all ve V,

Proof. We introduce new smoothing operators with the function q>o(v) = q>(J!~(v» by putting

for yefI,

and

forge~,

and we consider the modified Nash process with these smoothing operators. Then, with the norms Ilyll: = IIq>OkYI!'. and Ilgll: = IIq>Ok-s-l g ll"" we get all our old esti-


mates and thus the operator ~-l as well. Furthermore, for real analytic Xo and g, we obtain these estimates in all small obtuse neighborhoods (see 2.3.4) U c en of [Rn c en, since the real analytic operators M and L' satisfy in small neighborhoods U the same estimates as on V, and since the estimates for Si extend to obtuse neighborhoods U (see 2.3.4). Therefore, the Nash process converges in the norm II II~o (see 2.3.4), and in particular, the limit x(oo) = ~-l(xo,g) is real analytic.

Exercise 2. Generalize this theorem to an arbitrary Can-manifold V and show, that COO-solutions x of the equation ~(x) = 9 can be approximated in the fine COO-topology by can-solutions, provided the operators ~ and M and the section 9 are real analytic.

Hint. Imbed V --+ [Rq; first extend the functions from V to a normal neighborhood N(V) --+ V, and then make them zero outside N(V) c [Rq. Now, analytically smooth the extended functions and then restrict them back to V. This gives a "good" analytic smoothing on V.

Exercise 3. Let V be compact and let Xo and 9 be real analytic. Show that Nash's process with Si = Id (that is Newton's process, see 2.3.3) converges, when directed by (xo, eg) for small positive numbers e, and it defines ~-l(XO' eg).

Hint. Pass to a complexification ev and use Cauchy's inequality. [Also see Gromov-Rochlin (1970), Green-Jacobowitz (1971).J

Final Remarks and References. Nash's theory has been developed and generalized in several directions (see Schwartz 1960; Moser 1961; Clarke 1970; Jacobowitz 1972; Sergeraert 1972; Gromov 1972; H6rmander 1976; Hamilton 1982), but unfortunately, there is no single general theorem that would cover all interesting possibilities.

2.3.7 Infinite Dimensional Representations of the Group Diff( V)

A vector bundle G --+ V is called natural if the action of the group of COO -diffeomorphisms, Diff(V) on V, lifts to fiberwise linear action on G.

Examples. Trivial bundles V x [Rn --+ V with the obvious action of Diff(V) are natural. Invariant subbundles and factor bundles of natural bundles are natural. The bundle Hom(G1 , G2 ) for two natural bundles G1 and G2 is natural. The jet bundles G(r) --+ V, r = 1,2, ... , for natural G are natural.

All classical bundles oftensors on V, for example, the tangent and the cotangent bundles of V come from these constructions, but there are some "non-classical" examples as well.

Exercise 1. Construct a continuum of one dimensional natural bundles over V = [Rl.


Hint. Use one-dimensional linear representations of the structural group of the tangent bundle T(~l).

The group Diff(V) acts by linear transformations on the spaces of sections of natural bundles, and such an action is called (algebraically) irreducible if there is no non-trivial invariant subspaces.

Exercise 2. Consider the action of Diff( V) on the space COO (V) of Coo -functions V -+ ~

and prove the following five facts (a) If V is a compact connected manifold without boundary then the only

non-trivial invariant subspace consists of constant functions. (b) If V = ~n, n ~ 2, then there are exactly six (non-trivial) invariant subspaces,

and for n = 1 there are ten of them. (c) If V has a non-empty boundary then there are uncountably many invariant

subspaces. However, if V is compact, then there are only countably many spaces of Can-functions invariant under Diffan(v).

(d) The number of invariant subspaces is finite iff V has no boundary and it has only finitely many ends.

(e) The action of Diff(V) on COO (V) has a cyclic vector [that is a vector in coo(V) whose orbit spans the whole space Coo(V)] iff there is no discrete infinite orbits of the action of Diff(V) on V and the number of compact orbits of this section on Vis finite.

Now, look at the space f§OO of quadratic differential Coo-forms on V and at the (Diff)-invariant subspace f§0' of the forms with compact supports. Denote by Diffo the subgroup in Diff(V) of those Coo-diffeomorphisms, that are fixed outside compact sets K c V.

Lemma. Let 9 E f§OO be a positively definite form (that is a Riemannian metric on V). Then the span A of the Diffo-orbit of 9 contains the space f§0'.

Proof Take a form go E f§0' and a small neighborhood U in V of its support supp(go) c V. Then, for m = 1,2, ... , we consider manifolds (W,h) = XT(V,g) and let x: V -+ W be the diagonal imbedding. If we take a small perturbation Xo of x which equals x outside U, then the induced form x~(h) on V is contained in A, since the projections Pj a Xo: V -+ Vm , j = 1, ... , m, are diffeomorphisms in Diffo and x~(h) = Lj!.1 (Pj a xo)*(g). Now, for a large number m, generic maps Xo are free on U (see 1.3.2) and by the implicit function theorem of 2.3.2 there is a small number A. =F 0, such that the form x~(h) + A.g can be induced by a Coo-map Xl: V -+ (W, h) that is C1-close to x~ and that agrees with Xo outside U. Therefore,

m

go = A.-1 L «PjOx~)(g) - (Pjox~)(g»EA. j=l

Theorem. If V is a compact connected manifold without boundary then the action of Diff(V) in the space of quadratic differential Coo-forms on V is (algebraically) irreducible. [See (D) of 2.3.8 for generalizations.]

Proof Take an arbitrary form g E <'§OO such that g(vo) f:. 0 for some point Vo E V and first consider the diffeomorphisms that keep this point fixed. Then we obtain an action on the tangent space T"o(V) = [Rn, that amounts to the standard action of the linear group GLn on [Rn. Since the induced action of GLn on the space of quadratic polynomials on [Rn is irreducible, the span A of the (Diff)-orbit of g contains a form which is positive at Vo and therefore, positive in a small neighborhood Vo c V of va. Then, by the lemma, all forms with supports in Vo are contained in A. Finally, we cover V by diffeomorphic images of Vo and with a partition of unity we put all forms into A.

Exercise 3. Let the group Diff(V) be transitive on a non-compact manifold V and let dim(V) ~ 2. Show that the only non-trivial Diff(V)-invariant subspace in <'§OO consists of forms with compact supports. Analyze the case of dim(V) = 1.

Warning. If a form go E <'§OO has infinite support, then, in the space <'§OO with the fine COO-topology, the convergence A --+ 0 does not imply Ago --+ O.

Example. Consider the differential operator that sends COO-functions f: V --+ [Rl to the forms (dff E <'§OO induced by f from the standard quadratic form (dxf on [Rl.

This operator commutes with the action of Diff(V) and the span of its image consists exactly of those forms in <'§OO which can be induced by maps into pseudo-Euclidean spaces,

V --+ ( [Rq"q2, h = ~(dXi)2 - ~(dXY). I t follows, that every Coo -form g can be induced from some pseudo-Euclidean space. This result also can be obtained by applying the implicit function theorem to isotropic maps x: V --+ [RQ"q2 for which x*(h) == O.

(Green 1970). If there is a free COO-smooth isotropic map X o of a compact manifold V into [RQl,Q2, then any given Coo-form g on V can be induced by a Coo-map V --+ [RQ"Q2.

Indeed, Ag --+ 0 = x6(h) as A --+ 0, and for some A> 0 one can induce Ag by a map x that is close to xo, and so g is realized by the map A -2 x.

There is an easy way to get free isotropic maps V --+ [RQ"Q2 for ql = q2 = q = tn(n + 5), n = dim(V), by taking two copies of the same free map, one of V to [RQ,O

and the other to [R0,Q. But isometric imbeddings V --+ [Rq,q can be obtained for so large a q even without the implicit function theorem. Namely, for a form g on V and for an arbitrary free map xo: V --+ [RQ we first solve the linearized equation (L) of 2.3.1

(L) L(xo,y) = g,

and then we observe (see 2.3.1) that the form (xo + y)* induced by the map (xo + y): V --+ [RQ satisfies the following identity,

(xo + y)* = X6 + y* + L(xo,y) = g + X6 + y*.


Now, to realize a given form go by a map V --+ ~q,O Ee ~O,q, we solve L with g = go - x~ and get (xo + y)* = go + y*, so that the map (xo + y,y): V--+ ~q,O Ee ~O,q induces the form (xo,y)* - y* = go.

Notice that this algebraic construction works for C2-forms go, while the implicit function theorem only applies to CO"-forms for (J > 2. However, this theorem applies for smaller ql and q2' where (known) algebraic constructions fail. Namely, free isotropic maps exist locally iff ql ~ n, q2 ~ nand ql + q2 ~ n(n + 5)/2, (see 3.3.4). Then the theorem of Hirsch yields free isotropic maps of all open parallelizable manifolds into these spaces ~qloq2; It follows that if such a Vis compact, then every CO"-form, (J> 2, on V can be induced by a free CO"-map V --+ ~qloq2 for ql + q2 =

n(n + 5)/2 and min(q 1, q2) ~ n. In the general case with no parallelizability condition, the sheaf theory of 2.2 leads to a slightly less precise result, namely to the existence of immersions V --+ ~qloq2 with ql + q2 = !(n2 + 7n + 2) for min(ql' q2) ~ 2n - 1. (See 3.3.)

Positive Forms. Consider in 'DOO the subspace of positive definite forms. These constitute a convex open cone, 'Dr: c 'Doo , invariant under the action of DitI(V). We shall see later, in 3.1.3, that there is no nontrivial convex Diff(V)-invariant subcone in 'Dr:, and, in particular, all positive forms g on V are induced by maps x: V --+ ~q,

for q large, that is, g = Lf=l (dxY, for (Xl"'" xq ) = x. The reduction to the implicit function theorem is more complicated when it comes to maps to ~q (rather than maps V --+ ~qloq2, min(ql' q2) ~ n) and there is no purely algebraic construction of isometric immersions V --+ ~q whatsoever.

Exercise 4. Show, that there is no algebraic immersion of the hyperbolic plane H2 to ~q for any q.

Hint. The Poincare metric is given by rational functions and the existence of an algebraic isometric immersion H2 --+ ~q would imply polynomial growth of (the area of) the balls B(p) c H2 for p --+ 00.

2.3.8 Algebraic Solution of Differential Equations

We show in this section that many under-determined systems of linear P.D.E. can be solved by purely algebraic manipulations and we obtain in particular, the infinitesimal invertibility for generic "under-determined" non-linear differentia operators. (Compare Nash's theorem in 2.3.1.)

We denote by q and q' respectively the dimensions of our COO-smooth vector bundles X and G over V. We use the same symbol, say L, for linear differential operators, L: f!('+1Z --+ 'DIZ, and for the corresponding vector bundle homomorphisms, L: Xl') --+ G, so that L(x) = L(J~) for x: V --+ X. We call the operator L and the system L(x) = g under-determined if q > q'. An inversion of Lis, by definition, a linear differential operator M: 'DS+IZ --+ f!(1Z, such that Lo M = Id. Observe that the existence of such an M is a purely local problem. Indeed, if some operators M" invert L over neighborhoods U" c V, then, with a partition of unity {P,,}, we set M(g) = L" M,.(P"g) and obtain for all sections g: V --+ X


(LoM)(g) = ~)LoMI')(Pl'g) = LPl'g = g. I' I'

Now, we assume V = /R" and we start with the following example.

(A) Operators with Constant Coefficients. Let first q' = 1. Then the operators L send maps x = (Xl"'" Xq): /R" --. /Rq to functions g: /R" --. /R, g = L(x) = L1=1 LiXj), and one represents such an L = (L l , ... , Lq) by q (characteristic) polynomials, i l , ... , i q , on /R". If an inversion M = (Ml, ... ,Mq), M(g) = (Ml(g), ... ,Mq(g)), also has constant coefficients and is represented by polynomials All' ... , Alq, then the relation L 0 M = Id turns into the identity

q ~.~ L Lj 1V1j = 1. j=l

This identity says that the ideal generated by the polynpmials i j equals the whole ring of the polynomials on /R", that is the (non-linear) algebraic system

il(w) = 0

iq(w) = 0

has no solutions WEe"::::> /R". Now, for generic polynomials i j, this system is solvable iff q ~ n and so generic operators L with constant coefficients are invertible for q ~ n + 1.

Example. Let n = 1 and q = 2. Take Ll(X) = alx + bl(dx/dt) + (d2x/dt2) and L2(x) = a2x + (dx/dt) for some real constants al , bl , a2 and for tE /R l . Now, the system (*) reduces to the following two equations in the unknown WEe!,

a l + bl W + w2 = 0,

a2 + W = O.

These equations have a common solution iff A = al - bl a2 + a~ = 0 and so the genericity condition above amounts to A =F O. Then for Ml(g) = A-lg and for M2(g) = A-l (a2 - bdg - A-1(dg/dt) we have L1 oMl + L2 oM2 = Id, that is L1 (M1 (g)) + L2(M2(g)) = g for all g.

Now, in the general case of q' ~ 1, an operator L of order r is given by a matrix with polynomial entries, (iij)' i = 1, ... , q',j = 1, ... , q and this matrix is invertible by another such matrix iff rank (iij(w)) = q' for all WEe". It follows that generic operators with constant coefficients are invertible iff q ~ q' + n.

Exercise. Show that if an operator L with constant coefficient is invertible by an operator with non-constant coefficients, then L is also invertible by an operator with constant coefficients.

Remark 1. The genericity condition above can be specified as follows. Recall, that an algebraic relation between some given real numbers 11 , ... , IN is, by definition,


a non-zero polynomial Q = Q(z 1, ... ,z N) with integer coefficients such that Q(ll, ... ,lN) = O. The numbers 11, ... , IN are called k-independent, if there is no relations Q whose degrees and the absolute values of the coefficients are bounded by k.

Now, each polynomial Lij of degree r is determined by its (n + r)!jn!r! coefficients and so we have N = qq'(n + r),ln!r! numbers 11 , ... , IN that determine the matrix (Lij), and the invertibility statement above holds if the genericity is understood as k-independence of these numbers for a sufficiently large k, for example for k = expexpN.

Remark 2. If q = q' then the invertibility of L amounts to the relation

Det(Lij(w)) = const # 0,

and so we have an infinite dimensional group of polynomial maps £: IW --+ SLq that correspond to invertible operators.

(B) Ordinary Differential Equations. Let L(x) = 9 be a first order system of ordinary differential equations, that is L = A + B(dldt), where A = A(t) and B = B(t) are matrix functions in t E IR, A, B: IR --+ Hom(lRq, IRq'). Then, with x = (Xl, ... , Xq) and 9 = (g 1, ... ,gq')' we express our system by

(1) dx

Ax + Bdt = g.

Next we introduce a new function h = (h 1 , ••• , hq,), we consider the following algebraic system,

(2) Bx= h,

and by differentiating (2) we obtain

dx dh B-=--B'x

dt dt (2') ~or B' = dB(t)

dt .

Now, the joint system (1) + (2) can be written as

Bx= h

(3) (A - B')x = 9 - ~~.

Example. Take the following single equation

dX 1 dX2 Tt + b(t)Tt = 9 = g(t).

The system (3) in this case reads

Xl + bX2 = h

_b'X2=g_dh dt'

for b' = db(t) dt .


and so the functions

Xl = b(bTl (g - ~~) + h

X 2 = (b'rl G~ -g)

151

satisfy our equation for all h = h(t). This gives a regular solution (Xl (t), x 2(t)) as long as the derivative b'(t) does not vanish. Furthermore, if b'(t) has some zeros of finite multiplicity, then in order to have x l (t) and x 2 (t) regular, we choose the function

. ~W . h(t) such that the dlfferenceTt - g(t) has the same multIple zeros as b'(t).ltfollows

dx l (t) dx 2 (t) that the operator L: (x l (t); x 2 (t)) H-d- + b(t)-- is invertible if the zeros of the

t dt derivative b'(t) have finite multiplicities. In particular, L is invertible if b(t) is a nonconstant real analytic function or if it is a generic Cn-function.

Now, we return to the system (3) and we suppose that the matrix (A ~ B') (t) has constant rank = min(q,2q'). Then we express all linear relations between the rows of this matrix by another matrix R = R(t) with 2q' columns and with at most 2q' - q rows. Thus we reduce the system (3) to the consistency condition expressed by the system

(3')

In particular, for q ;;::: 2q', the system (3') contains no equations at all, and so this process delivers a zero order operator M for which LaM = Id, for L = A + B(d/dt), as in the example above. In the general under-determined case, for q > q', the system (3') contains less equations than the original system (1) and so by repeating this elimination process several times we again produce an operator

d dS

M = MO + Ml_ + ... + M S _

dt dt S '

for some s s ~ - 1,

such that La M = Id. The coefficients Mi = Mi(t) of this operator are q x q' matrices whose entries are rational functions in the entries aij and bij the matrices A = (aij(t)) and B = (bij(t)) and in the derivatives of these entries of order s q/2.

Exercise. Show that the common denominator of these rational functions, A =

A (aij' bij' a;j, ... ), is not the identically zero polynomial. Prove, that for generic Cn-functions aij(t) and bij(t) the function A(t) = A(aij(t), ... ) has zeros of finite multiplicity and show in the generic case that the under-determined operator L can be inverted by an operator M with COO-coefficients.

Hint. Use the same argument as in the example above. Also see section (E) where a more general case is studied.


(C) Lie Equations. Let G -+ V be a vector bundle with a flat connection. Then the covariant derivative VLx equals the Lie derivative Lx for all vector fields L on V and all sections x: V -+ G. We say that a system of vector fields, (L l , .•. ,Lp), is large if there are some Coo -functions r l' ... , r p on V for which If rjLj = ° and such that the function If Ljrj does not vanish.

000 Example. Let V = IR", and let Ll = -;-, ... , L" =~, L"+l = Ii Ui-;-. Then the

uU I u~ u~

system (L I , ... , L,,+tl is large.

Exercise. Show that generic systems (L I , ... ,Lp) are large for p ~ 2n + 1.

Now, we consider the following P.D.E. system in unknowns Xj: V -+ G, j = 1, ... ,p,

and we observe that if If rjLj = 0, then the sections Xj = rjx for x = (If Ljrjflg satisfy (*). Therefore the operator L = EBf=1 Lj: (x I, ... , Xp) H If Ljxj is invertible for large systems of vector fields Lj.

Exercise. Generalize this to the operators VL . for an arbitrary (non-flat) connection J

Vin G.

Denote by .P the Lie algebra generated by the fields Lj • Then, for any vector fields L. E.P, v = 1, ... , N, solutions of the equation

N

"LL.xv = g I

yield solutions of (*). For example, if [LI,L2JX = g then the tensors Xl = !L2 x and X2 = -!LIX satisfy the equation LIXI + L2x2 = g.

It follows, that if the Lie algebra .P is large, i.e. if it contains a large system of vector fields, (L I , ... , Lv> ... ,LN ), LvE.P, then the operator L = EBf L j is invertible.

Example. Let V = 1R2, and take LI = O/OU1' and L2 = (u l + U2)(O/OU2)' Then the system (L l ,L2, [L l ,L2J = O/OU2) is large, while the original system (L l ,L2) is not large.

Exercise. Show that two generic fields Ll and L2 on an arbitrary manifold V generate a large Lie algebra, and so the system LlXl + L 2 x2 = g (containing q' = dim G equations and 2q' unknowns) is generically solvable.

(D) Inversions of Zero Order. Consider a differential operator L: X(1) -+ G, and express it in local coordinates UI , •.• , u" in V as

" ox L(x) = Ax + i~ Bi OUi '


where A and Bi are q' x q matrix functions in the coordinates U 1 , ... , Un for q' = dim G, q = dim X, and where x is a vector function with q components. Consider all those sections x: V ~ X for which

LUx) = fL(x),

for all Coo -functions f: V ~ R The condition (*) is expressed in the local coordinates by the following system of nq' algebraic equations in x,

(**) Bix=O, i=l, ... n.

Assume the matrix B ~ (::l to have constant rank, eall it " and

consider the (q - r)-dimensional subbundle in x, call it X* c X, whose sections x: V ~ X* are exactly the solutions of (*).

Exercise. Show for q ~ n(q' + 1) that generic matrices B have constant rank r = nq' and then dimX* = q - nq' ~ n.

The condition (*) implies that the operator L, restricted to sections V ~ X*, has order zero and so it is given by a homomorphism LIX* = L*: X* ~ G. Indeed, by differentiating the equations Bix = ° we get Bi(ox/ou;) = - B;x for B; = oBJoui and so, on the solutions x of (**), the operator L is given by the matrix A * = A - II B;, that is

Lx = A*x for x: V~X*.

We call the operator L free if the homomorphism L * is surjective, that is if the

matrix (:*) has rank r + q' everywhere. If L is free then it is invertible, since the

system n ax

Ax + IBi-=g 1 aU i

reduces to the algebraic system

Bx =0

A*x = g.

(Compare with Nash's theorem of 2.3.1.)

Exercise. Show that generic operators L are free for q ~ nq' + n + q'.

Families of Operators. Denote by H ~ V the bundle Hom(X(l), G) and let yP be the space of Coo-sections V ~ H, that is the space of Coo-operators X(1) ~ G. Consider a linear subspace 2 c yP and take some operators Lv E 2, v = 1, ... , N,

n a Lv = Av + .L BiV~'

,=1 uU,

154

Let L = EEft Lv: (EEft X)(1) -t G, that is

N

L(Xl,···,XN ) = L Lv(xv), v=l


Denote by r(v), VE V, the maximal value of rank ii(v), for ii = (B iv ), over all choices of operators L 1, ••• , LN in 2 and over all N = 1, ... , and suppose that r(v) does not depend on VE V, that is r(v) = r = r(2) ~ nq'.

Exercise. Show, that one needs at most N = r + n ~ n(q' + 1) operators Lv to achieve this maximal value r at every point v E V.

Let us call the subspace 2 large if there are some operators Lv E 2, v = 1, ... , N, such that L = EEft Lv is a free operator.

Example. With a fixed operator L: X(1) -t G one gets a linear space of operators I acting on functions f: V -t ~, by taking l(f) = L(fx) for all sections x: V -t X. This space is large iff L is a free operator.

Remark. Observe, that the "large" property depends only on the I-jets of sections in 2, namely on the span of the one jets J1: V -t H(l) for all L E 2. In particular, if these jets span the whole bundle H(l) -t V, then the space 2 is large.

Exercise. Show that a generic subspace 2 c Je of dimension N ;;::: q'n + q' + n is large.

More on Natural Bundles. Let G -t V be a tensor bundle and let X -t V be the tangent bundle T(V). For a fixed tensor g E G, the Lie derivative x(g) = ogjox, for tangent fields x: V -t X, defines a linear first order operator L g : X(l) -t G, namely Ly(x) = x(g). These operators L g for all g: V -t G form a linear subspace 2 in Je. We call the tensor bundle G large if this space 2 is large.

Exercise. Show that the exterior powers of degrees < dim V and the symmetric powers of the cotangent bundle T*(V) are large bundles. (Probably, all tensor bundles are large with a few obvious exceptions.)

Now we take a COO-section g: V -t G and ask ourselves under what condition the span ~' = ~'(g) c ~oo of the Diff(V) orbit of g is equal to the whole space ~oo, that is the space of COO-sections V -t G. First we observe the following obvious necessary condition that is reminiscent of the h-principle.

If~' = ~oo, then, for each r = 0, 1, ... , the jets J;,: V -t G(r), for all g' E~', span the bundle G(r).

Theorem. If V is a compact manifold without boundary, if the bundle G -t V is large, and if the 2-jets f;,: V -t G(2), for all g' E~' = ~'(g), span the bundle G(2) -t V, then ~' = ~oo.

Proof Let 2' denote the linear subspace in 2 of the operators L g ,: X(l) -t G for all g' E ~'. The correspondence g H Lg is a first order operator, G(l) -t H, and since the


jets J;,: V --+ G(2) for g' E '§' span the bundle G(2) the I-jets of sections L': V --+ H for all L' E,2' span the same subbundle in H(l) as the I-jets of all sections L E,2 ::::l ,2',

Now, as ,2 is a large space, we conclude with the above remark that ,2 is also large, It follows, that the operator L = EB~ L g, is free for sufficiently many sections gl' . ." g., ... , gN of the form gv = dv(g), where dvEDiff(V) are generic diffeomorphisms. Therefore, by solving the system (***) above we invert all these operators L by some zero order operators M whose coefficients smoothly depend on diffeomorphisms dv and on their derivatives.

Finally, define a first order non-linear operator ~ by ~(dl"'" dJ = I~ gv = L~ dv(g), The linearization of ~ at (d l"." dN), dv E Diff(V), is exactly the operator L, and so the inversions M for the linearizations near (d l , .. " dN) yield an infinitesimal inversion of ~. Hence, the operator ~ has open range, and so '§' = ,§OO,

(compare with 2.3.7).

Exercises. Prove that the action of Diff(V) in the space of symmetric differential forms of degree k ~ 2 is (algebraically) irreducible.

Let '§OO be the space of exterior Coo-forms on V of degree k ~ O. Show that every proper (i= '§OO) Diff(V)-invariant subspace in '§OO consists of closed k-forms. Establish a one-to-one correspondence between invariant subs paces in '§OO and invariant subspaces in the cohomology group Hk(V, IR),

Question. How does one classify invariant subspaces of sections of an arbitrary natural bundle G --+ V?

(E) Generic Under-determined Operators. We now denote by H the bundle Hom(X(r), G) for some r ~ 0 and we consider its Coo-sections that are differential operators L: x(r) --+ G. Then we take an open set (relation) A c H(r+s) for some s ~ 0, we denote by ,2 = ,2(A) the space of COO-solutions L: V --+ H of A and we consider differential Coo-operators M = M(L, g), M: ,2 x '§oo --+ :roo, that have order s + r in L and that are linear of order s in g. Such an operator M is called a universal right inversion over A if for all L E !£' it satisfies

LoM(L, )=Id,

that is L(M(L, g)) = g for all Coo-sections g. One also defines universal left inversions for operators L: G(r) --+ X as differential operators M(L, x), of order s in L and in x, such that M(L, ) 0 L = Id for all solutions L of a given open relation in the space of s-jets of the bundle Hom(G(r), X),

Observe that right inversions solve equations L(x) = g, while with left inversions one gets the uniqueness of solutions for the homogeneous equations L(g) = O.

If one has two universal right inversions, MI over A I C H(r+s) and M2 over A2 c H(r+s), then with a partition of unity in A = Al U A2 one constructs an inversion over A, and so there is a unique maximal open set A = A(r + s) c H(r+s) over which universal inversions exist. The same conclusion holds true for universal left inversions.

Example. If r = 0, then differential operators are homomorphisms L: X --+ G and they are universally right invertible over the set A(O) c Hom(X, G) of surjective


homomorphisms. The complement to this A(O), called E* = H\A(O), is a stratified set of codimension q - q' + 1 for q = dim X and q' = dim G, and so the set A (0) is dense as well as open for q ~ q'. Futhermore, for q ~ q' + n we have codim E* > n = dim V, and so generic homomorphisms X -+ G are right invertible.

Let now r ~ 1. Then the operator L: x(r) -+ G is, definitely, not invertible, at the points where it is not surjective as a homomorphism, and so the complement E*(r + s) = H(r+')\A(r + s) must have codimension ~p - q' + 1, for p = dimX(r) = q«n + r)!jn!r!).

Exercise 1. Show that determined operators, (i.e. q = q') have no universal inversions at all. Moreover, generic determined operators of order r ~ 1 are not invertible. (Neither from the right nor from the left.)

Theorem. If q > q', then for all sufficiently large numbers s ~ so(q, q', r, n), the set A = A(r + s) c H(r+s), over which universal right inversions exist, have the following properties.

(i) A is dense as well as open; (ii) if r ~ 1, then the complement E* = E*(r + s) = H(r+s)\A has codimension at

least n + 1; (iii) both statements, (i) and (ii), hold true for universal left inversions of operators

G(r)-+x.

Remark. The statement (ii) sharpens (i) and we also shall see in the course of the proof that E* is, in fact, a stratified set. The inequality codim E* ~ n + 1 implies invertibility (from the right) of generic operators L: x(r) -+ G, since their jets, J£+': V -+ H(r+.), do not intersect E* (see 1.3.2).

Exercise 2. Let H -+ V be an arbitrary fibration and let Ho be Coo-submanifold in the jet space H(r). Show, that there is a non-empty open subset in Ho consisting of those points hE Ho, that can be represented by sections V -+ H whose r-jets V -+ H(r) meet Ho without tangency. For codimHo ~ n this is equivalent to the existence of a non-empty open set of sections V -+ H whose r-jets intersect Ho. If codim Ho > n then, for every such hEHo, there exists a section L: V -+ 11, such that JL(v) = h for some v E V, while the image of the differential T,,(v) -+ T,,(H(r» of this jet has only zero intersection with T,,(Ho) c T,,(H(r».

Let us indicate three corollaries of the theorem. [Compare Ritt (1950).] (1) Take some linear differential operators L 1 , ••• , Lk acting on functions V -+ IR

and consider the right ideal spanned by L10 ... , Lk in the ring of differential operators. If k ~ 2, then for generic operators L 1 , ••• , Lk this ideal is equal to the whole ring, and the same is true for the left ideal spanned by L 1 , ••• , Lk •

Remark. A single operator of positive order never spans the whole ring.

(2) A generic non-linear under-determined operator fJ}: f![r+a -+ t'§a of order r ~ 1, admits an infinitesimal inversion over an open dense set t'§ C f![2r+s for some sufficiently large s.


(3) Generic over-determined operators,!?J: '§'+« -+ f!£«, enjoy the following rigidity property.

For every x E f!£OO the set of solutions of the equation !?J(g) = x is discrete, that is there is no smooth non-constant families of solutions, gt E '§OO for t E [0, 1].

Observe, that for an arbitrary over-determined Coo-operator !?J, the equation !?J(g) = x is COO-unsolvable for generic sections XEf![OO.

The Proof of (i). We assume our fibrations to be trivial, X = ~q X ~n -+ ~n and G = ~q' x ~n -+ ~n, and we write operators L: Xl') -+ G as

L = L Ll(}l, 1119

where I = (il"'" in)' (}1 = (}1/(}uil ••• (}u!n, III = il + ... + in> and where Ll = (L!p) are q' x q matrix functions in the variables u1 , ... , Un' called the coefficients of L.

The coefficients of the composition P = L 0 M for M = LIJI~" M J (}J are bilinear functions Pff", for jl, v = 1, ... , q' and IKI ::;; r + s, in the coefficients L!p and in the derivatives (}IM:«; therefore, the relation L 0 M = Id is expressed by a system of partial differential equations in the unknowns M:«. On the other hand, the left inversion equation, MoL = Id for a given operator L: G(') -+ X, is expressed by a system of linear algebraic equations in the unknowns M;p, while the coefficients of these equations are linear combinations of the derivatives (}J L~«. This last algebraic system is much easier to handle and the following classical operation of conjugation shows that the solution of the P.D.E. system L 0 M = Id is equivalent to finding solutions M of the algebraic system Mol = Id, where M and I are the (formal adjoint) operators conjugate to M and L. Namely, the operator I = I(g) = I(gl, ... ,gq') for L = LU(}I, L = L(x) = L(x1 , ... ,xq), is given by

I(g) = L (_1)III(}I«LIYg), 1/19

where ( Y denotes the transposition of matrices, and M is defined in the same way. Observe that the functions I~«, for I = LIII~,I~«(}I, are combinations of partial derivatives (}1' L!p for II'I ::;; I JI, that is the operation L r-+ L is, in fac~ a linear partial differential operator, H(') -+ ii = Hom(G', X). On the other hand, I = Land L 0 M = Mol, so that the equations L 0 M = Id and Mol = Id = Id are indeed equivalent.

Now we concentrate on the left inversion equation, MoL = Id, and we write it as

(*) ~~~~.~ ....... ~ .~~~~~~~.~ .~~~~ ~~~.~ ~ ~ ~.~ .~~~~~~.1. ~.~~~~~~~.~.~ {

MOLo + ... + MJ(}JLO + ... + M"(}"Lo = fJ

M"L' = 0

where (}JLI, IJI::;; s, Ill::;; r, are given q x q' matrix functions on ~n, where M J are unknown q' x q matrix functions and where fJ denotes the diagonal q' x q' matrix with the unit diagonal entries. Observe, that the first line in (*) contains (q')2 equations, the second line represents n(q')2 equations and the last one, M"L' = 0, schematically expresses (q')2«n + r + s - l)!/(n - 1)!(r + s)!) equations.

There are (q')2((n + r + s)!jn!(r + s)!) equations in (*) altogether and the number of unknown functions M:p is equal to qq'((n + s)!jn!s!). The coefficients of these equations are linear combinations of the derivatives (Y L~~, and let us interpret these derivatives as independent variables, that parametrize the fibers of the fibration jj(s) = IRN X IRn ~ IRn for jj = Hom(G(r), X), where N = N(s) =

qq'((n + r)!(n + s)!j(n!fr!s!). Denote by .At the field of rational functions in the variables aJL~~.

Lemma 1. If s ~ n((qjq')l/r - Itl, then the system (*) admits a solution (M:fJ ), 0( = 1, ... , q', f3 = 1, ... , q, IJI ::;; s, in the field .At.

Remark. Since the coefficients of (*) are (linear) polynomials on IRN there is a uniquely defined Zariski open set A c IRN, such that (*) admits a solution by some COO-functions M: on A, and there is no such solution near the points of the complement r =P IRN\A. It follows that the set I* c H(r+s) (of the statement of the theorem) is indeed a stratified set which is algebraic over each coordinate neighborhood in V. The lemma implies that r is contained in the poles of the rational functions M:p on IRN that satisfy (*) and so the set I*, as well as r, must have positive codimension.

Proof of the Lemma. For every 0(0 = 1, ... , q' we take the row M:o = M~op in each of the matrices M:fJ and then the system (*) splits into q' independent systems

(*0(0) {.~~~~~ .~.::: .~ .. ~~o.~~~.o .. ~. ~.(~:.~~~ M!oL' = 0,

where <5(0(,0(0) equals one for 0( = 0(0 and is zero otherwise. There is only one non-homogeneous equation in this system, namely

M O L 0 + ... + M S oSLo = 1 £lalla CCo CCo '

where L~o denotes the column L~~o' and to prove the existence of a solution (M:) of (*0(0) in the field .At we must only show that the "non-homogeneous row,"

is not a combination with coefficients in .At of the rest of the rows, called "homogeneous rows," of the system (*0(0)' Observe, that the entries of these "homogeneous rows" are linear combinations with integer coefficients of the variables aJL~~ and aJL~~ for III > 0. Furthermore, a variable aJL~~, IJI = 0, ... s, appears with a nonzero coefficient in an entry of a "homogeneous row" only if this entry is contained in a column that follows the column containing this very aJL~~ in the "nonhomogeneous row". For example, the variable L?~o appears as the first entry in the "non-homogeneous row," but it does not appear anymore in the first column of the system (*0(0)'

Now, suppose that the "non-homogeneous row" is represented as a combination with some coefficients fl' ... , hE.At, k = q'[(n + s + r)!jn!(s + r)!] - 1, of the "homogeneous rows". Then every entry aJL~~o of the "non-homogeneous row" can

be expressed as a polynomial in the variablesfl' ... ,h and oJL~afor III> 0. Indeed, we start with the first column in (*0(0) and get L8ao as a linear combination of some L{a, for III > 0, with the coefficients fl, ... ,k Then we substitute L~ao' as it appears in the following columns, by this combination and express the second entry in the "non-homogeneous row" as a polynomial quadratic in fl' ... ,h and linear in some oJL~a for III> 0, and so on.

As the entries oJL~ao are independent variables, in %, their number, q((n + s)!jn!s!), can not exceed the number k of the "homogeneous rows" and so we must have

(n + s)! , (n + s + r)! q < q ,

n!s! n!(s + r)! but this inequality is incompatible with our assumption,

s> n[(q/q')l/, - lrl. Q.E.D.

The Proof of the Inequality codimE* > n. Take an operator L: Xl') - G, for r ~ 1 on V = IRn, namely L = LI/I,,;rUOI, and a COO-function P: IRn -IR. Let us expand L(P'x) according to Leibniz' formula and write

L(P'x) = L,(x) + PL'(x),

where L, and L' are differential operators in x whose coefficients are polynomials in L!p and 01 P. Observe, that L, is, in fact, a zero order operator that is a homomorphism L,: X - G, whose coefficients are linear functions in LI for III = rand polylinear of degree r in the first derivatives of P. For example, if P(ul , ... , un) = Ul' then L, = r!L1 for I = (r, 0, ... ,0). Let Vo c V denote the zero set of P and let us observe that over each point v E Vo the homomorphism L" that is the linear map called L,(v): Xv - Gv, depends only on the differential dP(v) and on U(v) for III = r. If dP(v) = 0, then L,(v) = 0, and for dP(v) ¥= ° we call the operator L transversal to the tangent space 'T,,(Vo) c 'T,,(V), or briefly, to Vo at v, if the map L,(v) is surjective. We say, otherwise, that L is tangent to Vo at Vo. This definition agrees with the usual notion of tangency of vectorfields, L = L/=lli(O/OUi), to Vo.

A rational function is called regular at some point if its denominator is not zero at this point. If we have a rational function F = FWf,J in the partial derivatives of some functions fll on IRn, we call F regular at v E IRn, if it is regular at (olfiv)).

The following lemma is an algebraic extract from the Cauchy-Kovalevskaya theorem.

Lemma 2. Suppose that L is transversal to Vo at v E Vo. Then, for every p = 0, 1, ... , there are two operators M: G(s) - X and M: G(s) - G, for some s = s(r, p), whose coefficients are regular at v rational functions in the derivatives oJLap and aJp for I J I ~ s, such that in a neighborhood U c IRn of v one has L 0 M + PPM = Id, and this identity holds for small perturbations of Land P.

Proof Let us expand L(P'+Px) as above and get

L(P'+Px) = pP L,p(x) + pp+l L~(x).

Observe, that Lrp = [(r + p)!/r!]Lr and so the homomorphism Lrp: X --+ G is invertible near v. Now, if we already have M and iff such that L 0 M + pp iff = Id, we take

M' = M + pr+p L -1 0 M rp ,

M' = -L' oL-1 oM p rp ,

and obtain

LoM' + Pp+1M = Id.

The proof is concluded with an obvious induction, starting with M = 0 and M = Id for p = o.

Let now Vo be a submanifold in V = ~n of codimension k > 0, and let us call Vo characteristic for L if L is tangent to all those hyperplanes in T,,(v) for all v E Yo, which contain the tangent spaces T,,(Vo) c T,,(v).

Lemma 3. Generic under-determined operators L have no characteristic submanifolds of codimension > o.

Proof First let Vo be given by the equations U1 = 0, ... , Uk = 0 and consider the k coefficients Llj, j = 1, ... , k at the derivatives alj = ar/auJ for j = 1, ... , k. If Vo is characteristic, then these coefficients have rank <q' for all points VE Yo. The conditions rank Llj(V) < q' for j = 1, ... , k define a subset of codimension k(q - q' + 1) in the fiber Hv c H = Hom(x<rl, G), and by taking derivatives along Vo we get for every s = 0, 1, ... , a subset Es E H~Sl of codimension c(s, k) =

k(q - q' + 1) (~ -_ \; ~)! , such that Es contains the s-jets of all those operators n .s.

L: V --+ H for which Vo is a characteristic submanifold. Next we consider all submanifolds that pass through VE V and that are given by equations Uj = .fj(Uk+1'

... , un),j = 1, ... k, and we observe that the space of(s + 1)-jets of these submanifolds, that is the space of(s + 1)-jets of the correspondingmapsf = (f1, ... ,fk): ~n-k --+ ~k,

has dimension d(s, k) = k e~ -=-:~: : 11~! -1). Finally, for the set E: c msl of the

jets of those operators L who have characteristic submanifolds that pass through the point v E V, we get

codim E: ~ c(s, k) - d(s, k),

and so for q' < q and for large s we get codim E: > n = dim V. Q.E.D.

Exercise 3. Show for all s ~ 0 that if q = 1, then

(n + r - 1)' codimE: = (q - q' + 1) ( _ 1)' , . = c(r),

n .r.

and prove that codim E: > c(r) for q' ~ 2 and n ~ 2. Prove that generic determined operators (q = q') have no characteristic submanifold of codimension ::;; 2 for q' ~ 2.


r

L = L L £I(8I t. m=O III=m

To prove our inequality codim I* > n, we first observe that Lemmas 1, 2, and 3 extend to differential operators on en with holomorphic coefficients. Now we work in the complex analytic context. The derivatives aJLI for IJI ::;; s, III ::;; r, are inter-

h . . N (n + r)!(n + s)! preted as t e coordmates m e ,for N = N(s) = qq' 2 ,and let us denote

(nO r!s! by &(s) the ideal in the ring of polynomials on eN that consists of those polynomials P = p(aJLI) for which the equation LoM = P Id has a solution M whose coefficients are polynomials in aJLI. Denote by I(s) c eN the zero set of this ideal and observe that the natural projection II;': eN(s') ~ eN(s) for s' > s, sends I(s') to I(s), and that according to Lemma 1 we have codim I(s) > 0 for a large. Let Io c I(s) be an irreducible component of I(s) such that codim Io = k ::;; n.

Lemma 4. For some s' > s the pullback I~ = I(s') n (II:')-l(Io) c eN(s') satisfies the inequality codim I~ ~ k + 1.

Proof Take some polynomials PI , ... , Pk that vanish on I(s) and such that their differentials are linearly independent at some non-singular point 0"0 E Io. Then we take a generic operator L on en with holomorphic coefficients LI such that for some point Vo E en the vector (aJLI)(vo E eN(s) is in Io close to 0"0' and such that the unduced functions

pf = Pf(v) = PI ((aJL!p)(v)), ... , pt = Pf(v) = Pk((aJL!p)(v)),

v E en, have independent differentials at the point Vo. Then the zero set of the function pf, ... , pt, called IL(S) c en, is a non-singular variety near Vo E IL(S) and by Lemma 3 it is not characteristic. Therefore, there is a linear combination pL = Al Pf + ... + AkPt, such that the operator L is transversal to the zero set of pL, called Vo c en, at some point VI E IL(S) C Vo close to vo. Next, by Hilbert's Nullstellensatz, for some number p the polynomial pP = (AI PI + ... + AkPdP is contained in &(s) and so we have an operator Mo such that L 0 Mo = PPId. Finally, according to Lemma 2, we have L 0 M + PPM = Id, for some operators M and £1, and with P = pL, and so for M = M + Mo 0 £1 we get L 0 M = Id. The coefficients of this operator M, of some order s > s, are rationalfunctions in the derivatives aJu for IJI ::;; sand these functions are regular at VI EIL(S) c en. Since the point (aJU(Vl))EeN(S) projects to I o, the common denominator of these functions, call it Q, is not identically zero on the pullback to = (II:rl(Io). On the other hand, the operator M' defined by M'(g) = M(QS+l(g)) has for the coefficients some polynomials in aJLI for IJI = s' = 2s, and since L 0 M' = QS+lId we get QS+1 in the ideal &(s') and thus we prove our lemma.

Now, with this lemma we get codim I(s) > n for large s, so we have the same inequality for our set I* c H(s+r) and the proof of the theorem is completed.

162

Exercise 4. Show that

Hint. See Exercise 3.


(n + r - 1)! codimE* ~ (q - q' + 1) ( _ 1)' , .

n .r.

(E') Non-free Isometric Immersions. Let us return to the system (L) of 2.3.1 in the unknown map y: V -+ ~q,

(L) \:~;' :~) + \:~ :~) = g;j.

If the map x: V -+ ~q is free, then this system is algebraically solvable and so for q ~ [n(n + 1)/2] + n we have infinitesimalinvertibility of the metric inducing operator ~ over a non-empty open set of maps x: V -+ ~q. This system (L) is still underdetermined for [n(n + 1)/2] + n > q > n(n + 1)/2, but unfortunately, it is not "sufficiently generic" to use our general theorem.

Problem. Is the operator ~ infinitesimally invertible over a dense (or at least non-empty) open set of maps x: V -+ ~q for all q > n(n + 1)/2?

Let us indicate an approach to this problem for [n(n + 1)/2] + n ~ q ~ [n(n + 1)/2] + n - m, where m ~..jifi. We modify the system (N) of 2.3.1 by introducing new unknown functions hi: V -+ ~q, i = 1, ... n, and then we write

(Nh) \:~;,y) = hi'

Next, we differentiate this system and reduce (L) to

(N*h) / 02X ) 1 (Oh. oh· ) \ou;ou/ y = 2" OU; + ou~ - g;j ,

as in 2.3.1. The [n(n + 1)j2] + n linear algebraic equations (Nh) + (N*h), are solvable in y,

iff the right hand sides of these equation satisfy the consistency condition that is expressed for generic maps x by a system of P.O. equations in the unknowns hi' [compare with (B)]

(h) n oh

Ah+ LB;~=g, ;=1 uU;

where h = (hi" .. ,hn) and A and B; are some m x n matrix functions on V for m = q - [n(n + 1)/2] - n, whose entries are some rational functions in the derivatives ox/ou; and 02X / OU;OUj. These functions are regular as long as the derivatives span the whole space ~q, q = [n(n + 1)/2] + n - m. The system (h) contains m equations and so it is under-determined for m < n, but it fails to be generic since the n(n + 1)m entries of the matrices A and B; are expressed in the derivatives of q


functions, and there must be at least n(n + l)m - q differential relations between these entries. Now, if we "truncate" the system (h) by letting hm+2 = 0, ... , hn = 0, we get another system in the unknowns (hl, ... ,hm+l) = h,

(h) Ah + f 13ih = g, i=l

where the total number of the entries in A and 13i is (n + l)m(m + 1), and so for q = [n(n + 1)/2] + n - m ~ (n + l)m(m + 1) the system (h) has a fair chance to be generic for generic maps x: V --+ IRq.

Exercise. Apply the proof of our "generic theorem" to the system (h), for q ~ qo = [n(n + 1)/2] + n - .;;ii, and prove {0 to be infinitesimally invertible over generic maps x: V --+ IRq for q ~ qo.

(F) Completely Integrable Systems. A first order system of P.D.E. in unknowns (Xl, ... , xq ) = X: IRn --+ IRq is called complete if it can be written as

(0) c1i/l(X' u1 , ••• , un) = 0, Jl = 1, ... , m,

(1)

and so the I-jets of X are uniquely determined by its O-jets. The system (0) + (1) is called integrable if (oc1i/llox) 'P; + (oc1i/llou;) = 0 and if the identities o'P;(x, ... ,)/ouj = o'lj(x, ... ,)/OUi' i,j = 1, ... , n, hold for all solutions x of the system (0). Frobenius' theorem says in the integrable case that O-jets of solutions x of (0) uniquely extend to germs of solutions of the joint system (0) + (1).

The integrability condition may also be expressed by saying that the O-jets of solutions of (0) extend to 2-jets of infinitesimal solutions of(O) + (1). More generally, take a differential relation 99 c x(r) and suppose 99 is a Coo-submanifold in x(r). The completeness condition now says that the projection 99 --+ x(r-l) is an embedding, and we slightly relax this condition by allowing this projection to be an immersion. Then we consider the "lift" 991 c x(r+1) (compare 1.1.1) that consists of (r + I)-jets of germs of those sections V --+ X, whose r-jets V --+ x(r) are tangent to 99. To get a better picture, let us first call a tangent n-plane, • E TAx(r») for x E x(r) and n = dim V, holonomic if there exists a holonomic (see 1.1.1) section cp: V --+ x(r) tangent to ., that is • equals the image of the differential dvcp: T.,(V) --+ TAx(r») for v = pr(x) E V. Then the space x(r+1) is canonically isomorphic to the space of all holonomic n-planes in x(r) and 99 is equal to the space ofholonomic planes in T(99) c T(x(r»).

If the projection 99 --+ x(r-l) is an immersion, then at each point x E 99 there is at most one holonomic n-plane in -4(99), and we call such a relation 99 integrable if holonomic n-planes exist in all spaces 4(99), x E 99, or equivalently, if the map 991 --+ 99 is surjective as well as injective. In this case we get a smooth field of holonomic n-planes tangent to 99 and by Frobenius' theorem for every point x E 99 there is a unique germ of a holonomic jet cp: (Dft(v) --+ 99 for v = pr(x) E V, such that cp(v) = x, and so we have a local solution of 99, f: (Dft(v) --+ X for which J; = cp. Therefore, the whole manifold 99 is, in fact, foliated by holonomic germs cp that may


be interpreted as germs of leaves of this foliation; the global leaves project to V by immersions, and when such an immersion is a diffeomorphism we get a global solution of ~ over V.

There are two essential obstructions to the existence of global solutions: a continuation of a local solution goes to infinity in finite time, as for the equation dx(t)/dt = x2 (t) + 1, or solutions become multivalued over V for 7t1 (V) # 0, as for the equation dx(t)/dt = 1 for t E Sl = ~/Z.

Let us indicate several examples where global solutions of integrable relations ~ do exist.

(1) The projection pr: 9l ~ V is a diffeomorphism. Then the section (pT1: V ~ 9l is holonomic iff ~ is integrable, and so the integrability is sufficient for the existence of global solutions of 9l. Of course, there is only one solution in this case.

(2) Let X ~ V be a vector bundle and let ~ c x(r) be an affine subbundle of the bundle x(r) ~ V. Then, if V is simply connected, all local solutions of ~ extend to global ones.

(3) Let V and W be Riemannian manifolds and let X = V x W ~ V. Suppose that the projection 9l ~ X is a proper map and let all local solutions of ~ be short maps (see 1.2.2(B) (9jt(v) ~ W, v E V. If V is simply connected and W is complete, then all local solutions uniquely extend to global solutions V ~ W.

Now, let 9l c x(r) be a submanifold, that admits at every point x E 9l a unique tangent holonomic n-plane, but the projection ~ ~ x(r-1) is not necessarily injective. Take the lift ~1 C x(r+1) and observe that the projection 911 ~ ~ is a diffeomorphism, but the induced field of n-planes on ~1 may not be holonomic. Then we inductively define ~s = (9lS - 11)1 c x(r+s), assuming that ~s-l C x(r+s-1) is a sub-manifold, and we denote by 9ls c 9l the projection of 9lS back to~. If for some s ~ 2 we have 9ls = ~s-l then the relation 9lS - 1 is integrable as well as complete, so 9ls- 1

is also complete and integrable, and, if ~s is non-empty, we get local solutions of the relation 9l.-1 •

Example. Consider two Riemannian manifolds, (V, g) and (w, h) of dimension n, call a germ xo: (9jt(v) ~ W, for VE V, an infinitesimal isometry of order s if the induced metric go = x~(h) satisfies J;o(v) = J;(v), and denote by ~(s) c X(s+l), for X = V x W ~ V, the set of the (s + I)-jets of such germs for all v E V. Observe that ~(O) coincides with the isometric immersion condition, 9l(0) = 9l c X(l), that is the principle O(n)-fibration, 9l ~ X, whose fiber 9lx , for x = (v, w), consists of the isometric linear maps T,,(v) ~ Tw(W), Then we also have 9l(1) = ~1 and 911 = 9l, that is each isometric map 1: T,,(V) ~ Tw(W) extends to an infinitesimal isometry of order one. Futhermore 9l(2) = 9l2, but 912 c ~ may be not equal to ~. In fact, the map I extends to a second order infinitesimal isometry iff it establishes an isomorphism between the respective curvature tensors, of V in T.,(V) and of Win Tw(W), For example, if W = ~n, then the fibration 9l ~ V is a principal Is(~n)-fibration and the field of holonomic n-planes tangent to 9l is called the canonical affine conection of V. The identity 912 = 9l here amounts to the vanishing of the curvature of V and so, with Frobenius' theorem, we come to the following classical fact, V is locally isometric to ~n, iff its curvature is everywhere zero.

A Riemannian manifold V is called infinitesimally homogeneous of order so, if

for any two points VI and V2 in V there is an infinitesimal isometry (9jZ(v l ) ~ (9jZ(v2) of order So, sending VI to v2 • Observe that all manifolds have such homogeneity of order one, and that homogeneous manifolds are infinitesimally homogeneous for all So. If V is infinitesimally homogeneous of order So, then for all s :::;; So we have £Jl(s) = f7ts and all subsets f7ts c £Jl are subfibrations of the fibration £Jl ~ V. Futhermore, if the projection £Jl(so) ~ X = V x W is onto, then each subset £Jl(s), for s :::;; So, is also a subfibration ofthe fibration f7t ~ X whose fiber f7tx (s) c £Jlx = O(n), x E X, is a closed subgroup in O(n), called G(s) c o (n).

Exercise. Show that f7t(s) always contains f7ts but in general, f7t(s) ¢ f7ts for s ~ 3. Denote by b(n) the maximal length of descending chains of connected subgroups

in O(n).

Exercise. Show that b(2) = 2, b(3) = 3, b(4) = 5 and, in general, b(n) < !n.

Theorem. If for every two points, V E V and WE W, there is an infinitesimal isometry (9jZ(v) ~ (9jZ(w) of order So > b(n), which sends V to w, then (a) there is a local isometry that sends v to w. (b) If V is simply connected and if W is complete, then there is a global isometric immersion V ~ W that sends a given point v E V to a given point WE W

Proof Under our assumptions, the manifold V is infinitesimally homogeneous of order So and since So > b(n) we have, for some s < so, dim G(s + 1) = dim G(s) and so dim £Jls+I = dim f7ts. Therefore, the connected component of f7ts that contains some component of £Jls+I is integrable. We get (a) by applying Frobenius' theorem and for (b) we use (3) above.

Question. What is the minimal So for which the theorem holds? It is known that the theorem fails for So = 2 if n is large. (Ferus-Karcher-Munzner 1981.)

Exercise. Generalize (a) to pseudo-Riemannian manifolds and also generalize (b) under the additional assumption of W being compact and simply connected.

Corollary (Singer 1960). Complete simply connected Riemannian manifolds are homogeneous iff they are infinitesimally homogeneous of a sufficiently high order.

A pseudo-Riemannian manifold V is called weakly locally homogeneous if for any two points VI in V2 in V there are arbitrary small isometric neighborhoods, (9jZ(v l ) and (9jZ(V2). One calls V locally homogeneous if there is an isometry between (9jZ(vd and (9jZ(V2) which sends VI to V2.

Exercise. Show, that if a pseudo-Riemannian COO-manifold V is weakly locally homogeneous then it is infinitesimally homogeneous (of any given order) and so it is locally homogeneous.

Question. Does the last conclusion hold for C'-manifolds for some r :::;; b(n)?

Exercises. Show that compact simply connected locally homogeneous pseudoRiemannian C -manifolds are homogeneous for r ~ 4.

Hint. Show, with (2) above, that local Killing fields extend to global ones.

Show that complete simply connected locally homogeneous Riemannian CO_

manifolds are homogeneous.

Hint. See the book of Montgomery-Zippin (1955).

Question. What is the regularity condition under which the (weak) local homogeneity of a geometric structure on a compact simply connected manifold implies homogeneity?

(G) Generic Over-determined Systems. Consider a (non-linear) Coo-operator fi}:

!![,,+r ~ t§" and let AS: x(r+s) ~ G(s) denote the associated maps of jets. These maps are generically surjective in the (under) determined case, for dim X = q ~ q' = dim G, but now let q < q'. Then, for large s, we have dim x(r+s) = q«n + r + s)!/ n!(r + s)!) < q'«n + s)!/n!s!) = dim G(s) and so the map AS is far from being surjective. Therefore, there is a non-vacuous consistency condition for solvability of the equation .@(x) = g. Namely, the jet J;: V ~ G(s) must send V into the image of the map AS for all s. If we denote by 9l(g,s) c x(r+s) the pullback of the image J;(V) c G(r) under the map AS, that is 9l(g, s) = (ASr l (J;(V)), then this consistency condition can be equally expressed by saying that the projection 9l(g, s) ~ V is onto. Observe, that for every s' > s, the projection p:': x(r+s') ~ x(r+s) sends 9l(g, s') into 9l(g, s). Let us stabilize the conditions 9l(g, s) by intersecting the images p:' (9l(g, s')) c 9l(g, s) and put

9l[g,s] = n p:'(9l(g,s')) c 9l(g,s) c x(r+s). s':2:;s

Now, for a generic operator .@, we have left inversions of its linearization Lqp ,

and it follows [as in the "rigidity corollary" to the theorem of (E)] that the projections pS: 9l[g, s] ~ V are zero-dimensional maps, that is every "fiber" (pS)-l(V) c

9l [g, s], for v E V, is a zero-dimensional set.

Exercise. Let A: ~l ~ ~2 be an immersion with a non-empty set of transversal double points and let fi}(x) = A 0 x, for functions x: ~ ~ ~l. This operator .@ has order zero and it sends functions x to maps g: ~ ~ ~2. Analyse the consistency condition for the equation fi}(x) = 9 and study the sets 9l[g, s]. Show, that the consistency condition is not sufficient for solvability of this equation.

Let us sharpen the consistency condition by requiring the existence of a Coo _ section V ~ 9l [g, so] for some So ~ O.

Exercise. Show that if fi} is a generic operator and So is sufficiently large, for example So ~ exp exp(n + r + q'), then the sharpened consistency condition is sufficient for the solvability of the equation fi}(x) = g, for any given COO-section g: V ~ G. Show that the equations .@(x) = g abides by the h-principle.


There are many interesting "not quite generic" cases when the projections 9f[g, s] -+ V are not zero dimensional maps but the following two properties hold.

Stability. For all sufficiently large So the projections 9f[g,s] -+ 9f[g, so] are injective for all s > So.

Regularity. For all s > So the subsets 9f [g, s] c x(r+s) are smooth submanifolds, the projections 9f[g, s] -+ V are smooth fibrations and (9f[g, S])[1] = 9f[g, s + 1].

For such stable and regular relations 9f[g, s] one may apply Frobenius' theorem above and thus solve the equations £&(x) = g.

Example. Let £& be a linear operator, such that for g = ° the conditions 9f [g = 0, s] are stable and regular for large s, i.e. each 9f[g = 0, s] c x(r,s) is a smooth subbundle of the bundle x(r+s) -+ V and the dimension dim 9f [g = 0, s] does not depend on s for s z So. Then, under the consistency condition for an equation £&(x) = g, the sets 9f[g, s] c x(r+s) are also stable and regular for s > So, and by Frobenius' theorem, consistent equations £&(x) = g are globally solvable if V is a simply connected manifold.

Exercise. Let £& = (L 1 , . .. , L q.) for operators L; with constant coefficients who act on functions JRn -+ JR. Show that the corresponding sets 9f [g = 0, s] are stable iff the characteristic equations L1 (w) = 0, ... , Lq.(w) = ° have only finitely many common solutions WEen, and prove in this case the solvability of the system £&(x) = g, that is the system L;(x) = g;, i = 1, ... , q', under the consistency condition.

Further Examples and Exercises. Let (V, g) be a Riemannian manifold of dimension n z 3. Then the isometric immersion equation, £&(x) = g, for maps x: V -+ JRn+1 is over-determined.

Show that there is no stability if g is a flat metric. Show, for metrics g of positive sectional curvature, that under the consistency

condition the equation £&(x) = g is locally solvable, and if 1[1 (V) = 0, it is also globally solvable.

Give an example of a COO-metric g of non-negative curvature on V = S3, such that the equation £&(x) = g is consistent, moreover it is solvable near each point v E S3, but it is not globally solvable. Show that there are no can-metrics with such properties. Prove, that if a can-metric on a compact simply connected manifold V of dimension n z 3 without boundary admits a local isometric Coo-immersion into JRn+1 at every v E V, then there is a global isometric Can-immersion x: V -+ JR"+1. Show, this is in general not true for complete non-compact manifolds.

The stability condition for a general relation 9f c Xl') says, in effect, that dim 9fS does not depend on s for s large. In some cases, the condition 9f is acted upon by a large symmetry group that makes this stabilization impossible, but Frobenius' theorem still works, when applied to an appropriate space of orbits. For example, consider the operator £& that relates to a Riemannian metric g its curvature tensor R = £&(g). For R = ° this equation is consistent and it is also solvable, though the stabilization does not occur.

Questions. Is the consistency condition for the equation £&(g) = R always sufficient for solvability?


As another interesting example consider the differential relation ~ expressing the property of infinitesimal homogeneity of order 2 for Riemannian metrics on V.

Does the dimension ofthe space of orbits, ~s/Diff(V), stabilizes for large s? Does Frobenius' theorem apply?

Let Vo be a non-complete manifold that is infinitesimally homogeneous of order 2. Under what condition there is a complete manifold V which is infinitesimally, of order 2, isometric to Vo?

Exercise. Give an example of a non-complete locally homogeneous Riemannian manifold Yo, such that no complete manifold V is locally isometric to Yo.

2.4 Convex Integration

The global techniques of the previous sections only apply to a rather narrow class of differential relations ~. The continuous sheaves theory (see 2.2) depends upon the invariance of ~ under some diffeomorphisms of the underlying manifold V, while the removal of singularities of maps V -+ IRq (see 2.1) exploits the symmetry of ~ under certain transformations of IRq.

In this section we approach more general relations ~ and we establish hprinciple under mild geometric assumptions on ~.

2.4.1 Integrals and Convex Hulls

The key idea of the method of convex integration can be seen in the following

(A) Example. Let f: Sl -+ IRq be a C1-map whose derivative qJ = df/ds: Sl -+ IRq sends the circle Sl into a given subset A c IRq. Then the convex hull Conv A contains the origin 0 E IRq.

In fact, the path connected component Ao in A which receives Sl satsifies Conv Ao '3 o. Indeed, JSl qJ(s) ds = 0 and this integral is the limit of Riemann sums that are positive linear combinations of some vectors qJ(s) E qJ(Sl) C Ao. Q.E.D.

The converse is not true. For instance, the (connected!) ark A = {xi + x~ = 1, Xl ~ 1} in the plane 1R2 admits no continuous maps qJ: Sl -+ A for which JSl qJ(s)ds = 0, although OEConv A.

However, if the convex hull of some path connected subset Ao c IRq contains a small neighborhood of the origin, Conv Ao => lPfiO, then there exists a map f: Sl -+

IRq whose derivative sends Sl into Ao.

Proof Take some points al> ... , ak in Ao whose convex hull contains lPfiO. Since Ao is path connected there exists a continuous map t/!: Sl -+ Ao such that t/!(Si) = ai for some Si E S1, i = 1, ... , k. Let a continuous measure dJ.li on Sl (i.e. the density function dJ.lJds is strictly positive and continuous) approximate in the weak topo-


logy the Dirac measure b(s - sJ ds. Then the integral bi = ISI ljJ(s) d/li E IRq is close to ai = ljJ(sJ = ISI ljJ(s)b(s - sJ ds for i = 1, ... , k. Since the origin ° E IRq lies in the interior of the convex hull Conv{aJ, i = 1, ... , k, and since small perturbations of vectors only slightly move their convex hull, we have OEConv{bJ, i = 1, ... , k, as the vectors bi are close to ai· Now write ° = I~=l Pibi for some Pi 2 0, L~=l Pi = 1, and observe that the continuous measure d/l = I~=l Pi d/li satisfies

I, ljJ(s) d/l = it Pi I, ljJ(s) d/li = it Pibi = 0.

Finally, we integrate the measure d/l to a C1-diffeomorphism /l*: Sl -t Sl which pushes forward the Lebesgue measure ds to d/l. Namely, we assume both measures to have total mass one, we put /l(s) = I~o d/l and we take the inverse ofthe diffeomorphism s f---+ /l(s) for /l*. Then the compositions of maps, cp = ljJ 0/l*, has ISI cp ds =

ISI ljJ d/l = ° and the map cp: Sl -t Ao integrates to the required map f: Sl -t IRq, whose derivative dfjds = cp sends Sl to Ao.

We need a generalization of this construction to families of maps.

Definition. We say that a map ljJ: V x S -t IRq strictly surrounds a given map fo: V -t

IRq if the vector fo(v) E IRq is contained in the interior of the convex hull ofthe ljJ-image of the fiber Sv = v x S c V x S, that is

for all v E V.

(B) The Reparametrization Lemma. Let V be a smooth manifold and S = [0,1]. If a C-map ljJ: V x S -t IRq strictly surrounds a C-map fo: V -t IRq then there exists a fiber preserving C -diffeomorphism /l*: V x S -t V X S ["fiber preserving" means /l*(Sv) = Sv for all v E V], such that the composition cp = ljJ 0 /l*: V x S -t IRq satisfies,

f cp(v,s)ds = fo(v) for all VE V.

Proof. As (!Jjzfo(v) c Conv ljJ(Sv), there exist some measures on the fiber Sv, say d/li = d/li(v), i = 1, ... , k, on Sv for all VE V, such that the density functions d/l;/ds are positive and COO-smooth in the variables s and v and such that the integrals I;(v) = k ljJ(v, s) d/li E IRq strictly surro.unds the vector fo(v) E IRq. Namely,

Conv{I;(v)}i=l ..... k::::J (!Jjzfo(v) for all VE V.

Then, there is a C-smooth partition of unity Pi: V -t [0, 1], I~=l Pi == 1, such that If=l pi(v)Ii(v) = fo(v) for all VE V, and so the measure d/l(v) = If=l pi(v)d/li(v) on Sv has k ljJ(v, s) d/l (v) = fo(v) for all v E V. Finally, there exists a fiber preserving C-diffeomorphism /l* of V x S which pushes forward Lebesgue measure ds = ds(v) on Sv to d/l(v) for all VE V. (This /l* is unique if we require /ll V x ° = Id.) Clearly, Ib cp(v, s) ds = fo(v). Q.E.D.

The following proposition (which is not used in the sequel) explains the geometric significance of the lemma.


Convex Decomposition. Let A be a connected C'-submanifold in IRq for some 1 ;;:: 1 and let fo be a C' -map, r ::;:; 1,. of a compact manifold V into lnt Conv A c IRq. Then fo is a convex combination of some C' -maps ({Jv: V --+ A, that is fo = Lv Av({Jv for some constants Av ;;:: 0, Lv Av = 1.

Proof. As A is path connected there obviously exists a Cr-map tjJ: V x S --+ A which strictly surrounds fo and so there is a C'-map ({J: V x S --+ A whose integral I = Js({J(·,s)ds: V --+ IRq equalsfo. Since Vis compact, this integral is the uniform limit of Riemann sums I = N-1 L~=l ({J(., sv) and so each C'-map fo: V --+ lnt Conv A admits a C' -approximation by the sums I that are convex combinations of some C'-maps ({J(., sv): V --+ A.

Since the (connected!) manifold A has lnt Conv A "* 0, some tangent spaces T,..(A) c T,..(lRq) = IRq, ai E A, i = 1, ... , q, linearly span IRq. It follows with the implicit fu~ction th~orem that every C'-map 15: V --+ IRq which is CO-close to the constant map xo: V --+ t L1=1 ai is a convex combination of C'-maps V --+ A which are Co_ small perturbations of the constant maps V --+ a;, i = 1, ... , q.

Now, for small oe > 0, the map fo is the convex combination, fo = oexo + (1 - oe)fo, where the map fo sends V into lnt Conv A. Hence, fo = oexo + (1 - oe)(I' + 15') where I' is a convex combination of maps V --+ A and where 15' is an arbitrarily small C'-map. Then the map 15 = Xo + [(1 - oe)joe] 15' is also a convex combination of maps V --+ A and so fo = oeI5 + (1 - oe)I' is such a combination as well.

Exercises. Generalize the convex decomposition to those subsets A c IRq which are images of C'-maps of connected manifolds into IRq.

Let A be a disconnected compact submanifold in IRq, such that no component Ao of A has Conv Ao = Conv A. Prove the existence ofa C'-mapfo: V --+ IntConv A which is not a convex combination of continuous maps V --+ A.

Questions. How does one estimate (from above and from below) the minimal number N for which some convex decomposition fo = L~=l Av({Jv exists? Is the convex decomposition possible for non-compact manifolds V? What happens to pathconnected subsets A c IRq which are not Cl-submanifolds?

C1.-Approximation over Split Manifolds. Let a smooth manifold V split into the product, V = V' x [0,1]. If we fix local coordinates u1, ... , Un- l in V', then the rth order jet of a Cr -map f: V --+ IRq is given by the totality of the partial derivatives in the variable U1, ... , Un-I' t. Namely, J; = {a~a:J}, where K = (k1, ... ,kn-d, such that IKI = kl + , ... , + kn- 1 ::;:; r - I, and where a~a: stands for the partial derivative (ak , + .. ·+kn -, +I)j(au~' ... aU!~l' at'). We assemble in the notation J1. those derivatives a~a: for which I ::;:; r - 1 and thus we split the jet J; into J; = Jf $ a;f. This splitting Jr = J1. $ a; only depends on the splitting V = V' x [0,1], but it does not depend on the local coordinates U 1 , ••• , Un- l in V'. We introduce C1.-convergence of maps as the CO-convergence of the respective jets J1..

(C) The C1.-Approximation Lemma. Let V be a compact split Coo-manifold and let S = [0,1]. Let fo: V --+ IRq be an arbitrary C2r_map and let tjJ: V x S --+ IRq be a

C-map which strictly surrounds the derivative a;fo = arfo: V -+ IRq. Then the map ilt r

fo admits an arbitrary close C~-approximation by those maps f, whose derivatives a;f: V -+ IRq lift to C-sections V -+ V x S. That is a;f(v) = t/I(v, I'/(v)) for some C-function 1'/: V -+ s.

Proof We may assume with the reparametrization lemma that g t/I(v, s) ds = a;fo(v). Then we also have for IKI S; r,

(*) f a~t/I(v,s)ds = a~a;fo(v). Next, for every positive a < !, we choose some disjoint a-subintervals in [0,1]

of total length ~ 1 - a, and let he: [0,1] -+ S = [0,1] be a COO-function which linearly maps every chosen subinterval onto [a, 1 - a] c [0,1] = S. This h, is given on each subinterval, say on [to, to + a], by one of the two formulas:

(1)

(2)

h(t) = at + b, a = a-1 - 2, b = a - ato;

h(t) = - at + b', b' = 1 - a + ato.

Set I'/,(v',t) = h,(t), where (v',t) = VE V, take g,(v',t) = t/I(v',t,I'/,(v',t)) and show that the derivatives a~g" IKI S; r, satisfy

(+ ) a-1 a~g,(v', t)dt -+ a~a;fo(v', to). 1/0 +'

,-0 10

Indeed, in case (1) (the second case is left to the reader) the substitution t = h-1 (s) = (s - b)/a gives the identity

110+' 1 11 -' a-1 a~t/I(v',t,at + b)dt = -1 _ a~t/I(v',to + b"s)ds,

10 2a ,

(s - a) where D, = -- -+ 0. Therefore,

1 - 2a

and then ( + ) follows from (*).

a -+ 0,

The map fo(v', t) is C~-approximated by the r-fold integral in t of g,(v', t) for a -+ 0. This integral, called !e(v', t), is defined by the conditions a;!e = g, and Jf, IV' x ° = Ji;IV' x 0, which implies for i = r, ... , 1,

a:-1!e(v',t) = a:-1fo(v', 0) + I a:!e(v',t)dt.

The uniform in a bounds II a~ g, II S; const and the relation (+), which applies to a;!e = g, over the a-subintervals of the total length ~ 1 - a, show that for IKI S; r,

Ila~a;-l!e - a~a;-lfo II -+ 0, a -+ 0.

Then we obtain with successive integration in t the uniform convergences a~a:!e -+

o~o:fo, e --.. 0, for all i = r - 1, ... , 0, and IKI ~ r. This yields the required Cl.-convergence /. --.. fo.

Finally, the derivative o;/' = g.lifts to a section V --.. V x S by the very definition of g. and so the proof is concluded.

The Jet Space Xl.. Let p: X --.. V be an arbitrary smooth fibration with qdimensional fibers and let. be a continuous tangent hyperplane field on V that is a codimension one subbundle of the tangent bundle T(V). Then, there are for each r = 1, 2, ... , a unique manifold Xl. and continuous maps pr: X(r) --.. Xl. and P!:-l: Xl. --.. x(r-l), such that Pr~l 0 p~ = P;-l for the natural projection P;-l: x(r) --.. x(r-l) ofthe jet spaces, and such that the Jl.-jet Jl. d,;! p~ 0 Jr satisfies the following conditions for all pairs of (germs of) sections fl and f2: V --.. X,

Jl. = Jl.<;!>DJr- 1 1. = DJr- 1 1. II h II h'

for the differentials

DJ;~l and DJ;:l: T(V) --.. T(x(r-l».

The maps x(r) --.. Xl. and Xl. --.. x(r-l) carry natural structures of affine bundles which are compatible with the affine bundle x(r) --.. x(r-l). The fibers of the bundles x(r) --.. Xl. for all fields • on V form a certain distinguished class of affine qdimensinal subspaces in the affine fibers x;r) c x(r), YEx(r-l) which are called principal subspaces in x(r).

The Convex Hull Conv, (Bl). Consider a subset Bl c x(r), take the convex hull of the intersection Blx = Blnxi') c Xi') for Xi') = (pD-1 (x), XEXl., and denote by Conv,(Bl) the union ofthese convex hulls over all x E Xl..

(D) The Cl.-Dense h-Principle. A differential relation Bl c x(r) satisfies the Ci-dense h-principle, for i = -1, if the following four conditions are satisfied.

(1) The field. is integrable; (2) The subset IR c x(r) is open; (3) Conv,(Bl) = x(r); (4) The intersection Blx = Bl n Xi') is connected for all x EXl..

In particular, if there is a section Xl. --.. Bl then every cr -section V --.. X admits a fine Cl.-approximation by solutions of the relation Bl, where the Cl.-topology by definition is induced from the CO-topology in the space of the Jl..jets V --.. Xl..

Proof We may assume the fibration X --.. V to be a vector bundle as a small neighborhood of (the image of) each section V --.. X admits a vector bundle structure.

Since the field. is integrable, every point v E V lies inside a small split submanifold V = V' x [0,1] c V, for which the slices V' x t, t E [0,1] are tangent to the field ., and over which the fibration is trivial, XI V = V x IRq --.. V.

Let us prove the h-principle over V by solving the extension problem over V for all small split submanifolds V c V (compare 1.4.3) as follows.


Take a C-section fo: V --+ X and a CO-section qJo: V --+ flt, such that It =

Pi. 0 qJo and such that 1;0! Vo = qJo! Vo for a given closed subset Vo c V. To prove the C1.-dense h-principle for extensions (of solutions of flt from Vo to V) we must homotope qJo to a holonomic section 1;,: V --+ f2.l, such that fl is C1.-close to fo. Moreover, the homotopy in question must be constant near Vo while the projection of this homotopy to X 1. must be nearly constant over all of V (compare 1.2.2).

The conditions (2)-(4) obviously yield a COO-homotopy of (non-holonomic) sections, CPs: V --+ flt, s E S = [0, 1], with the following three properties.

(a) The homotopy qJ: V x S --+ flt c x(r) lies over It, that is Pi. 0 qJs = It for all SES.

(b) The homotopy qJ strictly surrounds the section 1;0. Namely, the vector 1;0 (v) EXf), for x = It(v) EX1., lies in the interior of the convex hull of the path SHqJ.(V)EXf), SES, for all VE V.

(c) The sections qJs!lDjZVo for all SES are as close to qJo!lDjZVo as we wish. (One even may have qJs!lDjZVo = qJo!lDjZVo if "strictly surrounds" in (b) is relaxed to "surrounds").

Now, the splitting of jets, l r = 11. EB a;, induces a splitting of the jet bundle, x(r) = X 1. X IRq over U. The IRq-component of the homotopy qJ: U x S --+ x(r) is a Coo-map, say 1/1: U x S --+ IRq, to which the C1.-approximation lemma applies. Thus we obtain a section f: V --+ X whose jet 1; nearly factors through a section V --+

U x S, that is l;(u) is CO-close to qJ(u, 1J(u) for some function 1J: U --+ S. This implies the existence of a homotopy of sections U --+ flt between CPo and 1;. Moreover, as the sections qJs!lDjZUo, SES, are close to qJo!lDjZUo, the jet l;!lDjZUo is also close to CPo, and so the above homotopy can be chosen almost constant on lDjZVo. Hence, a small C-perturbation f of fl equals to fo on lDjZVo and the jet 1;,: U --+ flt admits the required homotopy to qJo. Q.E.D.

(D') Remark. The results in 2.4.4 yield the Ci-dense h-principle for all i = 0, 1, ... , under the above assumptions on flt.

(E) Corollary. Let V be an arbitrary manifold and let L c x(r) be a closed stratified subset of codimension ~ 2. Furthermore, assume that the intersection of E with every principal subspace R ~ IRq in x(r) has codimension ~ 2 in R. Then E-nonsingular sections V --+ X satisfy all forms of the h-principle (see 1.2.1, 1.4.3, 1.5.2).

Proof First let V admit an integrable field T. Then the complement flt = x(r\E satisfy the assumptions of (D) and we get with (D') all non-parametric h-principles for E-nonsingular sections that, by definition, are solutions of f2.l. To get the parametric h-principle, we interprete families of sections CPp: V --+ X, PEP, as sections of the fibration X x P --+ V x P and then apply (D') to the respective problem over V x P for all manifolds P.

Now, if V is open, it admits a function h without critical points (see 1.4.4) and the above applies to the kernel T of the differential dh. If V is closed, then we have our field T on V minus a point. Since the h-principle is trivially true at every single point Vo E V, the extension h-principle over V\vo yields the h-principle over all of V. Q.E.D.

Remark. The condition codim(R n E) ~ 2 is generic for singularities E of codimension ~ 2. But this condition is not satisfied in most geometrically interesting cases, such as immersions, free maps etc. These classes of maps are covered by the techniques developed in the following sections.

2.4.2 Principal Extensions of Differential Relations

A differential relation over the jet space x(r), for a fibration X -+ V is by definition an arbitrary topological space qt with a given map qt -+ x(r). This generalizes our notion of a relation in x(r), where the structure map is the inclusion qt 4 x(r). A relation qt over x(r) also lies over every space Y under x(r), where "under" means a fixed map x(r) -+ Y and qt goes to Y by the composition of maps. For example, qt lies over X('), s ::;; r, and over V.

-----. Sections V -+ qt are those continuous maps for which the composed map V -+ qt -+ V is the identity Id: V -+ V. A section V -+ qt is called holonomic if the composition ~

V -+ qt -+ x(r) is a holonomic section V -+ x(r). A relation qt over x(r) is said to satisfy the h-principle if every section V -+ qt admits a homotopy of sections to a holonomic section V -+ qt. We define in a similar way the dense h-principle, the parametric one etc (compare 1.2).

Extensions. We call an extension of a differential relation p: qt -+ x(r) another relation p: ~ -+ x(r) which comes with an embedding E: qt -+ ~ and with a retraction (or projection) II: ~ -+ qt, such that the diagrams

E: qt -+ ~ and II: 11-+ 9l

\/ \1 x(r) x(r-l)

commute.

(A) Example. The relation IIr = {aEqt,XEx(r)lp(a) = P;-l(X)}, p: (a,x)Hx, extends qt for E: a H (a, p(a» and II: (a, x) H a. This ~r trivially satisfies the C-1-dense h-principle. Indeed, sections of ~r are pairs (cp, r/I), where cP: V -+ qt is an arbitrary continuous map and r/I: V -+ x(r) is a section. Hence, any homotopy of r/I to a holonomic section V -+ x(r) also makes the section (cp, r/I): V -+ ~r holonomic.

An extension ~ :::J qt is called h-stable (h for holonomy) if every holonomic section CPo: V -+ ~ admits a homotopy of sections CPt: V -+~, t E [0,1], such that CPl is a holonomic section V -+ qt c ~ and such that CPt(v) = CPo(v) for all those points v E V for which CPo (v) E qt and for all t E [0,1]. We do not require the sections CPt for ° < t < 1 to be holonomic. However, one can construct a homotopy of holonomic sections CPt, t E [0, 1], with little extra work for all h-stable extensions which appear in this chapter (see "convex extensions" below and in the following section).

The important (and obvious) property of h-stable extensions ~ :::J qt (which does

not depend upon the holonomy of CPt for 0 < t < 1) is expressed by the following implication,

[the C-dense h-principle for 9l] = [the Ci-dense h-principle for ~],

for all i = 0, ... , r - 1.

Convex Hull Extensions. Let r be a tangent hyperplane field on an open subset V c V and let P't: X~) --> X & be the associated affine bundle whose fibers are principal subspaces R ~ IRq in x(r). Take a point in the pullback p-l(R) c ~ for p: ~ --> x(r), say a E p-l(R), and denote by ConvR (~, a) the convex hull in R of the p-image of the path connected component in p-l(R) of the point a E p-l(R). In other words, the convex subset Conv R (~, a) c R consists of those points in R which are surrounded by homotopies of the point a over R.

Now we define the r-convex hull 9l = Conv.(~) as the set of those pairs (a, x) E

~ X x(r) for which either x = pea), or x E Conv R (~, a) for some principal subspace R c x(r) associated to r. This ,~ lies over x(r) for p: (a, x) --> x, and the relation p: 9l--> x(r) extends p: ~ --> x(r) for E: a 1--4 (a, pea)) and JI: (a, x) 1--4 a.

Remark. This elaborate convex hull is needed to distinguish the convex hulls of different path connected components of p-l(R). The impossibility to do this inside x(r) is the main reason for the "over" generalization.

Open Relations over x(r). A relation ~ over x(r) is called open if the implied map p: ~ --> x(r) is a microfibration (see 1.4.2). For example, if p is a submersion, then ~ is open over x(r). Observe that the openness of the map p is strictly weaker than the openness of the relation ~ over x(r).

(B) The h-Stability Theorem. If ~ --> x(r) is an open relation, then the extension ~ = Convr(~) :::J ~ is h-stable for every continuous hyperplane field r on an open subset V c V. In particular if V = V and if Conv R (~, a) = R for all principal subspaces R associated to 1" and for all a E P -1 (R) then the relation ~ satisfies the C.l-dense h-principle. [Compare (D) in 2.4.1.]

Proof Sections of open relations ~ over x(r) may be dealt with as if ~ were an open subset in x(r) because of the following

(B/) Properties of Open Relations p: ~ --> x(r). (a) If a section cP: V --> x(r) lifts to ~ (i.e. cP lifts to a section ip: V --> ~, p 0 ip = cp), then also small (in the fine CO -topology) perturbations of cp lift to ~.

(a/) If ~ is homeomorphic to a finite polyhedron, then there exists a neighborhood of ~ in the extension ~ x x(r) :::J ~ [see (A)], say (!)fz~ c ~x(r), such that the map (!)fz~ --> x(r), (a, x) 1--4 x, lifts to a map a: (!)fz,o/l-> ~ such that po a(a, x) = x for all (a,x)E(!)fz~ and a(a,p(a)) = a for all aE~.

(a") Let p: ~ --> x(r) be an arbitrary relation. Then ~ is open if and only if (a/) holds true for all maps of polyhedra into ~, say for p: P -->~. Namely, there is a neighborhood (!)fzP c P X x(r) of P = {p, p 0 P(p)} c P X x(r), such that the map

176 2. Methods to Prove the h-Princ!ple

(9fzP - x(r), (p, x) ~ x, lifts to a map 12: {9fzP - flt for which po iX(p, x} = x and iX(p, p 0 p(p}} = p for all (p, x) E {9fzP.

(b) The sheaf of holonomic sections V - flt, called cJ>h(9i), is microflexible (compare 1.4.2, 2.2). Recall that the sheaf of all continuous sections, cJ>c(9i}, is flexible for all (not necessarily open) relations 9i _ x(r).

(b/) The map p: 9i _ x(r) induces microjlexible homomorphisms (see 2.2.4) of sheaves, cJ>c(flt) - cJ>c(x(r)} and cJ>h(9i) - cJ>h(x(r)}. Namely, the maps ,,= ,,(A, B) defined in 2.2.4 are microfibrations for these homomorphisms. Furthermore, the images of these homomorphisms are the subsheaves cJ>c(p(9i)) c cJ>c(x(r)} and cJ>h(P(9i}} c cJ>h(x(r)} respectively for the image p(flt) - x(r), and the corresponding homomorphisms cJ>c(9i} - cJ>c(p(9i)) and cJ>h(9i} - cJ>h(P(9i}} are microextension (see 2.2.4).

(c) The extension ~ = Conv,(flt) is an open relation over x(r) for all open relations 9i - x(r) and for all hyperplane fields" on a given open subset U c V.

The proof of (a") is immediate with the definition of a microfibration (see 1.4.2), and (a") obviously implies (a), (a/), (b) and (b /). The proof of (c) is straightforward.

Now, we obtain the h-stability of the extension ~ = Conv,(9i} ::l 9i by generalizing the proof of the C~-dense h-principle in (D) of 2.4.1 as follows. We are given a holonomic section V - ~ c 9i X x(r), that is a pair of sections ({Jo: V - flt and fo: V - x(r), where the section fo is holonomic and such that

(i) the section fo equals po ({Jo outside the open subset U c V on which the field " is defined,

(ii) pl. 0 fo = pl. 0 p 0 ({Jo for the affine bundle pl.: X~) - X &, (iii) fo(u}EConvR(flt,({Jo(u)) for all UEU and for the fiber R = (pl.t1(X) for x =

p~ 0 p 0 ({Jo(u).

We must construct a homotopy of sections «({Jt,ft): V - ~ - 9i, t E [0,1], which is fixed outside U and such that f1 = po ({J 1: V - x(r) is a holonomic section. In fact, we only need the homotopy ({Jt: V - 9i of ({Jo to a holonomic section ({J1: V - 9i. The homotopy offt then obviously follows with the linear homotopy «({Jo, (1 - O}fo + O(p 0 ({Jo)): V -~, 0 E [0,1] which brings the section «({Jo,Jo) to 9i c ~ for 0 = 1.

Obviously, there exists a homotopy of (non holonomic!) sections ({J.: U - flt, s E S = [0, 1], whose projection to x(r) surrounds the section fo in the fibers of the fibration pl.: X~) - X&, and then (an obvious modification of) the proof of (D) in 2.4.1 works in the present case provided" is a split integrable field. This means the existence of a split submanifold Vo = V~ x [0,1] c V which contains U, such that Vo is a disjoint union of countably many compact submanifolds in V and such that the field" is tangent to the submanifold V~ x t c Yo, at all points (v', t) E U.

Next, we reduce the general case of the h-stability theorem to the split integrable case by first localizing the problem to a small neighborhood of each point U E U and then by approximating the field" near U by an integrable split field.

Localization. Let open subsets Ui c U, i = 1, ... , k, cover U and let "i be a hyperplane field on Ui • Put flto = flt and 9ii = Conv" (flt i - 1) for i = 1, ... , k. Suppose that the fields "i are the restrictions "I U;, i = 1, ... , k, and let us "lift" a given


section iP = (({Jo,fo): V ~ II to a section ({Jk: V ~ ~ as follows. Fix a partition of unity b l , ... , ~ on U inscribed in the cover {UJ and assume the fibration X ~ V to be a vector bundle [compare the proof of (D) in 2.4.1]. Define a section 11: U ~ xg) by 11 = pO ({Jo + bl (10 - pO ({Jo) and then by induction take I; = 1;-1 + bi(fo - pO ({Jo) for i = 2, ... , k. Observe that the sections I; continuously extend to V:::l U by I;IV\U = IolV\U and that the (extended) pairs ({Jl = (({JO,fl), ({J2 =

(({Jl,f2),···' ({Jk = (({Jk-l,fd are sections of the relations ~1' ~2"'" ~k respectively. Furthermore,

k

Ik = pO({Jo + L Mio - pO({Jo) = 10' i=l

This amounts to the equality pO iP = Pk ° ({Jk for Pk: ~k ~ x(r), which by definition expresses the "lift" property of our construction iP -- ({Jk'

If the extensions ~i :::l ~i-l' i = 1, ... , k, are h-stable, then the extension £1Iik :::l ~o = ~ also is stable and so ({Jo admits the required deformation, provided 10 = Ik is a holonomic section V ~ x(r). Now, let .;, i = 1, ... , k, be sufficiently small (in the fine CO-topology in the space of the hyperplane fields on U) perturbations of the field. and let 'fi = .;/ Ui' Then the perturbed relation !ik still admits a section iiik such that Pk ° iiik = Pk ° ({Jk = po iP = 10 and such that iiik is close to ({Jk in the ambient relation ~ x x(r) x ... x x(r) which contains !ik as well as ~k' The existence of iiik

l J Y k

for small perturbations .; of. is immediate from the openness of the relations ~i' i = 0, ... , k. Moreover, the required "smallness" of these perturbations depends only on the original section iP = (({Jo,fo): V ~ ij but not on specific open subsets Ui and (or) the partition of unity bi' Indeed, let ({J;, i = 1, ... , k, be small perturbations of ({Jo which are equal to ({Jo outside U and such that ({J; agrees with 10 under the projection p; ~ p~: xg) ~ X &, which corresponds to the field .; as follows,

for i = 1, ... , k.

Then, the sum L~=l biP ° ({J: is close to L~=l hiP ° ({Jo = po ({Jo and so there is a small perturbation iiio of ({Jo such that pO iiio = L~=l biP ° ({J:.

Take it = poiiio + bl (fo - pO({JD and l = l-l + Mio - pO({J!) for i = 2,

... , k. The pairs iiil = (iiio,it), ... , iiik = (((Jk-l,h) are sections of the perturbations !i l' ... , !ik of ~ l' ... , ~k and the section iiik is a lift of 10 as

k

h = poiiio + L bi(fo - pO({J;) =10' i=l

Now, we take for .; (arbitrarily) small perturbations of • for i = 1, ... , k =

dim V + 1, for which there is a cover {UJ of U such that each field 'fi = .;/ Ui> i =

1, ... , k, is an integrable split field (i.e. 'fi is tangent to the slices V;' x t of some split manifolds V; = V;' x [0,1] :::l UJ The section 10: V ~ x(r) lifts to a section V ~!ik and so the h-stability theorem reduces to the split case which was considered earlier. Q.E.D.

Principal Convex Extensions. We define the (first) principal extension Prl~ of a relation p: ~ ~ x(r) as the subrelation in the extension iir of (A), such that (a, x) E

Pr 1 fll if and only if there is a principal subspace R in x(r) which contains the points p(a) and x E x(r). This R is uniquely determined by a and x unless p(a) = x. The principal convex extension Conv 1 fll of fll is the subset of those (a, x) E Pr 1 fll for which either x E Conv R (fll, a) or p(a) = x. Then we define by induction the principal extensions PrNfll = Pr1PrN- 1fll and ConvNfll = Conv1 ConvN-1 fll for N =

2,3, .... Next, we define PrNfll for N = 00 as the space of pairs (a, y = y(t)), where a E fll

and where y: [0,1] --+ x(r) is a piecewise principal path which issues from p(a)Ex(r). That is

(i) y(O) = p(a), (ii) there is a principal subdivision of the path y = y(t) by finitely many points, say

by 0 = to :;;; t1 :;;; ... :;;; tN = 1, such that each segment [ti- 1, t;], i = 1, ... , N, is sent by y into some principal subspace R = R(y[ti- 1, t;]) c x(r) and y(t) linearly

t· - t interpolates between y(ti-d and y(ti) for tE[ti- 1,t;], y(t) = I y(ti- 1) +

t - t;-1 . ---=--=-y(t;) for l = 1, ... ,N. t; - t i - 1

ti - t;-1

There is a unique minimal principal subdivision of y for which the number N is the least possible. This N is called the principal length of y and of (a, y). Every principal subdivision of y refines the minimal one.

The relation Prooflllies over x(r) for the map Poo: (a,y(t)) --+ y(1) and it extends fll for the embedding Eoo: aH(a,y(t) == p(a» and for the projection IIoo: (a,Y)Ha. Furthermore, every principal subdivision of (a,y) EProofll defines a point in the union UN<OO PrNfll as follows. If 0 = to :;;; t1 :;;; ... :;;; tN = 1 are the division points, then

(a,Y)H(a,Y)N = (a,y(td,···,y(tN»EPrN91·

Now, we define the extension Convoofll c Proofll of fll as the set of those pairs (a,y) EProofll, for which there is a principal subdivision into N segments, for some N = 1, 2, ... , such that the corresponding sequence (a, y)N E PrNfll is contained in ConvNfll c PrN 91. Observe that this definition is stable under refinements of subdivisions. Namely, if (a'Y)N 1 EConvN1 fll for some principal subdivision into N1 segments, then also (a, y)N2 E ConvN2 fll for every refined subdivision into N2 > N1 segments.

The space of piecewise principal paths y: [0,1] --+ x(r) comes with the CO_

topology and thus we have natural topologies in the spaces Proofll and Convoofll. Next, a continuous map f of a locally compact space P --+ Proofll is called locally finite if the principal length off(p)EProofll is bounded from above by some positive function on P which is continuous in pEP. All maps P--+Proofll as well as maps P --+ Conv 00 fll c Pr 00 fll are assumed from now on to be continuous and locally finite. In particular, sections V --+ Pr 00 fll and homotopies of these are always assumed locally finite as well as continuous.

If 9l is open, then the relations PrNfll and ConvN91--+ x(r), N < 00, are open by (c) in (B') and also Proo and Convoo are open relative to the quasitopology defined by locally finite maps (compare 1.4.2).

(C) Principal Stability Theorem. The principal convex extensions ConvN.cJt ~ .cJt, N = 1,2, ... , 00, are h-stable for all open relations .cJt -+ x(r).

Proof Every section (cp,f): V -+ Pr1.cJt for cp: V -+.cJt and f: V -+ x(r) defines a unique hyperplane field r on the open subset U c V of those points v E V for which po cp(v) "# f(v). This is the field which corresponds to the field of principal subspaces R in x(r) through the points po cp(v) and f(v) in x(r). The section (cp,f) is contained in Cony 1 .cJt c Pr 1 (.cJt) if and only if it is a section of Cony, (.cJt) and so the h-stability theorem implies the h-stability of the extension Cony 1 .cJt for open relations fJt. Then, we conclude with an obvious induction to the h-stability ofConvN.cJt for N < 00. If the manifold V is compact, then the h-stability of Cony N fJt for N < 00 implies that for N = 00, as the sections V -+ Cony OCJ .cJt in question have bounded principal length and so they lie in some ConvN.cJt for N < 00. If V is non-compact, then the above applies to compact parts of V and the h-stability of Cony OCJ fJt is obtained with a compact exhaustion of V. Q.E.D.

Applications of the Theorem (C) depend upon the structure of piecewise principal homotopies of sections fr: V -+ x(r) which by definition are (continuous and locally finite!) sections

V -+ (the space of piecewise principal paths [0, 1] -+ x(r»),

where v 1-+ !t(v), t E [0,1].

(D) Lemma. If two sections fo and f1: V -+ x(r) have equal projections to x(r-1), that is if P;-1 0 fo = P;-1 0 f1' then fo and f1 can be joined by a piecewise principal homotopy of sections it: V -+ x(r), t E [0, 1]. Moreover, every continuous homotopy, whose projection to x(r-1) is constant in t, admits a piecewise principal approximation.

Proof We must show that there are "sufficiently many" piecewise principal paths in the fiber X~) of the affine bundle x(r) -+ x(r-l). We need for this "sufficiently many" principal subspaces in X~). The required sufficiency of principal subspaces is immediate from the following

(D') Sublemma. Let (X~), 0) be the vector space with some point 0 E X~) taken for the origin in the affine space X~). Then the principal subspaces in X~) which pass through o linearly span the space (X~), 0).

Proof The space (X~), 0) is naturally isomorphic to the space of maps Q: IRn -+ IRq (where n = dim V and q is the dimension of the fiber of the underlying fibration X -+ V) whose components Q;, i = 1, ... , q, are homogeneous polynomials of degree r. Such a Q E (X~), 0) lies in a principal subspace R c X~) associated to a hyperplane r c IRn = T.,(v) if and only if Qi = c)r, i = 1, ... , q, where I: IRn -+ IR is a linear form for which Ker I = ! and where C; are arbitrary constants. Now, the sublemma follows from the existence of a decomposition of every homogeneous polynomial IRn -+ IR of degree r into a linear combination Lil clll; for some linear forms III on IRn.


(D") Corollary. The relation Proo!1t _ x(r) trivially [compare (A)] satisfies the Cr- l_ dense h-principle for all relations !1t _ x(r).

Proof Let «((J,fr) be an arbitrary section of Proo!1t for ((J: V -!1t and fr: V _ x(r),

tE [0, 1], and let f2: V - x(r) be a holonomic section whose projection to x(r-l) equals to that offl. We lift the sectionf2 to Proo!1t by joiningfl andf2 by a piecewise principal homotopy of sections V - x(r) and thus we obtain the required holonomic section V - Proo!1t which is homotopic to the original section «((J,fr). Q.E.D.

2.4.3 Ample Differential Relations

A differential relation !1t - x(r) is called ample over x(r) if Conv 00 !1t = Pr 00 IR. As !1t lies over the jet spaces X(·) for s ~ r, one may also speak ofthe ampleness over X(·).

We say that!1t is ample if it is ample over X(·) for all s = 1, ... , r. We shall frequently use the following

Sufficient Condition for the Ampleness. If Conv 1 !1t = Pr l!1t (over x(r)} then !1t is ample over x(r).

Indeed, Pr 1 Conv 1 !1t c Conv 1 Pr 1 !1t for all !1t, and so the equality Conv 1 !1t =

Pr l!1t by induction implies Conv N!1t = PrN!1t for N = 2, 3, ... , 00.

(A) Theorem. If an open relation !1t - x(r) is ample over x(r) then it satisfies the cr-l-dense principle for extensions. Furthermore, if!1t is ample over all x(·) for s ~ r, then it satisfies the h-principle. Moreover the h-principle is C·-l-dense for s =

1, ... , r and the parametric h-principle holds true as well. In fact, !1t satisfies all the h-principles discussed in Chapter 1.

Proof The Cr-l-density immediately follows from the principal stability theorem (C) in 2.4.2 and the trivial h-principle for Proo!1t = Convoo !1t. [See (D") in 2.4.2. The latter h-principle obviously holds for extensions as the proof of (D") shows.]

Next we observe that the Ci-l-dense principle for !1t - X('), where r ~ s > i > 0, formally follows from the Ci-l-dense h-principle for !1t _ X(·-l) and the C·-l-dense h-principle for !1t _ X(·). Thus by induction we obtain the C·-l-dense h-principle for !1t - x(r) which implies for s - 1 = ° the ordinary h-principle.

In order to prove the parametric h-principle we apply the above to families of sections fp: V - X, PEP, which are viewed as sections V x P - X x P for all manifolds P (compare 1.2.1). Q.E.D.

Examples of Ample Relations. We start with the immersion condition!1t c X(l) for X = V x W - V. The fiber X~l), x = (v, w), ofthe fibration X(1) - X is the space of linear maps IRn = T.,(V} - Tw(W} = IRq. A linear map Lo E X~l) belongs to !1t if and only if rank Lo = n = dim V. The principal subspace R = R(Lo, -r} c X~l) through Lo for some hyperplane L c IRft = T.,(V} consists of those linear maps L: IRft - IRq for which LI-r = Lol-r. If LoE!1t, then the intersection Rn!1t equals R minus a subspace R' c R of dimension n - 1 = dim -r. If dim R = q > n, then codim R' ~ 2

and so the intersection R n Yl is path connected as well as dense in R and so Conv I Yl = Pr 1!J1l. Hence, !J1l is ample for q > n (!J1l obviously is not ample for n = q) and so we conclude to the Hirsch immersion theorem in the extra dimension case.

Exercises. Derive Hirsch's theorem for equidimensional immersions of open manifold from the Theorem (A).

Hint. Restrict the immersion condition to the (n - 1)-skeleton of an appropriate triangulation of V (compare 1.4.1).

Prove Feit's k-immersion theorem by showing Convl !J1lk = PrIYlk, k =F dim W, for the differential relation Ylk c X(I) which governs maps V --+ W of rank ~ k.

Show that the map PI: Pr l!J1lk --+ X(!) sends Pr I Ylk onto Ylk- I c X(I) and that PrNYlk goes onto !J1lk-N for all N = 1,2, ....

Let A --+ V and B --+ W be vector bundles and let IX: A -+ T(V) and p: T(W) --+ B be homomorphisms such that rank p ~ ko for some integer ko.

Prove the h-principle for those CI-maps f: V --+ W for which the composed homomorphism po Df 0 IX: A --+ B has rank::;; k for a given integer k < ko.

Free Maps. Let W be a Riemannian manifold of dimension q. Then the freedom condition for C2-maps V --+ W (see 1.1.4) is ample in the extra dimension case, that isforq ~ [n(n + 1)/2] + n + 1. This also is true (and equally obvious)forkth-order free maps, k = 2, 3, ... , in the extra dimension case. Thus we obtain the h-principle for free maps (of any order) in the extra dimension case.

Systems of Sections of D-rank ~ k. Let Y and Z be vector bundles over V and let D: rr(y) --+ rO(Z) be a differential operator of order r which is given by a continuous vector bundle homomorphism ,1: y(r) -+ Z (compare 2.1.2). The D-rank of a system of C-sections Ii: V -+ Y, j = 1, ... , q, is the infimum of the dimensions of the subspaces span {D.fj(v)} c Zv over all fibers Zv c Z, V E V. We have indicated in 2.1.2 the h-principle for systems of D-rank ~ k < q for COO-smooth homomorphisms ,1. This h-principle can be recovered (for all continuous ,1) with the convex integration as the corresponding differential relation Yl c x(r), for X = Y EB ... EB Y, is ample for

"'---y------.J q

k < q. This is an exercise for the reader. Moreover, the convex integration allows the case k = q for the homomorphisms L1 of principal rank ~ 2, which means that the rank of ,1 on each principal subspace in y(r) is at least 2. Indeed, the relation Yl c x(r) is ample in this case as the following consideration shows. Every principal subspace ReX:;), V E V, is associated with some hyperplane r E T.,(V) and R is the Cartesian product of q principal subspaces Rj c 1:,<'), j = 1, ... , q, which are also associated to r. The homomorphism ,1 sends the subspaces Rj c ~:), j = 1, ... , q, which are also associated to r. The homomorphism ,1 sends the subspaces Rj on mutually parallel subspaces in Zv, say onto Rj c Zv and dim Rj ~ 2, j = 1, ... , q, provided principal rank (,1) ~ 2.

Lemma. Let R' denote the Cartesian product of the subspaces Rj c Zv,j = 1, ... , q, and let 1:' c R' denote the subset of the q-tuples of those vectors Zj E Rj c Z"' j =


1, ... , q, which are linearly dependent in the ambient space Zv' If d = dim Ri ~ 2, then the convex hull of every connected component of the complement R'\l:' equals R' in so far as I' -=1= R'.

Proof Let first d = 2 and dim Zv = q. Choose some vectors a, band Cj in Zv, j =

1, ... , q, such that every vector Zj E Ri is a combination, Zj = xja + yjb + cj for some real coefficients Xj and yj' and such that the vectors a, b, C2' c3 , ••• , cq are linearly independent in Zv' The exterior product Zl A •.. A Zq is a (non-homogeneous) quadratic polynomial Q(Xj,Yj) = AX1Y2 + BX2Yl + "', where A, B -=1= 0, and the subset I' is given by the equation Q = O. The complement to the hypersurface AX1Y2 + BX2Yl = const in 1R4 has two connected components whose convex hulls equallR4, as A, B -=1= O. Hence, the complement to I' in R' has the same property.

Now we still let d = 2 but let q' = dim Zv > q. For q = 1 the lemma is obvious. If q ~ 2, then we take a linear map Zv -+ IRq which sends the planes Ri to parallel planes, say onto iij c IRq. The above case of q = q' applies to the corresponding singularity f' c R' = R' and then the lemma follows for I' as I' c f'.

Finally, let d > 2. If Z = (Zl"" ,Zq) is an arbitrary point in R'\l:' and if Rj c Ri are parallel planes through the points Zj E Ri, then, by the above, the convex hull Cz

of the component of Z E R'\l:' contains the Cartesian product R" c R' of these planes. Take a linear projection of Zv :::J Ri onto a hyperplane H c Zv such that the subspaces Ri go to (d - I)-dimensional subspaces, say onto iii = Ri n H c H, and the planes Rj go to lines iij = Rj n H c H. The projections Zj E H of the vectors Zj span a subspace in H of dimension ~ q - 1, and so we may assume that the vectors Zl' ... , Zq-l are linearly independent. We apply, by induction, the lemma to the subspaces iii c H, j = 1, ... , q - 1, and we conclude that the convex hull Cz c R' goes onto the product of the subspaces iii c H, j = 1, ... , q, under the projection Zv -+ H. Therefore, the inclusion R" c Cz implies Cz = R'. Q.E.D.

The lemma shows that Conv1 £Jl = Pr 1 £Jl and so £Jl is ample.

Examples. A differential operator D by definition is elliptic if the principal rank equals dim Zv and so the relation £Jl is ample for elliptic operators unless dim Zv = 1.

Let Ybe an exterior power of the cotangent bundle, Y = Am(V),letZ = Am+l(V) and let D be the exterior differential d. It is clear,. that the principal rank of d is at least n - 1, unless m = 0 or m + 1 = n = dim V, and so we have by the h-principle the following

Theorem. Let Wj' j = 1, ... , q, be differential (m + I)-forms on V which are linearly independently at every point v E V. If 1 ~ m ~ n - 1, then there exist exact independent forms wj,j = 1, ... , q on V, such that the systems of forms {wJ and {Wj} can be joined by a homotopy of linearly independent forms. In particular, if the manifold V is parallelizable, then it supports q exact (m + l)-forms (1 ~ m ~ n - 1) for q = n!f(m + l)!(n - m - I)! which are linearly independent at every point VE V.

Corollary (Divergence Free Vector Fields). Let Q be a nonvanishing n-form on V and let L 1, .•• , Lq be linearly independent vector fields on V. If dim V ~ 3, then there exist


q independent fields I j on V whose flows preserve Q and such that there is a homotopy of independent fields between {IJ and {Lj }. In particular, every parallelizable manifold supports n = dim V independent divergence free vector fields.

Proof The flow ofa field L on VpreservesQifandonlyifthe(n - l)-formw = LAQ is closed. Recall the definition:

w(t1,···, tn-d = Q(L, t 1 ,.··, tn- 1 ).

As the correspondence L 1-+ W is induced by an isomorphism T(V) --+ A n- 1(V), exact (n - l)-forms give us the required divergence free (i.e. preserving Q) fields on V. Q.E.D.

Exercises. Generalize the above h-principle to divergence free fields on nonorientable manifolds V.

Show that the h-principle for non-vanishing divergence free fields fails for the 2-torus as every such field on T2 is homotopic to a standard linear field.

Let us sketch another construction of non-vanishing divergence free fields on split manifolds V = V1 X V2 for dim V1 :2': 2 and dim V2 :2': 2. We may assume (see Moser 1966) that the volume form Q also splits, Q = Q1 EB Q2 and then we deform a given non-vanishing field L on V to a non-vanishing divergence free field I as follows. The splitting V = V1 X V2 induces the splitting of the tangent spaces,

and thus the splitting of fields, L = L1 + L 2. Since the form Q splits, this splitting of fields respects the div = ° condition. We assume the original field L = Ll + L2 to be generic, such that the zero set L"(Ld c V of the field Ll meets every submanifold V1 x V2 C V, V1 E V1 , at (at most) finitely many points. Then we deform L to a split field L' = Ll + L~, where L~ is a divergence free field, which does not vanish on L"(L1) and such that the intersection L"(L~) n (V1 x V2) is finite for all V2 E V. The existence of L~ is an easy "zero dimensional" (compare 1.4.4) problem. Then we pass from L' to the required field I = I1 + L~ where I1 is a divergence free field which does not vanish on L"(L~). Q.E.D.

An advantage of this new proof is ~ possibility of applications to complex analytic and algebraic manifolds (compare 2.1.5).

Yet, another construction of non-vanishing divergence free fields is due to Asimov (1976). Amisov uses his round handle decomposition of V and he produces fields with controlled dynamic properties.

Exercise. Fill in the details in the construction of the fields L~ and Il and generalize the construction of I to all (non-split) manifolds V of dimension :2': 4.

2.4.4 Fiber Connected Relations and Directed Immersions

A continuous map between two topological spaces, f: A --+ B, is called a 01ibration if the homotopy lifting property holds for maps of zero-dimensional polyhedra. That

is every path g: [0, 1] ~ B admits a lift to a path G: [0, 1] ~ A which issues from a given point a E A over g(O) E B. We say that f is fiber connected if the pullback f- 1(b) c A is path connected for all bE B. The following two properties of fiber connected microfibrations f: A ~ B are obvious.

(i) If the intersection of the image f(A) c B with an open subset U c B is path connected then the pullback f-1(U) c A is path connected.

(ii) The map of A onto its image, f: A ~ f(A) c B is a O-fibration.

Consider two relations p: fJl ~ Xl') and jj:!j ~ Xl') and let U:!j ~ fJl be a continuous map such that jj = po u. Let UN: PrN!j ~ PrNfJl, N = 1,2, ... ,00, denote the induced maps of the principal extensions.

(A) Lemma. Let the map U be a 01ibration and let (a,x)EPr1~ and (a, x) EPr1fJl be points, such that u(a) = a. Then the point (a, x) is contained in Cony 1 !j C Pr 1 ~ if and only if (a,x)EConv1fJl.

Proof. The inclusion u 1 (Conv 1 !j) C Cony 1 fJl is obvious. Now, if (a, x) E Cony 1 fJl, then by definition there is a path IX: [0, 1] ~ fJl, IX(O) = a which lies over the principal subspace R c Xl') through the points p(a) and x in Xl'), such that the point x E Xl') is surrounded by the path po IX: [0, 1] ~ R. This IX lifts to a path a. in !j with the same image jj 0 a. = po IX in R. Q.E.D.

(A') Corollary. Let u be a 0libration. Then the map UN is a 0libration of ConvN~ over ConvN fJl for N = 1, ... , 00. Furthermore, if fJl is ample then ~ also is ample. In particular, if jj: ~ ~ Xl') is a 0libration then !j is ample.

Proof First, the map u 1: Cony 1 !j ~ Cony 1 fJl is a O-fibration. Indeed, if (a(t), x(t» is a path [0, 1] ~ Cony 1 fJl, then there is a lift a(t) of a(t) to !Jt for every given point a(O) E!j over a(O) and by the previous lemma the path (a(t), x(t» lies in Cony 1!j. Then by induction we obtain the same result for all N = 2, 3, ... , and for N = 00.

Also by this argument a point (a,X(t»EProo!j lies in ConvoofJl if and only if (a, x(t» E Cony 00 fJl for a = u(a). Q.E.D.

(B) Lemma. Let a: [0, 1] ~ fJl be a continuous path whose projection y = po a to Xl') is piecewise principal. Let b = (a(l),x = x(t» and b* = (a(O),z = z(t» be two points in Pr 00 fJl, where the path z is the composition of paths, z = y * x. Then [b E Convoo fJl] <::> [b* E Convoo fJl].

Proof The implication => is immediate from the definition of Cony 00 and the implication <= follows from =>. Q.E.D.

Two points ao and al in fJl by definition lie in the same principal component of fJl if they can be joined by a path whose projection to Xl') is piecewise principal. Each principal component in fJl is sent by the map p: fJl ~ Xl') to some fiber XZ) of the fibration Xl') ~ X(,-l), X E X(,-1). If the relation fJl is 'open, then the principal components of fJl are exactly and only the path connected components of the


pullbacks p -1 (X:) c fYt for all x E x(r-l), as paths [0, 1 J -+ p -1 (X:) may be perturbed to those which have piecewise principal projections. Lemma (B) now implies the following

(B') Corollary. Let the image of the map u: ~ -+ fYt meet every principal component of fYt. If ~ is ample over x(r) then fYt also is ample over x(r).

(C) Recipes for Checking the Ampleness. A relation fYt -+ x(r) is ample over x(r) if and only if each path connected component of the intersection p -1 (X:») is ample for all XEX:- 1). (Observe that empty relations are ample). Thus the verification of the ampleness reduces to those path connected relations fYt which lie over a single fiber, fYt -+ X:) c x(r), X E x(r-1). The fiber X:) is naturally isomorphic to the space Hn = H~(q) of homogeneous polynomial maps IRn -+ IRq of degree r. The ampleness of fYt -+ Hn depends on how fYt interacts with principal subspaces R ~ IRq in Hm where "ampleness" is understood (here and below) as the ampleness over x(r). We obtain with (A') and (B') the following list of implications which helps us to establish the ampleness of relations p: fYt -+ Hn.

(a) fYt is ample ¢> each path connected component of fYt is ample. (a') 9t is ample ¢> each principal component of fYt is ample.

[If fYt is open then (a) is equivalent to (a').J (b) fYt is ample¢>Conv1 fYt is ample. (b') fYt is ample ¢> Conv 00 fYt is ample. (c) Let fYt be a O-fibration over its image fYt' c Hn. Then

fYt is ample ¢> fYt' is ample. (d) Let fYt consists of a single principal component and let 9t' c fYt -+ Hn. Then

fYt' is ample => fYt is ample, unless fYt' is an empty set. Finally, the empty relation and fYt = Hn are ample.

One can generate with (a)-(d) a great deal of ample relations over (as well as in) Hn and so over x(r).

Example. Triangular Relations. Let r = 1. [See (C) in 2.4.4 for r 2': 2.J Then every linear embedding IRn- 1 -+ IRn induces an obvious linear map hn: Hn -+ Hn- 1 whose fibers are principal subspaces in Hn. By induction we define a class of relation fYt cHi' i = 0, 1, ... , n, which are called triangular, as follows. If i = ° then all relations in Ho = ° (i.e. fYt = 0 and fYt = 0) are triangular. A relation fYt c Hi is triangular if and only if there is an embedding lR i - 1 c lR i such that the corresponding projection hi: Hi -+ Hi- 1 satisfies three conditions,

(i) the image hi(fYt) c Hi- 1 is triangular; (ii) the map hi is a O-fibration of fYt over hi(fYt). [For example, the map hi: fYt -+ hi(fYt)

is a fiber connected submersion.J; (iii) the convex hull of each path connected component of fYt n (hi- 1 )(x) equals hi- 1 (x)

for all x E hi(fYt).

Triangular conditions are ample [use (b) and (c)J and so we have the following


Theorem. Let fli c X(1) be an open relation such that each connected component of the intersection fli n X~l) contains a non-empty triangular relation for all x E X. Then fli satisfies the dense h-principle.

Let us apply this theorem to immersions of an oriented n-dimensional manifold, f: V ~ IRn+l, which are directed (compare 1.4.4), by a given open connected subset A c Grn(lRn+1) = sn, that is Gf(V) c A for the oriented normal map G/ V ~ sn. The corresponding differential relation fli = fli(A) c Hn = Hom(lRn ~ IRn+1) by definition consists of those injective linear maps L: IRn ~ IRn+1 for which L(lRn)EA. Take a small open e-ball Bo in sn, fix a hyperplane 1R~-1 C IRn, and denote by flio c Hn the set of those maps L: IRn ~ IRn+1 which send 1R~-1 into some hyperplane bE Bo c

Grn(lRn+1) = sn. A straightforward check up shows,

If every great circle Sl in sn, which meets Bo, intersects the subset A over an arc of length > n, then the relation fli n flio c Hn is triangular.

Now fix a point bo E sn and consider the pencil of great circles Sl C sn through boo Let A satisfy the following condition.

(D) The intersection Ansi contains an arc of length > n for all Sl in the pencil.

This (D) obviously implies the existence of a triangular subset flio c fli = fli(A) and so fli is ample. Hence we have the following

(D') Theorem. If A satisfies (D) then A-directed immersions abide the CO-dense h-principle. In particular every continuous map of a parallelizable manifold V into IRn+1 admits a fine approximation by A-directed immersions V ~ IRn+1 (notice that V is necessarily parallelizable if there exists an A-directed immersion V ~ IRn+1 for A -# sn).

The condition (D) is satisfied, for instance, in the following two cases. (1) The complement ~ = SR\A is a finite subset with no pairs of opposite points. (2) The subset A contains a closed hemisphere.

Exercise. Give a direct geometric construction of an immersion of the n-torus yn ~ IRn+1 whose oriented spherical image is contained in a small neighborhood of the hemisphere S~ c sn.

Question. Is there a "simple" immersion T2 ~ 1R3 whose spherical image misses the four vertices of a regular tetrahedron in S2?

(E) Necessary Conditions for the Existence of A-Directed Immersions V ~ IRq. There is a wide gap between known suffficie~t and necessary conditions for the validity of the h-principle for A-directed immersions. For example the only known necessary condition for the existence of an A-directed immersion f: V ~ IRn+l for closed parallelizable manifolds V is as follows. The double cover map sn ~ pn sends every path connected component of A onto the projective space PR. Indeed, every linear function IRn ~ IR has critical points on f(V) c IRR.


Exercise. Show that there is no A -directed embeddings of a closed manifold V -+ ~n+1 unless A = sn.

A similar necessary condition holds true for immersions f of closed manifolds V -+ ~q for all q > n = dim V.

(i) If the non-oriented tangential map Gf : V -+ Grn(~q) sends V into a given subset A c Grn(~q) then A intersects the Grassmann manifold Grn(H) c Grn(~q) for all hyperplanes H c ~q.

There is an additional condition on A (for q ~ n + 2) which is due to the obvious fact that every exact n-form on V necessarily vanishes. Namely, let w denote an n-form in ~q with constant coefficients and let ..r(w) c Grn(~q) be the subset of those n-dimensional subspaces in ~q on which w vanishes.

(ii) The set A intersects ..r(w) for all n-forms w.

Questions. Let an open connected subset A c Grn(~q) satisfy (i) and (ii). Does the h-principle hold for A-directed immersions? Does there exist an A-directed immersion of at least one closed manifold Vo -+ ~q? Does the existence of a single Adirected immersion imply the h-principle for A-directed immersions of all ndimensional manifolds V -+ IRq?

The latter question is motivated by the following

Example. A C1-map E: Vo -+ Vo is called expanding if some iterate EN of E strictly enlarges the length of all non-zero tangent vectors in Vo relative to a fixed Riemannian metric on Yo. (If Vo is compact then the expanding property does not depend on the choice ofthe metric). For instance, the map x H 2x ofthe n-torus Tn = IRn/7l..n

onto itself is expanding. [See Shub (1969) for additional examples.]

Let a compact manifold Vo admit an expanding map E: Vo -+ Yo. If there exists an A-directed immersion fo: Vo -+ ~q for some open subset A c Grn(~q), then an arbitrary continuous map f: V -+ IRq can be uniformely approximated by A-directed immersions.

Proof We may assumethemapfto be Cl. Then themapsf + N-lfNforfN = foEN

are A-directed immersions for large N and the sequence f + N-1fN uniformely converges to f as N -+ 00.

Exercise. Show that every compact manifold Vo which admits an expanding map has IRn for the universal covering.

Remark. Farrel and Jones (1981) introduced a notion of a compact branched manifold Vo and produced many examples of expanding maps Vo -+ Vo such that the above discussion applies. Their results indicate the existence of many compact n-dimensional subpolyhedra P c GrilRq) such that immersions V -+ IRq which are directed by arbitrarily small neighborhoods (9/tP c Grn(lRq) satisfy the h-principle.

(E') Directed Immersions of Open Manifolds. If V is open, then the h-principle for A-directed immersions f: V --+ IRq holds true for all open subsets A c: Grn(lRq) (see 2.2.2). However, if we additionally require the immersions f to be complete, which means the completeness of the induced Riemannian metric g on V, then we can partially recover the necessary conditions (i) and (ii).

Examples. (a) Let V have no boundary and let f: V --+ IRn+l, n = dim V, be a complete immersion whose non-oriented normal map Gf : V --+ pn sends V into a proper closed subset ..4 c: pn. Then the map f is unbounded and the manifold V topologically splits, V = V' x IR. Indeed, the normal projection to a line IE pn \..4 gives us an infinitesimally enlarging (in the metric eg on V for some e > 0) map V --+ I which is a fibration (see 1.2.3).

(a') If GiV\ Yo) c: ..4 for some compact subset Vo c: V, then the map V --+ I is infinitesimally enlarging (in eg) at infinity. Hence, the map f is unbounded, the manifold V has at most finitely many ends and the intersection map on homology, H*{V) ® H*(V) --+ H*(V), has finite rank, provided V is connected. This implies for dim V = 2 the finiteness of the topological type of V.

Subexample. Let the map Gf : V --+ pn be an open map outside a compact subset in V. (For instance, Gf is an immersion or a branched covering at infinity). If the pullback Gjl (p) c: V is compact for all p E pn, then the image Gf{V\ Yo) c: pn is not dense for some compact subset Vo c: V and so (a') applies.

(a") If the closure of the image Gf(V\ Yo) c: pn misses a hyperplane in pn, then the projection of V to this "missing" hyperplane H c: IRn+l is infinitesimally enlarging (for eg) outside Vo and so the projection V --+ H is a covering map at infinity. Hence, each end of V is diffeomorphic to sn-l X IR.

Question. For which open subsets A c: sn is there a complete immersion of some oriented manifold, f: V --+ IRn+l, whose oriented Gauss map is a diffeomorphism of V onto A? [See Verner (1970) and Burago (1968) for partial results.]

Here is a related result by Burago: (b) Let V be a connected open surface without boundary and let f: V --+ 1R3 be

a complete immersion of infinite total area. If the Gauss map Gf : V --+ p2 has finite total area (i.e. the Jacobian J of Gf has Iv IJI < 00) then the map f is unbounded and the Euler characteristic of V is finite.

Proof Let Al (v) and A2(V) denote the principal curvatures ofthe immersion at the points VE V. Then the (intrinsic) Gauss curvature K(v) = Al (V)A2(V) has IvK(v) > - 00, and so by the Gauss-Bonnet theorem (see Huber 1957) the Euler characteristic of V is finite.

Now, suppose the image f(V) c: 1R3 is contained in a ball; say in B c: 1R3. Take a projective map P of B into the unite sphere S3. Then the principal curvatures A'l


and A~ of the immersion Pol: V -+ S3 satisfy A~ (V)A~{V) ~ const Al (V)A2{V) for all VE V and for some const > ° (compare 3.2.3).

The Riemannian metric induced by Pol is complete and its total area is infinite. Since the Gauss curvature is K'{v) = 1 + A'l (V)A~{V), we conclude

Iv K'{v)dv = Area V- const Iv K{v)dv = +00.

This contradicts a theorem of Cohn-Vossen (1933). (c) Let I: V -+ IRq, q ~ n + 1, be a complete bounded immersion, such that the

closure of the image of the non-oriented tangential map G/ V -+ Grn{lRq) misses the subset 17{w) c Grn{lRq) for some n-form w on IRq with constant coefficients. Then V has the exponential growth relative to the induced Riemannian metric,

lim infR-IlogVoIBJR) > 0, R-+oo

for the concentric balls Bv{R) around each point v E V.

Proof If ( *) fails, then

lim infVoloBv{R)/VoIBJR) = 0. R-+oo

Since the tangential image Gf{V) c Grn{lRq) lies away from 17{w),

1 r I*{w) 1 ~ b Vol Bv{R), JBv(R)

for all R > ° and for some b > 0. Let w = dA for some (n - I)-form A on IRq. Since 1 is bounded,

1 r 1* (A) 1 ::; constVoloBv{R),

JOBv(R)

and Stokes formula, fBI*{w) = fOB 1* (A), yields the contradiction.

Subexample. Look at immersions of surfaces V -+ 1R4 = ([2 and let w = dX I 1\

dYI + dX2 1\ dY2 for the coordinates Xl + iYI and X2 + iY2 in ([2. The form w does not vanish on complex lines in ([2 and so the set 17{w) c Gr2 {1R4 ) is disjoint from the complex projective line ([pI c Gr2{1R4 ). The immersions V -+ ([2 directed by ([pI are holomorphic curves in ([2, and we see that all complete bounded holomorphic curves in ([2, as well as nearly holomorphic curves whose tangential image is close to ([pI c Gr2 {1R4 ), have exponential growth.

2.4.5 Directed Embeddings and the Relative h-Principle

Consider relations 910 c x(r) and p: 9l-+ 910 c x(r). Denote by Pr l (9l19l0 ) c Pr l 9l

the subset of those pairs {a,x)EPr l 9l for which the segment (I - t)p{a) + tx, tE [0, 1], is contained in 9l0 • Then by induction we define

PrN(9l!I9lo) = Pr1(prN- 1 (9l!I9lo)l9lo),

for N = 2, 3, ... , and we stabilize to Proo(9lI9lo) c Proo9l! that is the space of those pairs (a,y(t))EProo 9l! for which the piecewise principal path y sends [0,1] into 9l!o.

The lemma (D) in 2.4.2 immediately implies the following trivial

h-Principle for 9l! Relative to (the h-Principle for) 9l!o. Let 9l!o c x(r) and P;-l 0

p: 9l ~x(r-l) be open relations. If the extension Proo(9lI9l!o) ::::l 9l is h-stable then

[the C-1-dense h-principle for 9l!o] => [the C-1-dense h-principle for 9l].

Next we define

ConvN(9lI9lo) = ConvN(9l!)nPrN(9lI9l!o),

for N = 1,2, ... , 00, and we call the relation 9l ample over 9l!o if Convoo (9lI9l!o) = Proo(9lI9l!o). The principal stability theorem [see (B) and (C) in 2.4.2] implies the h-stability of the extension Convoo (9lI9l!o) ::::l 9l! for open relations 9l! ~ 9l!o c x(r),

and so we have the relative h-principle for ample relations over 9lo. Now, the (obvious modification of) implications (a)-(d) in (C) of 2.4.4 hold true

over 9lo and so we have a list of sufficient conditions for the ampleness of 9l over 9l!o. Unfortunately, the inclusion Pr1 Conv1 c Conv1 Pr1 may fail over 9lo -::f. x(r)

and so we lose the sufficient condition Conv 1 = Pr1 (compare 2.4.3). However, this condition still works if modified as follows. We restrict to an individual fiber Xf) = Hn [compare (C) in 2.4.4] and we fix a Euclidean metric "dist" in Hn. Next we introduce another metric, distp (x, y) (P for principal), as the lower bound of the lengths of piecewise principal paths between the points x and y in Hn.

Observe that

dist ::; distp ::; const dist,

for const = const(dim Hn). We impose on relations 9lo c Hn and p: 9l ~ 9lo the following

A-Condition. If b = distp (p(a), x) ::; A -1 distp (x, o9lo), for the boundary o9lo of 9lo c x(r), then

for all (a, x) E Pr 1 (9l!I9lo), where Bx(Ab) denotes the distp-ball around x E 9l!o of radius Ab, and where A 2 1 is a given number.

Denote by Pr). c Proo(9lI9lo) the subset of those pairs (a,x(t)) EProo(9lI9l!o) for which the path x(t), t E [0, 1], has length < (A + It1 distp (x(I), o9l!o).

It directly follows from these definitions that

[A-condition] => CPr). c Convoo (9lI9l!o)].

(A) Corollary. If open relations 9l!o c Hn and p: 9l ~ 9l!o satisfy the A-condition for some A 2 1 and if the map Pr). ~ 9l!o is a 0-:fibration then the relation 9l is ample over 9lo. In particular, if Conv 1 (p -l(U)1 U) = Pr1 (p -l(U)1 U) for all open subsets U c 9l!o, then 9l! is ample over 9l!o.

(B) Examples. (a) Let E c x(r) be a closed stratified subset such that the intersection R n E is either equal to R or codim(R nEe R) ~ 2 for all principal subspaces R c x(r). Then the difference f7t = f7to \E c+ f7to is an ample relation over (in fact, in) f7to for all open subsets f7to c x(r), as the equality ConvdU\EIU) = Prl(U\EIU) is satisfied for all open subsets U c x(r). Hence, we get the h-principle for f7t relative to f7to. This applies, for instance, to the relations f7tk c f7to c X(l), X = V x W, which define strictly short maps V -+ W of rank ~ k between Riemannian manifolds V and W, provided k < dim W. Therefore,

Every strictly short map V -+ W, which is homotopic to a Cl-map of rank ~ k admits a fine CO-approximation by strictly short Cl-maps V -+ W of rank ~k, provided k < dim W. (Compare 1.1.5.)

(b) Let '"C c T(W) be a codimension n subbundle and let the relation f7to c X(1), X = V x W, correspond to Cl-maps V -+ Wwhich are transversal to '"C. The convex integration completely fails in proving the h-principle for such an f7to as the "convex hulls" Conv N f7to -+ X(l), N = 1, ... , 00, are not greater than f7to (in fact, their images in X(l) equal f7to). However, the h-principle does hold for f7to if the subbundle '"C is "sufficiently non-integrable" (see 2.2.3). Now, we take f7t = f7to \E for the above E and we derive the h-principle for f7t from that for f7to.

The proof of the h-principle for f7to depends upon Nash's implicit function techniques as well as on the formalism of continuous sheaves (see 2.2, 2.3). One expects a simpler argument which would incorporate the non-integrability of'"C into the convex integration scheme thus giving a direct geometric proof ofthe h-principle for f7to and for f7t.

(C) The Relative h-Principle over Embeddings. Consider a relation f7to c x(r) and let tPo be a subset in the space ofholonomic sections V -+ f7to. Take a relation f7t -+ f7to

r---J and denote by f7t n tPo the pullback of tPo under the map between the spaces of sections, r(f7t) -+ r(f7to) ::::> tPo. An extension f7tl ::::> f7t is called stable over tPo (or

r----...- r----...J

tPo-stable) if every section iio E f7t 1 n tPo admits a homotopy iit E f7t 1 n tPo, t E [0, 1], ~

such that ii l Ef7t n tPo and such that iit(v) = ao(v), tE [0,1], for those VE V, for which iio(V)Ef7t c f7tl (compare 2.4.2).

We fix a metric in the space x(r) and then we may speak of e-neighborhoods Ue(cp) c r(f7to) of sections cp: V -+ f7to, where e = e(x) stands for a continuous positive function on f7to. We also associate to such an e: f7to -+ IR the following

e-Condition. Let cP E tPo be an arbitrary section. Then there exists a fine CO-neighborhood of the section P;-l 0 cP: V -+ x(r-l), say U' c rO(x(r-l»), such that every holonomic section tjJ c Ue(cp), whose projection to rO(x(r-l») lies in U', is contained in tPo and there is a homotopy in tPo between cP and tjJ which is constant at those v E V where cp(v) = tjJ{v).

Example. Let X = V x W, let f7to c X(l) be the immersion relation and let tPo be the space of the I-jets of Cl-embeddings V -+ W. Then tPo satisfies the e-condition for a fixed metric on X(l) and for all sufficiently small e.

Indeed, if an immersion "': V -+ W is CO-close to a given embedding <p: V -+ W and if the angle between the tangent spaces T,,(<p(V» and T"(,,,(V» in T(W) is at most ~ - (j for all v E V and for a fixed (j > 0, then the immersion '" clearly is an embedding which is isotopic to <p.

We assume the metric on x(r) to be Euclidean on the fibers of the fibration x(r) -+ x(r-l) and we take a function e on gJo for which dist(x, iJgJo) > e(x) for all XEgJo. Let Conv1 (gJ,e) c ConvdgJ I gJo), for p:gJ-+gJo, denote the union over XEgJo ofthe subsets Conv1 (p-l(Bx(e(x))) IBAe(x))) c Conv1 (gJlgJo) for the e-balls BAe(x)). Then by induction we define ConvN(gJ,e) c ConvN(9llgJo) for N = 2,3, ... , and as earlier we define Convoo (gJ, e) c Convoo (gJlgJo).

The principal stability theorem [see (C) in 2.4.2] implies:

If r[>o satisfies the e-condition, then the extension Convoo (gJ, e) :::> gJ is CPo-stable.

Next, the proof of (A) implies the following result for open relations gJo c x(r)

and p: gJ -+ gJo.

If Conv1 (p-l(U)1 U) = Pr1 (p-l(U)1 U) for all open subsets U c gJo, then

Convoo(gJ,e) = Convoo(gJlgJo) = Proo(gJlgJo),

for all positive continuous functions e: gJo -+~, and so the extension Proo(gJlgJo) is r[>o-stable for all r[>o which satisfy the e-condition for some continuous function e > O.

Recall that a pair (a,x(t))E Proo(gJlgJo) by definition is contained in Convoo (gJ,e) if there is a subdivision of some N = 1, 2, ... , such that (a,x(t))EConvN(gJ,e) c PrN(gJlgJo) relative to this subdivision (compare 2.4.2).

Totally Real Embeddings. Let W be a smooth manifold with a complex (or quasicomplex) structure. A smooth map f: V -+ W is called totally real if the complexified differential Df : CT(V) -+ T(W) has rankeDf ~ min (dim V, dime W).

Every real analytic map f: V -+ Wextends to a holomorphic map Cf: CV -+ W for some (small) complexification CV :::> V. Totally real maps extend to holomorphic immersions C V -+ W for n = dim V ::; dime Wand to holomorphic submersions for n ~ dime W.

The total reality condition gJ c X(l), X = V x W, is of the form gJ = X(l)\E, where E is a stratified subset such that codim(R nEe R) ~ 2 for all principal subspaces R c X(l) which are not completely contained in E. This implies the h-principle for totally real maps V -+ W.

Now, let gJo c X(l) be the immersion condition, let r[>o be the space of the i-jets of embeddings V -+ Wand let gJ = gJo \E for the above E. The CPo-stability of the extension Proo(gJlgJo) leads to the following

Theorem. Let fo: V -+ W be a smooth embedding whose differential Dfo: T(V) -+ T(W) is homotopic (viafiberwise injective homomorphisms) T(V) -+ T(W)!) to a totally real homo-<Pl: T(V) -+ T(W) [i.e. ranke C<Pl ~ min(n, dime W)]. Then fo is isotopic to a totally real embedding fl: V -+ W.

Example. A totally real embedding sn ~ en exists if and only if n = 1,3.

Proof The multiplication by i = j=1 establishes an isomorphism between the tangent and normal bundles of a totally real immersion vn ~ en. As embedded spheres sn ~ ~2n have trivial normal bundles, the existence of a totally real embedding sn ~ en implies the parallelizability of sn. Hence n = 1, 3 or 7. [We refer to Bott (1969) for the basic algebraic topology which is used here and below.]

The case n = 1 is trivial as all immersions S1 ~ e1 are totally real. Let Stn denote the Stiefel manifolds of n-frames in ~2n = en and let DC: U(n) ~

SO(2n) and p: SO(2n) ~ Stn be the obvious maps. The above theorem shows that an embedding sn ~ en with a normal frame

v: sn ~ Stn is isotopic to a totally real embedding if and only if there exists a homotopy class JLE1tn(U(n» for which (poDC)*(JL) = v*[sn].

Ifn = 3, then DC*: 1tn(U(n» ~ 1tn(SO(2n» is an isomorphism while the homomorphism p*: 1tn(SO(2n» ~ 1tn(Stn) is onto. Hence, the standard embedding S3 ~ e3 is isotopic to the totally real one.

If n = 7 then the image of the homomorphism P* is divisible by 2 [in 1t7(SO(14» ~ £:], while 1t7(St7) = £:2. Hence (poDC)* = O. On the other hand, every embedding f: S7 ~ ~14 carries a normal frame v: S7 ~ St7 which is not homotopic to zero and so f is not isotopic to a totally real map S7 ~ ~14. Q.E.D.

Corollary. There exists a domain of holomorphy in e3 which is diffeomorphic to S3 x [R3.

Proof Take a real analytic totally real embedding f: S3 ~ e3 and then analytically continue f to a small complexification es3 ~ S3 x ~3 with a pseudo-convex boundary.

Remark. There are other interesting classes of maps besides embeddings to which the above techniques may apply. For example, one may try maps V ~ ~q which meet every k-dimensional subspace at no more than m points for given k and m [compare Szucs (1982)].

Exercise. The total reality condition for immersions V ~ en can be expressed by the non-vanishing of the angle 1: (t, iT) =I- 0 for all non-zero tangent vector t and T E T,,(V) and for all v E V. A stronger condition, say qtrz E X(l), is 1: (t, iT) > DC for given DC in the interval 0 < DC < l Prove the h principle for qtrz as well as the relative h-principle over embeddings. Show that every embeddingfo: V ~ en which satisfies qtrz admits an isotopy of embeddings fr, tE [0, 1], which also satisfy qtrz and such that f1 satisfies qtp for a given P in the interval DC ~ P < ~.

(C') Directed Embeddings of Open Manifolds. Take an open subset in the Grassmann bundle of n-planes in W, say A c Grn W, and let qt = qt(A) c qto be the subrelation in the immersion relation qto which distinguishes immersions f: V ~ W whose tangential lift Gf : V ~ Grn W sends V into A.

Theorem. If V is an open manifold, then fYl satisfies the relative h-principle over embeddings V -+ W Namely, every embedding fo: V -+ W whose tangential lift GIo : V -+ Grn W is homotopic to a map G1 : V -+ A c Grn(W) can be isotoped to an embedding fl for which GIl (V) -+ A.

Proof First, let V = V' X lR i , i ~ 1, and let fYl' = Pl1(V' X 0) for the projection PI: X(!) -+ V, where X = V x W -+ V. This fJ1l' naturally lies over X//) for Xo =

(V' x 0) x W -+ V' x 0 and the earlier consideration yields the h-principle for fJ1l' over embeddings V' -+ W; this implies the theorem for V = V' X lRi. Now, any open manifold decomposes into handles Dn- i x lR i for i = 1, ... , n, to which the above yields the h-principle for extensions from aDn - i x lRi to D n- i x lRi. This implies the theorem for all open manifolds V.

2.4.6 Convex Integration of Partial Differential Equations

We want to reduce the h-principle for a non-open relation p: fJ1l-+ x(r) to that for an auxiliary open relation p*: fJ1l* -+ x(r). We assume that fJ1l admits a metric, say dist on fYl, such that (fJ1l, dist) is a complete metric space. Denote by PfYl the space of pairs (a, x(t)), where a E fYl and where x(t), t E [0,1], is a path in x(r) such that x(O) = a and such that the path x(t) is piecewise principal on the semiopen interval (0,1]. We allow this path to have infinitely many principal segments in (0,1] which may accumulate to OE[O,l]. The space PfJ1llies over x(r) for (a,x(t))f-+x(l), and PfJ1l extends fJ1l for obvious maps E: fJ1l c PfYl and II: PfJ1l-+ fJ1l. The principal extension Pr 00 fYl clearly is a subextension of PfJ1l. Furthermore, there is a natural isomorphism Proo (PfJ1l):::::; PfJ1l, where (a,x(t),y(t) f-+(a, z(t)) for

z(t) = {X(2t), 0 ~ t ~ !. y(2t - 1), t ~ t ~ 1.

Consider a subset fJ1l' c PfJ1l, take s > 0 and denote by fJ1l~(s) c fYl', a E fYl, the intersection fJ1l' n II-I (Ba(s)) for the s-ball Ba(s) c fJ1l around a E fJ1l. Consider the union UaE9l Conv 00 fJ1l~(s) c Pr oo(PfYl) and let C~fJ1l' c PfJ1l denote the image of this union under the isomorphism Proo(PfYl):::::; PfYl.

In-extensions. A subextension fYl+ c PfYl of fJ1l is called an in-extension (of fYl) if the following three conditions are satisfied

(1) If fJ1l' c fJ1l+ is an arbitrary neighborhood of fYl c fYl+, then the sub relation fYl* = fYl+ \fJ1l c PfJ1l is contained in the image of Conv 00 (fYl' n fJ1l*) under the isomorphism Pr 00 (PfJ1l) :::::; PfYl.

(2) There exists, for an arbitrary s > 0, a neighborhood fJ1l' c fYl+ of fJ1l such that fJ1l' n fJ1l* c C~(fJ1l' n fJ1l*) for all neighborhoods fJ1l' c fYl+ of fJ1l c fJ1l+.

(3) The subrelation R+ c PfJ1l is invariant under those piecewise linear homeomorphisms h: [0, 1J -+ [0, bJ, 0 (a, x(h(t))) E fYl+.

(A) Theorem. Let ~+ c ~ be an in-extension for which the relation .o/t* = ~+ \~ is open over x(r). Then the extension ~+ :::J ~ is h-stable.

Proof. Take a sequence ei > 0, i = 1, ... , such that L~l ei < 00 and let ~i = ~ei [see (2) above]. Let ({Jo: V ---+ ~+ be a holonomic section which is to be deformed into ~ = ni~l ~i c .o/t+. Suppose for the moment that the image ({J(V) c ~+ does not meet ~ c ~+. Then the principal stability theorem (see 2.4.2) implies with (1) above that ({J admits a homotopy to a holonomic section ({J 1: V ---+ ~ 1 n ~*. Property (2) allows one to homotope ({Jl to a holonomic section ({J2: ~2 n ~*, then to ({J3: ~3 n ~* and so on. Moreover, the holonomic section ({Ji: V ---+ ~i n ~* can be deformed to ({Ji+1: .o/ti+1 n ~*, such that dist(11 0 ({Ji+1 (v), 11 0 ({Ji(V)) ::;; ei for the projection 11: ~+ ---+ ~ and for all v E V This is done by considering the ecneighborhood Ui = Ue/11 0 ({Ji(V)) c ~ and by applying the principal stability theorem to the extension Convoo (~i+1 n ~* n 11-1 (UJ). Since the space (~, dist) is complete, there exists a common limit of the sequences of sections 110 ({Ji: V ---+ ~ and ({Ji: V ---+

~* c ~+ which is the desired holonomic section limi->oo ({Ji = ({J: V ---+~. Now, if the section ({Jo: V ---+ .o/t+ meets~, we apply the above argument over the

complement V\({JOl (~) thus concluding the proof of the theorem.

(A') In-deformations. We have reduced the h-principle for ~ to that for ~+. But the relation ~+ ---+ x(r) may not be open and so we need an additional deformation of (sections of) ~ to ~*. Namely, we need an in-deformation of ~ to ~*, that by definition is a continuous map 0": ~ ---+ ~* such that 11 00" = Id. Such a map 0" is given by a continuous family of paths xa(t), a E~, such that xa(O) = a and (a, xa(t)) E

~* for all aE~. Now (A) implies the following

Corollary. If ~ admits an in-deformation to the open relation ~* = ~+ \~ for an in-extension ~+ :::J ~ then the h-principle for ~ reduces to that for ~*.

2.4.7 Underdetermined Evolution Equations

Let r be a continuous tangent hyperplane field on V and let p -L: x(r) ---+ X -L be the corresponding affine fibration whose fibers xf') ~ IRq, x E X -L are principal subspaces in x(r) (see 2.4.1). We study relations ~ c x(r) which are locally closed subsets in x(r) (i.e. ~ is closed in some open subset U c ~). Such an .o/t always carries some complete metric. If ~ meets each fiber Xf'), x E X-L, at a unique point a = ~ n xf') then the relation .o/t can be expressed by a determined evolution system of P.D.E. of Cauchy-Kovalevskaja form, a;f = IjJ(Jt), where f: V ---+ X is the unknown section, where ljJ(x) = a = ~ n Xf'), and where a; denotes the rth order derivative in the direction of some fixed vector field transversal to r. (Compare 1.1.1) Here we are concerned with relations ~ whose intersections ~ n Xf') have positive dimension and which correspond to underdetermined P.D.E. systems. [Compare Gromov (1973); Spring (1983,1984).]

Nowhere Flat Sets and Maps. A continuous map oc: A -+ ~q is called nowhere flat if the pullback of every affine hyperplane, oc-1(H) c A, is a nowhere dense subset in A. In particular, a subset A c ~q is nowhere flat if the intersection A n H is nowhere dense in A for all hyperplanes H c ~q. The empty set is nowhere flat according to this definition. Generic maps of manifolds A of dim A ~ 1 are nowhere flat.

A path x:[0,1]-+~q is called an in-path at aEA if x(O)=oc(a)E~q and if x(t) E Int Conv oc(U) for 0 < t < e, where U c A is an arbitrary neighborhood of a E A, and e = e(U) > O. For example, in-paths of a convex hypersurface A c ~n are those which approach A from inside.

Let A be a metric space and let the map oc: A -+ ~q be nowhere flat. Then the convex hull of the oc-image of the c5-ball around a E A has non-empty interior, Int Conv oc(Ba(c5)) # 0 for all a E A and c5 > 0, and so every value oc(a) E ~q can be approached by an in-path xa(t). In fact, let Ya(c5) be the barycenter of the set Convoc(Ba(c5)) c ~q. Then xU<t) = t-1 J~Ya(c5)dc5 is the required in-path which is continuous in the variables (t, a) E [0,1] x A.

We return to the fibration p~: Xl') -+ X~ and we consider a relation p: 9t -+ Xl') which satisfies the following four conditions.

(1) 9t carries a complete metric. (2) 9t is open over X ~. That is the map p ~ 0 p: 9t -+ X ~ is a microfibration. For

example, A is a submanifold in Xl') which is transversal to the fibers of the fibration p ~: Xl') -+ X ~. .

(3) The subset 9tx = p -1 (Xr») = (p ~ 0 ptl(X) C 9t is locally path connected for all XEX~.

(4) The map p: 9tx -+ Xr) ~ ~q is nowhere flat for all XEX~.

Take a point aE9tx , consider the path connected component of a in 9tx and let Ca c Xr) be the interior of the convex hull (in Xr) ~ ~q) of the p-image of this component. Let 9t* c P9t be the subset of those (a, z(t)) E P9t, for which the path z(t) lies in Ca and which is an in-path at aE9tx for the map p: PAx -+ Xr). The conditions (2) and (4) show the relation 9t* -+ x(r) to be open. The condition (3) implies that 9t+ = 9t* U 9t is an in-extension of 9t. Thus, we obtain with (A)

The Local Solvability of fffi. If U c V is a small neighborhood of a point v E V which lies in the image of the map 9t -+ V, then there exists a holonomic section U -+ 9t.

Example. Let 9t be a submanifold in Xl') of codimension s which locally represents a system of s P.D.E. in the unknown map V -+ ~q.1f s ~ q - 1 and if 9t is a generic submanifold, then there is a principal subspace R ~ ~q in Xl') which is transversal to 9t and such that the intersection 9t n R is a nowhere flat submanifold in R of positive dimension. A small neighborhood in 9t of each point a E 9t n R satisfies the conditions (1)-(4) for some (locally defined) field t and so 9t admits a C'-solution U -+ X over some neighborhood U c X.

This local solvability differs from the similar Coo-result (see 2.3.8) in several respects.


(a) The genericity condition in 2.3.8 assumes ~ c x(r) to be a Coo-manifold (or at least a Ck-manifold for large k), while ~ is only C1-smooth here. (b) The "nowhere flat" condition excludes linear and quasilinear systems of P.D.E. for which ~x = ~ n X,r) is an affine subspace in the fiber X,r) of the fibration P;-l: x(r) -+ x(r-l) for all x E x(r-l), while the assumptions in 2.3.8 allow many linear and quasilinear systems. (c) Local solutions delivered in 2.3.8 are COO-smooth, provided ~ c x(r) is a coo _

smooth submanifold. But no (known) regularity assumption on ~ helps the convex integration to produce a Ck-solution for k ~ r + 1. (d) The space of local solutions U -+ X under the above assumptions (1)-(4) is "large": The cr-1-closure of this space has a non-empty interior in the space of cr-1-sections U -+ X. (In fact, the C.i-closure has a non-empty interior.) A similar result fails to be true in the context of 2.3.8. Indeed, linear P.D.E.-systems are functionally closed (see 1.2.3) and so their solutions are nowhere Ci-dense for all i = 0, 1, ....

One has with these remarks (a)-(d) several open

Questions. Let ~ c x(r) be a generic Ck-submanifold of codimension s ::;; q - 1. For which I = I(k, r, dim X) does there exist local C'-solutions U -+ X of ~? Under what assumptions are the cr+1-solutions U -+ X CO-dense in some open subset in the space of continuous sections U -+ X?

Now, we turn to the

Global Solution of r!Il. Let p: ~ -+ x(r) satisfy (1)-(4) and let the convex subset Ca c X,r) equal x,r) for all aE~x = p-l(X~) and for all XEX.i. Then the relation ~ abides the Ci-dense h-principle for i = 0, ... , r - 1, .i.

Proof There obviously exists a continuous family za(t) of in-paths in the fibers X,r), which provides an in-deformation of ~ to ~*. Then the h-principle for rYt is reduced with the above (A) and (A') to that for the open relation ~* to which the results of 2.4.2 apply.

Examples. Consider a relation ~ c x(r) and let Xl. X P -+ Xl. be a trivial fibration for some connected manifold P. Let p: Xl. x P -+ x(r) be a fiberwise map with the following two properties

(i) the image of p is contained in ~; (ii) the map of each fiber, p: x x P -+ X,r), x E Xl., is nowhere flat and

Conv p(x x P) = x,r) for all XEX.i.

Then every cr-1-section V -+ X admits a fine cr-1-approximation by cr-solutions V -+ X of the relation ~. Moreover every cr -section admits a fine C.i-approximation by cr-solutions V -+ X of~.

Proof Apply the above h-principle to the relation p: Xl. x P -+ x(r).

Let us describe a class of equations in two unknown functions f1 and fz: IRz --+ IR to which the general theory applies. Let <PI: IRZ --+ IR be a real analytic function without critical points such that the level {<PI (x, y) = c} is a connected curve in IRz

whose convex hull equals IRz for all c. For example, <PI (x, y) = x + y3. Consider the following single equation

<P _l_Z_<p (ar! ar!:) 1 au~' au~ - Z

h m.' b' . f' . h' aSf h were 'Pz IS an ar Itrary contmuous unctIOn WIt entnes u1 , UZ'::l ::l ,were uuf' UU~2

f = (fl,fz), s = 0,1, ... , r, and Sl = 0, ... , r - 1.

The above h-principle shows that C-solutions f: IRz --+ IRZ of (*) are C-1-dense in the space of C-1-maps IRZ --+ IRZ.

2.4.8 Triangular Systems of P.D.E.

Let ai' i = 1, ... , k, be continuous linearly independent vector fields on V. For example, V = IRn and ai = a/aui, i = 1, ... , n. Let <Pi be smooth vector valued functions, such that <Pi takes values in IRS; and <Pi has entries v, I, a1f, azf, ... , ad, where f is the unknown map V -+ IRq. In other words <Pi: V x IRq(i+1) -+ IRs;. Consider the following (triangular) systems of s = Lf=l Si P.D.E.

<PI (v, I, ad) = ° <Pz(v,f, ad, azf) = °

(A) Local Solvability. If Si :::; q - 1 for all i = 1, ... , k and if the functions <Pi' i = 1, ... , k, are generic, then the system (**) admits a local C 1-solution f: U -+ IRq, for some open subset U c V. Moreover, C1-s01utions U -+ IRq are CO-dense in some open subset in the space of CO-maps U -+ IRq.

This result is an immediate corollary of the Theorem (C) below. Observe that the system (**) may have n(q - 1) equations (for k = nand Si =

q - 1), and then it is highly overdetermined for n ;;::: 2 and for large q. However, the above result does not claim the local solvability of generic over-determined systems. In fact, the equation <Pi = ° is not generic for i < n, since <Pi is a generic function in the variables v, f, ad, ... , ad rather than in v, I, ad, ... , anI, as true genericity requires.

Question. Does there exist a generic map <P: V X IRq(n+1) -+ IRq (or rather an open subset in the space of Coo-maps <P: V X IRq(n+1) -+ IRq) for which local C1-solutions


of the (determined) system C/J( v, j, al j, ... , an!) = ° are CO -dense in an open subset in the space of CO-maps V ~ IRq for some neighborhood V c V?

(B) Triangular Differential Relations. Let ri' i = 1, ... , k, be continuous hyperplane fields on V which are r-independent in the following sense. If Ii: T(V) ~ IR are linear forms on V for which Ker Ii = ri' i = 1, ... , k, then the (symmetric differential of degree r) forms Ir are linearly independent, that is the (homogeneous of degree r) polynomials lr, i = 1, ... , k, on T,,(V) are linearly independent for all v E V. For example, [I-independence] = [independence]. Furthermore, if the forms Ii are independent then the forms lij = Ii + lj' 1 :s; i :s; j :s; k, are 2-independent, the forms Ii + lj + 1m , 1 :s; i :s; j :s; m :s; k, are 3-independent and so on.

Take a point ° E XZ) c x(r) for the origin in the (affine) fiber XZ) of the fibration x(r) ~ x(r-I) for some x E x(r-I) and let Ri c XZ) be the principal subspaces through ° E xZ) which correspond to the fields rio The (linear) subspaces Ri c (XZ),O), i = 1, ... , k, clearly are linearly independent for r-independent fields rio Put X~ = XZ) and X~-i = X~/Span(RI"'" RJ Thus we obtain affine bundles X k- i ~ x(r-I), i =

0, ... , k, with the fibers X~-i, and affine homomorphisms, say Pk- i: X k- i ~ X k- i- I . The fibers of the affine fibration (homomorphism) PH are (naturally identified with) principal subspaces (now in X k - i rather than in x(r») associated to ri+!O

Consider a locally closed relation [Jf = [Jfk C X k = x(r) and denote by [Jfk-i C

X k- i the image of.r?ll under the (composed) map X k ~ X k- i. Let us impose on [Jf the following

Triangular Condition. The subset ,IJlo c XO is open and the map Pk- i: [Jfk-i ~ [Jfk-i-I is a microfibration for all i = 1, ... , k.

Example. Start with an arbitrary open subset [Jfo c XO. Next, take a submanifold .r?lll c P11 (:Jlo) of codimension Sl :s; q which is transversal to the fibers P11(x), x E [Jfo, and is mapped by PI onto :Jlo. As the map PI: [Jf I ~ [Jfo is a submersion it also is a microfibration. We proceed with taking a similar submanifold [Jf2 c

P2- 1(.'Jld of codimension S2 and we continue up to a submanifold .'Jl = .'Jlk C

pk- 1 (.r?llk-l) of codimension Sk'

Notice that generic systems (**) locally (i.e. in some open subset in the jet space) satisfy the above transversality assumption and so they locally fit this example.

Next, we add the

Nowhere Flat Condition. The intersection of [Jfk-i c X k - i with every fiber R ~ IRq of the fibration X k - i ~ Xk-i-l is a locally path connected nowhere flat subset in R for all i = 0, ... , k - 1.

This property is satisfied for generic submanifolds :Jl in the above example, provided Si :s; q - 1 for i = 0, ... , k - 1.

In-paths. Consider a path z: [0, 1] ~ x(r) which consists of at most countably many principal segments (with the only admissible accumulation point t = 0), such that each segment lies in a principal subspace associated to some of the fields ri> i =


1, ... , k. This path is necessarily contained in Span{RJi=l, ... ,k' where Ri c x(r) is the principal subspace through z(O) E x(r) which is associated to 1:i' Then we use the canonical projection Span {R;} -+ Ri and we invoke the projection of Ri into X k-i+1 which isomorphically maps Ri onto some fiber, say i{ c X k -i+l, of the fibration X k-i+1 -+ X k- i. Thus, we send the path z to a path in Ri, say Zi: [0,1] -+ Ri.

A path z(t) with the above properties is called an in-path at a E fJfl if z(O) = a and if the path Zi(t) in Ri is an in-path at Zi(O) E fJflk-i+1 n Ri for the subset fJflk-i+1 n Ri c

Ri ~ IRq for all i = 1, ... , k.

(C) Theorem. Let fJfl c x(r) satisfy the triangular and the nowhere flat conditions. Then fJfl admits a local C -solution U -+ X for some open subset U c V, unless fJfl is empty. Furthermore, let fJflo = Xc, let the maps Pk- i: fJflk- i -+ fJflk- i- 1 be O-fibrations and let each path connected component of the intersection fJflk- i n Pk-=-\(x) c Pk~i(X) ~ IRq have convex hull = Pk-!i(X) for all i = 0, .. " k - 1, and for all X EfJflk-i-l' Then the relation fJfl satisfies the C'-dense h-principle for I = 0, ... , r - 1.

Proof. Let fJfl* c PfJfl be the subset of those pairs (a, z(t)) E PfJfl for which z(t) is an in-path at a = z(O) E fJfl and let fJfl+ = fJfl* U fJfl. Then fJfl* is open over x(r) and fJfl+ ~ fJfl is an in-extension. Hence, the local solvability follows from (A) in 2.4.6 (compare 2.4.7),

Now, the additional assumptions on fJfl* show the relation fJfl to be ample. Then an obvious in-deformation of fJfl into fJfl* (compare 2.4.7) allows us to apply (A') of 2.4.6 and to obtain the h-principle for fJfl. Q.E.D.

a (D) Examples and Exercises. (1) Take the differential operators .111 = -a x

U 1

( 0 0 ) 02 0 ( 0 0 ) z . -a + -a ,L112 = -a a and L122 = - -a + -a on IR and conSIder the U1 Uz U1 U2 oU2 U1 Uz

following system of three second order equations in the unknown map f = (f1,f2): IRz -+ 1R2,

(+ ) {

cP1(L111f1,L111fZ) = !/J1(Jj)

cP2(L1 12 f1,L1 12 f2) = !/Jz(Jj,L111 f)

cP3(L12zi1,L122fz) = !/J3(Jj,L1 11 f,L1 12 f),

h 1 ( f af af ) h ./.' b" were JJ = Ut> Uz, 'au1' aU2

,were 'I'i' 1= 1, 2, 3, are ar Itrary contmuous

functions, and where cPi = cPi(x, y), i = 1, 2, 3, are real analytic functions without critical points whose levels {cPi(x, y) = c} are connected and Conv { cPi(x, y) = c} = IRz for i = 1, 2, 3 and for all c E IR. The differential relation expressed by (+) is triangular for the line fields 1:1 = Kerduz, 1:z = Ker(du1 - du 2 ) and 1:3 = Kerdu1' The Theorem (C) shows CZ-solutions f: 1R2 -+ IRz to be C1-dense in the space of C1-maps 1R2 -+ 1R2,

(2) Let ai' i = 1, ... , k, be linearly independent C1-smooth vector fields on V. Set

for 1 :s;; i :S;;j:S;; k

for i = j = 1, ... , k.

Let Aij c: ~2, 1 :s;; i :s;; j :s;; k, be path connected, locally path connected nowhere flat locally closed subsets. Consider C2-maps f: V ..... ~2 whose derivative Aijf: V ..... ~2 sends V into Aij c: ~2 for all 1 :s;; i :s;; j :s;; k.

(2') If Conv Aij = ~q for 1 :s;; i :s;; j :s;; k, then every Cl-map V ..... ~q admits a fine Cl-approximation by the above maps f.

Proof Take linear differential forms Ii on V such that l;ilj = bij , 1 :s;; i, j :s;; k. Then the pertinent differential relation is triangular for the hyperplane fields

r .. = {Ker(li - Ij ) for 1 :s;; i :S;;j:S;; k 'J Ker Ii for i = j = 1, ... , k.

Observe that the above maps fare C2-solutions of a system of s = Ll:'5:i:'5:j:'5:kSij

differential equations of second order for Sij = codim Aij. The most overdetermined case to which (2') applies is k = n = dim V and Sij = q - 1 for all 1 :s;; i :s;; j :s;; n. This gives a system of S = n(n + 1)(q - 1)/2 P.D.E. in q unknown functions.

(3) Show the conclusion of (2') to hold true under the following weaker assumptions on Aij. Each Aij is an arbitrary subset in ~q which receives a continuous map (Xij: P ""7 Aij of a connected manifold P, such that

(i) the map (Xij is nowhere flat for all 1 :s;; i :s;; j :s;; k; (ii) Conv (Xij(P) = ~q for 1 :s;; i :s;; j :s;; k.

Hint. Prove (C) for triangular relations over (rather than in) x(r).

(3') Let the maps (Xij satisfy (i) and let each map (Xij strictly surround the origin in ~q, i.e. 0 E Int Conv (Xij(P). Prove the existence of at least one C2-map f for which (Aijf)(V) c: Aij for all 1 :s;; i :s;; j :s;; k.

Hint. Prove a relative version of (C) (compare 2.4.5).

(4) Let Aij = oA for 1 :s;; i :s;; j :s;; k. Prove (3) for these Aij in place of Aij under the additional assumption: the subset Aij c: ~q is open for 1 :s;; i < j :s;; k. (It is unclear if this assumption is necessary.)

(4') Show (3') to be false, in general, for Aij in place of Aij.

Hint. Consider the torus T2 with the standard fields 01 = %ul and O2 = 0/ou2. Take the unit disk {x2 + y2 :s;; 1} c: ~2 for Au and for A22 and take the exterior {x2 + y2 ~ 1O} for A 12 .

(5) Generalize (1)-(4) to pertinent derivations of an arbitrary order r.

2.4.9 Isometric CI-Immersions

Let V be an n-dimensional Riemannian manifold and let Oi' i = 1, ... , n be a frame of independent vector fields on V. The isometric immersion system for maps

202 2, Methods to Prove the h-PrincipJe

f: V --+ IRq (compare 1.1.5) is triangular in such a frame,

(1)

(2)

(n)

{<od,fJd) = <01,01)

{<ad, ad) = <01,02) <ad, ad) = <02,02)

{,~J'Onf) = <0;,0"), l - 1, "" n,

where the scalar products on the left hand side are in IRq and the right hand side ones are taken in the tangent bundle T(V), Unfortunately, the "nowhere flat" condition is not satisfied by the equations (k) for k = 2, ,." n. Indeed solutions Yk E IRq of the algebriac system

(k)'

for i = 1, ... , k and for fixed vectors Yl' ... , Yk-l in IRq form a round (q - k)dimensional sphere in IRq which is everywhere flat in IRq. However, successive convex hull extensions sufficiently enlarge these spheres for q - k ~ 1 and so the theorems (A) and (A') of 2.4.6 do apply to isometric immersions of V into Riemannian manifolds W, q = dim W > n = dim V, as follows.

The relevant jet space X(l) consists ofthe linear maps T.,(v) = IR" --+ IRq = Tw(W) and the isometric immersion relation is, in each fiber X~l) ~ Hom(lR" --+ IRq), x =

(v, W)EX, the subset of isometric homomorphisms IR" = (IR",g) --+ IRq, called Is =

ISg c HZ~ Hom(lR" --+ IRq), where g denotes the given Euclidean metric in IR". Let R c H~ be a principal subspace associated to some hyperplane, c IR". The

intersection Is n R is either empty or a round sphere sq-" c R = IRq obtained by "rotating" an isometry h: IR" --+ IRq, h EIs n R, around the "axis" hi, in IRq (compare 2.4.3). The ball Bq-"+l c R, bounded by this sq-", consists of those short homomorphisms hi: IR" --+ IRq, for which h'l, = hi" where "short" means the positive semidefiniteness of the quadratic form g - g' for the form g' on IRn induced by hi (from the given form on IRq). The form g - g' has rank = 1, unless hi E sq-n = oBq-n+l (and g = g'), and so g = g' + [2 for a unique (up to a sign) linear form [ on IRn which satisfies Ker [ = ,. This gives us a geometric description of the convex extension Conv1 (Is) ::::> Is as follows. If q = n, then Conv1 (Is) = Is, as the sphere SO c R consists of two single points and so the "component-wise convex hull" of So equals So. However, if q > n, then Conv 1 Is consists of the above pair (h, hi), where hEIs and where hi C Bq-n+1 c R for some principal subspace R c HZ through h.

Another useful description of Conv 1 Is starts with a homomorphism hi E H: such that the induced form g' equals g - [2 for some linear form [. Then, one takes a hyperplane " c Ker [(if g' 1= g, then, = Ker [ and so " is unique) and the associated principal subspace R' in H: through hi. Finally, one takes an arbitrary point in Is n R' ~ sq-n for h.

Denote by Conv~Is c PrNIs the subset of those (N + I)-triples (ho, h1 , ... , hN) for which (hi- 1 , hi) E Conv 1 I Sg'_I' for all i = 1, ... , N. Here gi-l is the quadratic form

induced by hi-I: [Rn -+ [Rq and ho E Is, that is go = g. The above description of Conv l Is immediately implies several useful facts.

(1) Conv; Is c ConvNls for N = 1,2, ... ,00.

(2) The image of the map Conv; -+H:, (ho, ... ,hN)l-+hN consists of those short homomorphisms h': [Rn -+ [Rq, for which the induced form g' satisfies rank (g - g') ~ N. In particular, the image of the map ConvNls -+ H: consists, for N ~ n, of all short homomorphisms [Rn -+ [Rq. (3) Let 9f c X(l) be the isometric immersion relation, that is 9f n X~l) = Is for all x E X, and let a section cP': V -+ X(l) correspond to a strictly short homomorphism T(V) -+ T(W). Then the quadratic form g - g' by definition is positive definite on T(V) (where g is the original form on T(V) and g' is induced by cp') and so g - g' = Lf:l If for some linear forms Ii on T(V). Hence, there is a piecewise principal path t/!: V -+ Convin9f c Conv2n 9f c Pr2n9f which terminates in cP', that is t/! = (CPo, CPl"'" CP2n = cp') for some section CPo: V -+~, and the pair (CPi-l' CPi) is contained in Convi 9fgi _ 1 for all i = 1, ... , 2n.

Now, we define a piecewise principal path z: [0,1] -+ H: to be an in-path at a = z(O) E Is, if the homomorphism z(t) E H: is strictly short for all t > 0 and if each principal segment, say [z(t), z(t')], for 0 < t < t' ~ 1, satisfies (z(t), z(t')) E Convi Is', where Is' = Isg , for the quadratic form induced by the homomorphism z(t')EH:. Here the path z may have countably many principal segments which accumulate at t = O.

We take the space of these in-paths in the fibers X~l), x E X, for the relation ~* c P~. It is obvious that 9f* is open over X(I) and that there is an in-deformation of 9f to 9f*. The above discussion shows that relation ~+ = 9f* U 9f to be an in-extension and we see with (3) above 9f* is ample over strictly short homomorphisms T(V) -+ T(W). Hence, the Theorem (A') implies

(A) The Theorem of Nash-Kuiper (compare 1.1.5). The h-principle for isometric Cl-immersions V -+ W is CO -dense in the space of strictly short maps V -+ W; provided dim W> dim V.

(B) Exercises and Generalizations. (1) Show that every strictly short embedding fo: V -+ W; dim W > dim V; admits an isotopy to an isometric C1-embedding. Moreover, let fo: V -+ W be an immersion with normal crossings. Construct a C1_ diffeomorphism g: W -+ W; such that the composed map go fo: V -+ W is isometric.

(1') Let ai' i = 1, ... , k, be linearly independently vector fields on V. Construct a C1-map f: V -+ [Rk+1, such that <oJ, ojf) = ~ij' 1 ~ i ~ j ~ k, where ~ii = 1 and ~ij = 0 for i < j. Construct for k = 1 a Coo-map f: V -+ [R2 for which <oJ, oJ) = 1, provided the field 01 is Coo-smooth.

Hint. Prove the h-principle for C1-maps V -+ [Rk+l which are isometric on a given k-dimensional subbundle r c T(V).

(2) Indefinite Forms. Let cP = Ll";i";j,,;qcPij(w)dxidxj be a quadratic differential form on W and let (q+ (w), q_(w)) denote the type (signature) of cPITw(W), WE W; so that q+(w) + q_(w) = rank cPl Tw(W) ~ q = dim W. The form cP is called non-

singular if q+(w) + q(w) = q for all WE W A linear map h: [Rn -+ Tw(W) is called IP-regular ifit misses Ker = Ker(IPITjW)) c Tw(W) which is a linear subspace in TjW) of codimension q+(w) + q_(w). This means the identity h-1(Ker) = O. A C1-map f: V -+ W is called IP-regular if the differential D/ T(V) -+ T(W) is IPregular on each tangent space T.,(v), v E V. If the form IP is non-singular, then IP-regular maps are exactly immersions V -+ W

Let g be a quadratic form on V oftype (n+ (v), n_ (v)), n+ + n_ :=:;; n = dim V, v E V, and let n - n_(v) + 1 :=:;; q+(w) and n - n+(v) + 1 :=:;; q_(w) for all VE V and WE W Prove the CO-dense h-principle for IP-regular isometric C1-maps f: V -+ W ("isometric" means f*(IP) = g) and derive the following

(2') Corollary. An arbitrary continuous map V -+ W admits a fine CO-approximation by isometric C1-immersions f: V -+ W in the following three cases.

(i) q+ (w) ~ 2n - n_(v) and q_ (w) ~ 2n - n+(v) for all (v, w) E V X W (ii) The manifold V is topologically contractible, the forms g and IP are nonsingular

and q+ ~ n+ + 1, q_ ~ n_ + 1. (Since the forms are non-singular their types are constant.)

(iii) The manifold V is para lie liz able, the manifold W is contractible q+ (w) == const+ ~ n + 1, q_(w) == consL ~ n + 1 for all WE Wand g == O.

(3) Forms of Degree d ~ 2.Let IP be a symmetric form of degree d ~ 2 on [Rq (that is a homogeneous polynomial on [Rq). Then each homomorphism hE Hom([Rn -+

[Rq) = H2 induces a form, called g = h*(IP) on [Rn. Let Sd = SA[Rn) be the linear space of forms of degree d on [Rn and denote by A = A<l>: H2 -+ Sd the map hf-+ h*(IP). A homomorphism hE H2 is called IP-regular if the differential D Lf: T(H2) -+ T(Sd) has rankDLl IT,,(H2) = s = s(n,d) = dimSd = (n + d)!/n!d!. We call h isometric for a given form g on [Rn if h*(IP) = g and we denote by Is = ISg c H2 the subset of IP-regular isometric homomorphisms. A straightforward calculation shows that rankDLfIR = s(n,d - 1) for all principal subspaces R c H2 = Th(H2) if and only if the homomorphism h is IP-regular. It follows that Is n R is a non-singular subvariety of codimension s(n, d - 1) in R for all R c H2 in so far as Is n R is non-empty.

Example. Let d = 3 and let g = gijk = g(ei ® ej ® ek) for a fixed basis e1, ... , en in [Rn. Then the relation h*(IP) = gis expressed by s(n, 3) = n(n + 1)(n + 2)/6 equations in Yi = h(ei) E [Rq,

1 :=:;; i :=:;; j :=:;; k :=:;; n.

If we restrict these equations to some principal subspace R = [Rq c H~ associated to the hyperplane Span(e1, ... ,en - 1) c [Rn, then we obtain s(n,2) = n(n + 1)/2 equations in Yn for fixed vectors Yl, ... , Yn-l,

Denote by lPi/ [Rq -+ [R the linear forms lPij: Z f-+ (Yi ® Yj ® z) for 1 :=:;; i :=:;; j :=:;; n. Then the IP-regularity of h is equivalent to linear independence of these forms lPij for Yi = h(eJ


We call a homomorphism h: IRn -+ IRq hyper regular if h extends to a cP-regular homomorphism h: IRn+1 -+ IRq for IRn+1 ::::) IRn, such that h*(cP) = p*(g) for the normal projection p: IRn+1 -+ IRn.

Example. Let g and cP be quadratic forms of types (n+, n_) and (q +, q_) respectively and let h: IRn -+ IRq be a cP-regular isometric homomorphism. Then h is hyper regular if and only if q+ ~ n - n_ + 1 and q_ ~ n - n+ + 1.

Now, let g and cP by symmetric differential forms of degree d on manifolds V and W respectively. A C 1-map f: V -+ W is called (hyper regular) isometric if the differential Df : T(V) -+ T(W) is (hyper regular) isometric on each tangent space T,,(v), v E V.

(3') Exercises. Prove the CO-dense h-principle for hyper regular isometric C1-maps f: V-+ W

Let the form cP on W be diagonal, cP = L1';;1 (Li)d - Lj;;;l (L/ for some linearly independent linear differentialforms Li and Lj on W Show for q+ ~ s(n + 1, d - 1) + nand q_ ~ s(n + I,d - 1) + n every continuous map V -+ W to admit a fine Coapproximation by hyper regular isometric C1-maps f: V -+ W for all V, dim V = n, and for all forms g on V of degree d.

Questions. How can one improve the above lower bound on q+ and q_? What is, for example, the minimal number q = q(n), such that every cubic form g on V is a sum of q cubes of exact linear CO-forms, g = L1=1 (dJ;)3? [The functions J; define an isometric map f = (f1,· .. ,h): V -+ IRq for the diagonal form cP = L1=1 (dxy on IRq.]

(4) Definite Forms of Even Degree. Let C c Sd = SilRn) be a nonempty convex cone which is invariant under the natural action of the linear group Ln(lR) on the space Sd. If some form g(x) = g(x1, ... , xn ) = axt + ... is contained in C then the form (sign a)xt is contained in the topological closure of C as A. -d g(A.x1' X2' • •• ,xn ) -+

(signa)x d for A. -+ 00. It follows that there are, for d = 2m, exactly two minimal closed convex invariant cones in Sd' called S: c Sd and Si = - Sd' where S; is the convex hull of the Ln(IR)-orbit of the form xt. Denote by S: c S; the interior of S; and call a form g positive definite (semidefinite) if it is contained in S; (in S;). For example, the form Li'=l Xid is positive semidefinite but it is not definite for d = 2m ~ 4, n ~ 2. In fact, if a form g is positive definite, then every even monomial, (Xi Xi , ... , Xi )2, enters g with a positive coefficient.

1 2 ~

There is a unique (up to a scalar) form go E Sd, which is invariant under the action of the orthogonal group On C LilR), namely go = (Li'=l xr)m. This go is positive definite. Indeed, the p-transforms gp E S:, PEOn, of an arbitrary form gp E S: average to go,

go = c fo gpdp, for c = c(g) > 0,

and for the Haar measure dp on On. It follows, in particular, that

( n )m q

i~ Xf = j~1 b)t, bj > 0, d = 2m,

for some linear forms lj = 2:1=1 aijxi and for q = s(n, d) - 1.

Remark (Hurwitz-Hilbert). Since the rational points are dense in the open cone Sj c Sd the identity (*) holds true for some rational coefficients bj > 0 and aij' This reduces Waring's problem of the representation of positive integers as sums of dth powers to that for mth powers, d = 2m, as every positive integer is 2:t=1 Xf, Xi E 7l., by Lagrange's theorem.

Let us concentrate on representations g = 2:1=1 It that correspond to isometric maps (IRn, g) --+ (IRq, if» for the diagonal form if> = 2:1=1 xt on IRq.

Exercises. Show that if>-regular homomorphisms IRn --+ IRq induce positive definite forms g on IRn.

Let a homomorphism h: IRn --+ IRq be given by q linear forms hj on IRn. Show that h is if>-regular for if> = 2:1=1 xt if and only if the forms h1-1 c Sd-1 span the space Sd-1' Prove generic homomorphisms h: IRn --+ IRq to be if>-regular for q ~ s(n, d - 1).

Show that a generic positive definite form g on IRn admits no isometric homomorphisms (IRn, g) --+ (IRq, if» for nq < s(n, d).

Prove that the boundary of the cone S; c Sd contains no affine simplices of dimension s(n, d) - n + 1. Using this, show the existence of an isometric homomorphism (IRn, g) --+ (IRq, if> = 2:1=1 xf) for all positive semidefinite forms g on IRn and for q = s(n, d) - n. Prove the existence of a if>-regular isometric homomorphism IRn --+ IRq for all positive definite forms g on IRn and for q = s(n, d) + s(n, d - 1) - n.

Now, let g be a differential CO-form on V of degree d = 2m which is positive definite on [the tangent spaces T,,(V) of V], and let if> be the diagonal differential form 2:1=1 (dx)2m on IRq.

Questions. Do if>-regular isometric maps f: V --+ IRq satisfy the CO-dense h-principle for large q ~ qo(n, d)? What is the minimal q for which an isometric C1-map f: V --+ IRq exists for all g on V?

Exercises. Prove for q = s(n, d) + s(n, d - 1) - n the existence of a if>-regular isometric C1-map f: U --+ IRq for a small neighborhood U c V of a given point v E V.

Show that every positive definite form g on V admits a decomposition g =

2:1=111 for some linear differential forms lj on V and for q = (n + 1)(s(n, d) - n). Prove generic COO -maps V --+ IRq to be if>-regular for q ~ s(n, d - 1) + n. Construct if>-regular isometric C1-maps V --+ IRq, q = (n + 1)(s(n, d) - n) +

s(n, d - 1) + n, for all positive definite forms g on V.

Hint. Consult 3.1.4 and Gromov (1972) for a similar study of isometric COO _ immersions of forms of degree d ~ 2.


(5) Nonsymmetric Forms and Further Generalizations. Let t/> and g be forms of degree d on Wand on V respectively having the same type of symmetry. The "isometric" map equation f*(t/» = g for f: V --+ W is of the triangular form for all t/> and g, but the convex integration may fail to prove the h-principle in certain cases. Namely, if the form t/> is antisymmetric (i.e. exterior), then the corresponding relation rH c X(!) is linear on every principle subspace R c Xl!): the intersection rH n R is an affine subspace in R for all R c X(1). For example, if d = 2, then the intersection rH nRc R = IRq is given by the following equations which are linear in the unknown vector YnER = IRq,

(Yi' Yn) = gin,

where Yi' i = 1, ... , n - 1 are fixed vectors in R. Hence, the convex extensions do not enlarge the relation rH. In fact, the h-principle fails for the equation f*(t/» = g unless the Stokes-De Rham theorem is taken into account (see 2.2.7 and 3.4.1).

If the form t/> of degree d ~ 2 is not antisymmetric, then the relation f*(t/» = g looks sufficiently ample and one expects the h-principle under appropriate regularity assumptions on f

Here are two further examples of differential equations to which the convex integration may apply.

(a) (o/Jjf, okod) = gijkZ, where f: V --+ IRq is the unknown C2-map, where Oi' i = 1, ... , n, are given C1-smooth vector fields on V and where gijkl are continuous functions on V for i ::;; j, k ::;; I and i ::;; k.

(b) Let Am denote the mth exterior power of the cotangent bundle of V and let (Amf denote the symmetric square of Am. Our equation is

q

I (dfj)2 = g, j=1

where jj: V --+ Am-1 are the unknown (m - 1)-forms and where g: V --+ (Am)2 is a given form of degree 2m on V.

2.4.10 Isometric Maps with Singularities

A continuous map between Riemannian manifolds f: V --+ W, is called isometric if it preserves the length of all rectifiable curves C in V.

Examples. (a) Let V be a closed n-dimensional manifold, such that V minus a point, V\vo, is a parallelizable manifold. (If n = 4, then this is equivalent to the vanishing of the first two Stiefel-Whitney classes, W l = 0, W2 = 0.) Then there exists an isometric C1-immersion V\vo --+ IRn+1 which obviously extends to a continuous isometric map f: V --+ IRn+1. This map may be extremely irregular near the point Vo E V, and if V admits no smooth immersions into IRn+1 (for instance, n = 4 and the first Pontryagin class P1 "# 0), then this singularity cannot be removed. However, the map f can be modified in order to have the following

(a/) Conical Singularity at Vo. Fix a diffeomorphism of a small neighborhood U c V of Vo into the tangent space, say eo: U --+ T.,o(V), such that the differential Deo: T.,o(V) --+ T.,o(V) is the identity map (in particular eo(vo) = 0) and such that ho is strictly short on U\vo. Next we take an isometric map fo: IR" = T.,o(V) --+ 1R"+1 which is the cone from the origin over some isometric C1-map between the unit spheres, say over g~: S"-1 --+ S", such that the immersion eo 0 fo: U\vo --+ 1R"+1 is regularly homotopic to fl U\vo. The existence of such a g~ is obvious with the h-principle for isometric C1-immersions S"-l --+ S". Then the map eo 0 fo extends to a map fo: V --+ 1R"+1 which is a strictly short immersion on V\vo. Finally, we take an isometric C1-map V\vo --+ 1R"+1 which finely CO-approximates fo on V\vo and thus we get an isometric map f1: V --+ 1R"+1 which is C1-smooth on V\vo and which has the isometric map fo for the tangent cone at the point Vo [compare (A') below].

(b) Let Co c V be a nowhere rectifiable curve, that is dimH(Co n C) < 1 for all rectifiable curves C in V where dimH stands for the Hausdorff dimension. Take a strictly short map fo: V --+ W which collapses Co to a single point in Wand then finely CO-approximate the map fo on V\Co by an isometric C1-immersion f: V\Co --+ W (this is possible, for example, if dim W ~ 2 dim V). Then the map f extends to an isometric map V --+ W which sends the curve Co c V to a single point. This collapse can be avoided with the following

(b/) Strongly Isometric Maps. Take two points x and y in V and consider chains of points, x = Xo, Xl' ... , Xk = Y in V, k = 1, 2, ... , such that disty (Xi' Xi+1) ~ e for i = 0, 1, ... , k - 1, and for a given e > O. Then we define dist. (x, y) to be the lower bound of the sums ItJ distw (f(Xd,J(Xi+1)) over all such chains of point (xo, .. ·, Xk) between x and y. Finally, we call a map f strongly isometric if lim ..... o dist. (x, y) = disty (x, y) for all x, y E V.

Exercise. Show every strongly isometric map to be isometric. Show that a strongly isometric map collapses no connected subset in V to a point.

(A) Stratumwise Smooth Maps. Let {ri}, iE I, be a C1-stratification of V (see 1.3.2). Then a continuous map f: V --+ W is called stratumwise smooth on {ri} if the restriction flr i : r i --+ W is C1-smooth for all strata ri. For example, if the stratification corresponds to a triangulation of V, then f is C1-smooth on the open simplices of all dimensions. Such an f may fail the piecewise smooth condition, that requires the C1-smoothness of f on all closed simplices.

Theorem. Let fo: V --+ W be a strictly short map, such that the restriction fo I ri: r i --+

W is homotopic to an immersion r i --+ W for all strata ri. If dim W > dim V then fo admits afine CO-approximation by isometric stratumwise smooth maps f: V --+ W.

Proof The convex integration does not directly apply here because the map f in question is not C1-smooth. However, an obvious induction by strata (compare 1.4.4, 2.1) reduces the problem to the following lemma to which the convex integration does apply (compare 2.4.9).

Lemma. Let U be an open subset in Vand let fo: U -+ W be a strictly short map whose restriction to a given submanifold I c U is homotopic to an immersion I -+ W. Then fo admits a fine CO-approximation by C1-maps U -+ W which are isometric on I and strictly short on U\I.

Exercises. Fill in the details of the above proof. Show with an appropriate stratification of V that every strictly short map

V -+ W, dim V < dim W, admits an approximation by continuous isometric maps V -+ W.

(A') Isometric Maps with Tangent Cones. Let f: [Rn -+ [Rq, f(O) = 0, be a continuous map for which the maps f).: [Rn -+ [Rq, f).(x) = 2 -1f(2x), uniformly converge on compact subsets in [Rn for 2 -+ O. Then the limit lim). .... of). is called the tangent cone Tof: [Rn -+ [Rq of f at 0 E [Rn. (If f is smooth, this is the ordinary differential of f.) This definition, being purely local, generalizes to maps between smooth manifolds, f: V -+ W, and so one may speak of (the existence of) tangent cones T,J: T,,(v)-+ Tw(W), w = f(v), at points VE V.

A map between Riemannian manifolds, f: V -+ W is called T 1-isometric (T for tangent) if the tangent cone T"f: T,,(v) -+ Tw(W) exists for all v E V and if the map T"f maps the unit sphere S:-1 c T,,(v), n = dim V, into the unit sphere S!-l c

Tw(W), q = dim W for all v E V.

Exercise. Let f: V -+ W be a T 1-isometric map. Show that f is strongly isometric and that the pull-back f-1(W) is a discrete subset in V for all WE W.

Next, by induction we define a T 1-isometric map f to be Ti-isometric, i = 2, 3, ... , n, if the map T"fIS:-1: S:-1 -+ S!-1 is Ti-l-isometricfor all v E V and w = f(v). Finally, T-isometric maps, by definition are Tn-isometric.

Exercise. Show the map constructed in the above Example (a') to be T-isometric.

Let us fix a smooth triangulation in V. Then the tangent spaces to the simplices at every point v E V partition the tangent space T,,(V) into finitely many convex (polyhedral) cones.

Exercise. Show that a T-isometric map f: V -+ W is piecewise smooth (for a given triangulation) if and only if the tangent cone map T"f: T,,(V) -+ Tw(W) is piecewise linear (i.e. linear on every cone of the above partition) for all v E V.

Take a point in an open k-dimensional simplex of our triangulation, v E Ak C V. The above partition of T,,(V) induces a triangulation of the unit sphere S:-k-l c

T,,~(Ak) c T,,(v), where T,,~(Ak) denote the normal space to Ak at v. Then by induction in n = dim V we define regular maps f All maps are assumed regular for n = 0 and we call f regular for n > 0 if it is T-isometric, stratumwise smooth (i.e. C1_

smooth on every open simplex Ak, k = 0, ... ,n) and the map T"fl S:-k-1 is regular for the above triangulation of S:-k-1 for all Ak and v E Ak.

Exercise. Combine the argument of the Example (a') with an induction by skeletons and prove the following

Theorem. If dim V < dim W then every strictly short map V -+ W admits a fine CO-approximation by T-isometric regular maps V -+ W

(B) Piecewise Linear Isometric Map. Let K be an n-dimensional simplicial polyhedronwithkOvertices,Ko = {vl,v2,···,vko},andwithk1 edges,K 1 = {e 1 ,e2,···,ek,}· An arbitrary simplicial map f: K -+ IRq by definition is linear on every simplex in K and so these maps are given by vectors (f(v 1 ), ••• ,f(Vko))E IRqkO. Every simplicial map f induces a flat metric on every simplex in K and so a (singular Riemannian) metric on the space K itself. Such a metric 9 is uniquely determined by the vector (g l' ... , gk') E IRk', where gi denotes the (lengthf of the edge ei. We study the map '!J: IRqkO -+ IRk' which assigns the induced metric 9 = '!J(f) to each simplicial map f: K -+ IRq. Since the map '!J is polynomial, the image '!J(lRqko) is a semi algebraic subset in IRk'. If q ~ q~ = kO - 1, then this subset '!J(lRqkO) c IRk' obviously is a convex cone in IRk' which does not depend on q ~ qo and which has dim '!J(lRqkO) = k1• But if q < kO - 1 then only few facts are known about (singularities of) the map '!J and the dimension of its image.

Examples. Let K be homeomorphic to the sphere S2. Then, obviously, k 1 = 3ko - 6. A theorem of Steinitz claims the existence of a simplicial map f of K onto a convex (with all dihedral angles <n) hypersurface in 1R3. The rigidity theorem of Dehn estimates the rank of the map '!J: 1R 3kO -+ IRk' at such "convex" maps f E 1R 3kO,

Hence, convex polyhedra f: K <=> 1R3 are rigid: every small deformation of f which does not change 9 = '!J(f) is a rigid motion. Since the map '!J is polynomial, the identity (*) holds on a (Zariski) open subset in 1R3kO. Hence, dim '!J(K) = k1 and so generic maps f: K -+ 1R3 are rigid (Gluck 1975). However, there are many (nongeneric) nonrigid maps. In fact, there exist, for certain K ~ S2, submanifolds F c 1R 3kO of positive dimension, such that the maps f E Fare embeddings K -+ 1R3 which are not rigid motions of a fixed map, but for which the induced metric is constant for all f E F (see Connelly 1979).

The definition of'!J generalizes to spherical simplicial maps f: K -+ sq which send every simplex in K onto a convex spherical simplex in sq. We write f = {j(Vi)},f(Vi)ESq, (= 1, ... , kO, and we assign to each edge el c K the (length)2 of the geodesic segment f(e ,) c sq that is (dist(f(vi),f(v)f for the ends Vi and Vj of el. Thus, we get '!J: (sqr -+ IRk' for the Cartesian product (Sq)kO = sq X ... x sq. This map '!J is real analytic on those f, for which length f(e ,) < n for alII = 1, ... , k'.

Since the metric in sq locally is a small analytic perturbation of the flat Euclidean metric, the identity (*) extends to an open dense subset of maps f: K -+ S3 for triangulations K of S2. In other words, generic polyhedra f: K <=> S3 are rigid as well as polyhedra in 1R3 for K ~ S2.

Now, let K be a triangulation of a 3-dimensional manifold without boundary and let f: K -+ 1R4 be a simplicial map. Then the tangent cone T"J of f at every


vertex Vi E K is a (spherical) simplicial map of the link L(Vi) c K (this is a triangulated 2-sphere which by definition consists of all simplices in K opposite to v;) into the unit sphere S3 c 1R4 around f(Vi)E 1R4. The rigidity of generic simplicial maps L(vi) -+ S3 for all Vi E K implies (this is an easy exercise) the rigidity of generic simplicial maps of connected three manifolds into 1R4. This argument equally applies to maps f: K -+ sn, dim K = n - 1, for n ~ 4 and then by induction on n we obtain the rigidity of generic maps f: K -+ sn for all n ~ 4. Finally we conclude to the rigidity of generic simplicial maps g: K -+ IRn+1 for all connected triangulated manifolds K, dim K = n ~ 3. Since the isometry group Is(lRn+l) has dimension (n + l)(n + 2)/2, this rigidity claims the following property ofthe map rJ: lR(n+1)kO -+ IRk'

(**) d · rO(ITb(n+l)kO) _ k ro _ ( l)kO (n + l)(n + 2) 1m 7;jf 11'\ - ran f 7;jf - n + - , 2

for all f in some (Zariski) open dense subset F c lR(n+l)kO.

The dimension of the image rJ(lR(n+1)kO ) c IRk' does not exceed kl = dim IRk'. Thus we arrive at the following lower bound [due to Barnette (1973)] on the number of edges in K,

(+ )

for all connected triangulated manifolds K of dimension n ~ 3. This ( + ) is false for n = 1 but it does hold true for n = 2 as

kl = 3ko - 3X ~ 3ko - 6,

for all closed surfaces K with the Euler characteristic X :s; 2. The above geometric proof of ( + ) can be reduced to a purely combinatorial

argument with the following definition. Let K be an arbitrary simplicial complex and let L c K be a subcomplex. The pair (K, L) is called q-rigid for some q > 0 if every subset of mo vertices in K\L has at least qmo edges in K which meet (the union of) these vertices. [Compare Kalai (1986).]

Exercise. Let f: K -+ IRq, q ~ n = dim K, be a generic simplicial map, such that no isometric deformation of f remains fixed on the subcomplex L. Show the pair (K, L) to be q-ngid.

Next, we call the polyhedron K itself q rigid if the pair (K, LIn) is q-rigid for all top dimensional simplices LIn C K.

The following three properties of this rigidity are immediate from the definition. (a) Let K be the simplicial cone over some (n - I)-dimensional complex L. If Lis (q - I)-rigid then K is q-rigid. (b) Let K be divided into a union, K = Kl U K 2, for some subcomplexes Kl and K2 in K such that dimK l U K2 = n = dimK (i.e. Kl n K2 contains some top dimensional simplex). If Kl and K2 are q-rigid complexes, then K also is q-rigid; (c) Let K be a connected complex of pure dimension n (i.e. each simplex in K is a


face of some top dimensional simplex Lr c K). If the link L(v;) is (q - 1)-rigid for all vertices Vi E K, i = 1, ... , kO, then the complex K is q-rigid.

In order to apply (c) to manifolds K of dimension n ~ 3 we need the following elementary fact. (d) If K is (homeomorphic to) a connected closed surface, then K is 3-rigid.

Proof Let L c K be the union ofthe (closed) 2-simplices in K which meet the given mo vertices in K\il2 for some triangle il 2 c K. The boundary 8L of L by definition is the union of those closed edges in L which do not meet the above mo vertices. The first Betti number of 8L is related to the number m~ of edges in 8L by the obvious inequality

b l (8L) ::;; tm~.

Let m l denote the number of those edges in L which meet the given mo vertices. Then the number 12 of the triangles in L satisfy 312 = 2ml + m~. The Euler characteristic of the pair (L,8L) is

X(L,8L) = mo - m l + 12 = mo - tml + tm~. On the other hand,

X(L,8L)::;; b2 (L,8L)::;; bl (8L)::;; tm~,

and so mo - tml ::;; o. Q.E.D. The propositions (c) and (d) by induction imply the (n + 1)-rigidity of closed

connected manifolds K of dimension n ~ 2 and then the inequality ( + ) obviously follows.

Let us return to the geometric rigidity problem and to the map '(§: [RqkO -+ [Rk'.

The algebra-geometric behaviour of'll depends on the combinatorial structure of the graph (KO, KI ) that is the 1-skeleton of the polyhedron K.

Question. Is there an efficient combinatorial description in terms of the graph (KO,K I ) of basic invariants of the map G? For instance, let Im = {XE [Rk'idim 'lI-I(x) ~ m}. What is the dimension of this subvariety Im c [Rk-l?

Example. Let Kd denote the 1-skeleton of the d-dimensional cube. If d ~ 3, then the generic maps f: Kd -+ [R2 are obviously rigid and so codim Im > 0 for m > 3 = dim(Isom [R2). Furthermore, the subvariety ,(§-l(X) c [R2kO, kO = 2d, clearly has dim ,(§-l (x) ::;; 2 + d for d = 1, 2, ... , and for all x E [Rk', kl = d2d- l . The subvariety I2+d c [Rk' is the principal diagonal in [Rk' and so the corresponding (most flexible) simplicial maps f: Kd -+ [R2 are those which give equal length to all edges in Kd. To see these maps we use the standard (unit) cube ]d c [Rd which is spanned by a fixed orthonormal basis {e l , ... , ed }. Every linear map f: [Rd -+ [R2 for which II f(ei)11 = 1, i = 1, ... , d, gives a simplicial map of Kd c ]d C [Rd to [R2 with edges of unit length. Thus we obtain the top dimensional component in 'lI-l(x), XEL2+d' that is the product [R2 x Sl X Sl X ••. X Sl.

l v~ __ ~1 d


Every linear map f: ~d -+ ~2 for which II f(ei) II = 1, i = 1, ... , d, gives rise to pretty configurations of unit circles in ~2 as follows. Let 7l.d be the integral lattice in ~d spanned by the vectors ei E ~d, i = 1, ... , d. Call a point z = (z l' ... ,Zd) E 7l.d even if the sum Z 1 + ... + Zd is even, and call z odd if this sum is odd. Let K c ~d be a i-dimensional polyhedron (graph) with vertices in 7l.d whose edges are straight unit segments in ~d parallel to the vectors ei . The above graph Kd c ~d is an example. Let K~v c K be the set of even vertices in K and let K~dd c K be odd vertices. Then the unit circles in ~2 around the points f(z) E ~2, Z E K~v' meet at the images of some odd vertices in KO. For example, if K = K3 is the i-skeleton of the 3-dimensional cube, then we get (a well known 2-parametric family of) four unit circles, such that every three of them have a point in common.

Exercises. (a) Show, that one cannot find the center of a unit circle in ~2 if the only operation one is allowed consists in drawing unit circles through pairs of points in ~2.

Hint. Every configuration of unit circles in ~2 comes from the above map f: 7l.d -+ ~2. Then, there is a perturbation f1 of J, for which Ilf1(ei)11 = II f(ei) II = 1, i = 1, ... , d, such that the images f1 (71.~v) and f1 (71.~dd) in ~2 do not intersect.

(b) Show generic simplicial maps f: Kd -+ ~q to be rigid for d ~ 2q.

Isometric Maps of Subdivided Polyhedra. The isometric map problem for piecewise linear metrics on K (for which every simplex in K is isometric to an affine simplex in ~n) approaches its Riemannian counterpart if one asks for piecewise linear isometric maps f: K -+ ~q which by definition are linear isometric on the simplices of some simplicial subdivisions of K. Two facts are known about these maps. (1) If dimK = n ~ 4, then K admits an [equidimensional ! Compare (B) below] piecewise linear isometric map into ~n (Zalgaller 1958) for an arbitrary piecewise linear metric on K. (2) Every compact oriented surface with a piecewise linear metric can be piecewise linearly isometrically embedded into ~3. (Burago-Zalgaller 1960.)

Exercise. Show that the torus yn with an arbitrary flat Riemannian metric admits a piecewise linear isometric map Tn -+ ~n.

Hint. Use maps with normal foldings along flat subtori in Tn [compare (B) below].

Question. Does every n-dimensional Riemannian manifold V admit a piecewise smooth (for some triangulation of V) isometric map f: V -+ ~n+1? (The solution requires a preliminary study of the tangent cones T,J: T,,(v) -+ ~n+1 which are piecewise linear isometric maps.)

Generalizations. Many partial differential relations admit interesting piecewise linear (and piecewise smooth) versions. Here are several sample questions.

(i) Take a simplicial polyhedron K and let us assign to each simplex A in K a subset alLJ in the space of projective maps J -+ ~q (one could take an arbitrary projectively flat manifold, for example sq, instead of ~q). Under what condition on


9l L1 can one find a continuous map f: K --+ IRq, such that the restriction fl A: A --+ IRq is a projective map and fl A E 9l,1 for all A E K?

(ii) If K' is a subdivision of K, then one extends the "relations" 9l,1 to the simplices A' in K' as follows. The subset ~,1' consists of those projective maps 1': A' --+ IRq such that I' = flA for A :::::J A' and for some f E 9l,1. Now, one takes a continuous map fo: K --+ IRq and one asks for a CO-approximation of fo by those piecewise projective maps f: K' --+ IRq of subdivisions K' of K for which flA' E~L1' for all A'EK'.

Example. A piecewise linear (in some triangulation) map f of a smooth oriented n-dimensional manifold V --+ IRq is called A-directed, for a given subset A in the Grassmann manifold Grn(lRq) of oriented n-dimensional subspaces in IRq, if the differential D/ T,,(V) --+ IRq sends the tangent space T,,(V) onto some subspace a E A for all regular points v E V of f (these are the interior points of the top dimensional simplices). For which A does an arbitrary continuous map fo: V --+ IRq admit a CO-approximation by piecewise linear A-directed maps f: V --+ IRq? If q = n + 1 then a necessary condition on A is the existence of an affine simplex An+1 c IRn+1

whose boundary aAn ~ sn ~ IRn+1 is directed by A (i.e. each oriented n-face of An+1

is parallel to some hyperplane a E A). Is this condition also sufficient for the approximation of continuous maps fo: V --+ IRn+1 by A-directed maps?

(iii) A continuous map f: V --+ IRq is called locally K -flat if each point v E V admits a neighborhood V c V whose image f(V) c IRq lies in some k-dimensional affine subspace in IRq. For which k, q and n = dim V does every continuous map fo: V --+ IRq admit a CO-approximation by piecewise linear locally k-flat maps f: V --+ IRq? Notice, that the k-flat condition cannot be expressed with the above "relatives" 9l,1.

2.4.11 Equidimensional Isometric Maps

Let Vand W be n-dimensional Riemannian manifolds. A continuous map f: V --+ W is called normally folded along a hypersurface Vo c V (compare 1.3.6) if the following condition is satisfied for every small open subset V c V which is divided by Vo into two submanifolds with boundaries, say into V' and V", such that V' n V" =

av' = av" = vn Yo.

The Folding Condition. The restricted maps I' = fl V': V' --+ Wand f" = fl V" are C1-smooth immersion (including the boundaries), whose differentials are related on the bundle T(V)I V n Vo by the orthogonal reflection in the subbundle T(Vo) I V n Yo, where "orthogonal" applies to the Riemannian metric in W Namely, if v E Tw(W), for w = f(v), VE Vo, is a normal vector to the hyperplane D!'(T,,(Vo)) = Df"(T,,(Vo)) c

Tw(W), then (DfT1(v) = _(Df"tl(V) for all VE Yo. (The Riemannian metric on V plays no role in this definition.)

If f is a normally folded map, then the Riemannian metrics induced by I' and f" agree on the bundle T(V)I Yo, and so the map f induces a continuous Riemannian metric, say f*(if» on V, where if> denotes the Riemannian metric on W


Suppose that the manifold V splits, V = V' x [O,IJ for a closed manifold V', dim V' = n - 1, and let fo: V ---+ W be a C1-map which is an immersion near the boundary av c V, av = (V' x 0) U (V' x 1), and which also is an immersion on the submanifolds V' x t c V, t E [0,1]. Consider the (positive semidefinite) quadratic forms go = fo*(rt» and 9 = go + a(dt)2 on V for a continuous non negative function a: V ---+ ~+ which vanishes on the boundary av c V.

Stretching Lemma. The map fo admits a uniform approximation by normally folded maps J.: V ---+ W, J. ---+ fo for I> ---+ ° such that the induced forms gE = J.*(rt» uniformly converge to 9 for I> ---+ 0, and such that each map J. equals fo on a small neighborhood UE c V of the boundary av c V.

The proof is given at the end of this section.

Now let Vbe an arbitrary n-dimensional manifold without boundary. A quadratic form g' on V is called elementary if there exists a split submanifold U = U' x [0, IJ c V, where U' is a finite or countable union of closed (n - I)-dimensional manifolds, n = dim V, such that g' is zero outside U and g'l U = a(dt)2 for some continuous function a: U ---+ ~+, which vanishes on au c U.

Decomposition Lemma. An arbitrary Riemannian metric on V is a finite sum of elementary quadratic forms.

This is a trivial corollary of the Nash-Kuiper theorem (see 2.4.9). An alternative proof is suggested in the following

Exercise. Let G be a nonempty set of continuous positive semidefinite quadratic forms on V with the following three properties:

(i) If 9 E G then also ag E G for all continuous functions a: V ---+ ~+; (ii) If d: V ---+ V is C1-diffeomorphism, then d*(g) E G for all 9 E G;

(iii) If theforms gi E G, i = 1,2, ... , have mutually disjoint supports, then Ligi E G.

Show that every positive definite form is contained in G and thus prove the decomposition lemma.

Proposition. Let 9 and go be Riemannian metrics on V, such that the form 9 - go is positive definite. Then the identity map V ---+ V admits a fine CO-approximation by isometric maps f: (V, g) ---+ (V, go)·

Proof Write 9 - go = If=l ej for some elementary forms ej. The stretching lemma implies the existence of Riemannian metrics gj on V and of normally folded strictly short maps jj: (V, gj) ---+ (V, gj-l ),j = 1, ... , N, which are CO-close to the identity, such that the forms gj - jj*(gj-l) and gj - gj-l - ej are CO-small for j = 1, ... , N. Thus we obtain a continuous map F1 : V ---+ V, namely the composed map Fl = flo· .. 0 fN, with the following three properties,

(i) The map Fl is (arbitrary!) CO-close to the identity map; (ii) The induced form g~ = Ft(go) is continuous and positive definite;

(iii) The form g - g~ is positive definite and it can be made (by some choice of Fd as small as one wishes.

Then we take g~ in place of go and construct a map F2 which also satisfies (i) and for which the metric g6 = Fi(g~) satisfies (ii) and (iii). We go on and obtain continuous positive forms gh on V, i = 1, 2, ... , and continuous isometric maps Fi+l: (V, gh+1) --+ (V, gh) (each Fi is a finite composition of some folded maps), such that the forms gh uniformly converge to g and such that the composed map Fl 0

F2 0'" 0 Fi uniformly converges for i --+ 00 to the desired isometric map f: (V, g)--+

(V, go)·

Corollary. Let V be an n-dimensional stably parallelizable manifold. Then V admits an isometric map V --+ IRn.

Proof Poenaru's folding theorem (see 2.1.3) yields a strictly short normally folded map fo: V --+ IRn (Compare the proof of the theorem below). Since the induced form go on V is continuous and since the form g - go is positive for the given Riemannian form g on V, there is an isometric map f: (V, g) --+ (V, go). Then' the composed map fo 0 f: V --+ IRq is isometric.

Theorem. Let fo: V --+ W be a strictly short map between Riemannian manifolds of dimension n. Then fo admits a fine CO-approximation by isometric maps f: V --+ W In particular, there exists an isometric map f: V --+ IRn for all Riemannian manifolds V of dimension n.

Proof We assume [compare (A) in 2.4.10] the map fo to admit a stratification {Ii}, i = 0, ... , k, such that the map folI i is an immersion Ii --+ W for all i = 1, ... , k. Then by induction we make the map fo isometric on the stratum Ii, j = 1, ... , k, by applying to U = V\ U{~6 Ii the following

Sublemma [compare (A) in 2.4.10]. Every strictly short map U --+ W homotopic to fo I U: U --+ W admits a fine CO -approximation by continuous maps U --+ W which are isometric on Ii c U and strictly short on U\Ii.

Proof First let dim Ii < n. Then the given map U --+ Wadmits a CO -approximation by another strictly short map, say by fo: U --+ W which is a C1-immersion on Ii. This follows from the relative h-principle (see 2.4.5). Then the above proposition yields an isometric map, say j: (Ii,g) --+ (Ii,io*(tJ»). This map fis obtained as a limit of strictly short maps Ii --+ Ii which can be chosen arbitrarily close to the identity, and then the composition fo 0 j: Ii --+ W admits a strictly short extension to U\Ii.

Now, let dim Ij = n, which amounts to Ij = U for our stratification. Then there is a strictly short normally folded map fo: U --+ W which approximates the given map U --+ W To obtain fo we take an arbitrary triangulation of U and we first

construct a strictly short map f~: U -+ W which is a C1-immersion near the (n - 1)skeleton. Here again we use 2.4.5 and the obvious induction by skeletons. Next, we perturb f~ inside every open n-dimensional simplex A as follows. Let S;-1, tE(O, 1), be smooth concentric spheres in A which approach the boundary 0,1 for t -+ 1 and which shrink to the barycenter SO EA as t -+ O. We require the perturbation f~': U -+ W to be a strictly short C1-map which is an immersion on the sphere S;-1 c A for all A and t E (0, 1), and which is also an immersion near the center So E A for all A. Recall that f~' equals f~ near the (n - 1)-skeleton, and so f~' is an immersion outside the cylinders sn-1 X [e,1 - e] c A, for all A c U and for some e > O. Finally, we apply the stretching lemma to the map f~'lsn-1 x [e,1 - e], for all cylinders sn-1 X [e,1 - e], thus obtaining a strictly short normally folded map fo: U -+ W. As the induced metric fo*(d'» is continuous, the above proposition yields an isometric map j: (U, g) -+ (U,lo*(d'>)) and then the composition fo 0 j: U -+ W also is isometric. Q.E.D.

Proof of the Stretching Lemma. We may assume with an obvious approximation that the map fo: V -+ W is COO-smooth and that the image fo(V) c W lies in the interior of W. We also assume the metric d'> on W to be COO -smooth and the function a(v', t) to be COO-smooth and strictly positive for 0 < t < 1. This allows us to use normal geodesic coordinates in (V, g) and in (w, d'» near a given submanifold V~ = V' x to c V, 0 < to < 1. Indeed, each point v E V which is close to V~ is uniquely expressed by a pair (v', t'), where v' E V~ is the g-normal geodesic projection of v to V', where t' E IR has It'l = distg(v, V~) and where sign t' is determined with a chosen orientation in the normal bundle of the submanifold V~ c V. Similarly, we use d'>-normal coordinates in W for the immersed manifold fo: V~ -+ Wand express the points w = f(v) E W, for maps f which are close to fo I V~, by pairs, w = (w', s), W' E V~, SE IR.

Next, we choose a small e > 0, such that [to, to + e] c (0,1), and we stretch the map fo on the submanifold V. = V' x [to, to + e] as follows. Denote bye' = e'(v) the g-distance, distg(v', v"), where v" E V' x (to + e) c V. is a (unique!) point whose normal projection to Vo equals v' which also is the normal projection of v to Yo, and denote by b' = b'(v) the s-coordinate of fo(V")E W. Now we consider the (stretched) map /.: V. -+ W, whose d'>-normal projection to the manifold V~ (immersed by fo to W) equals that of fo and whose s-coordinate s(v) = s(v', t') is uniquely determined by the conditions s(v', 0) = 0 and

v't' -ds ( ) {-1 for t' E [o:,e' - 0:],0: = He' + b'); dt' , - + 1 for t' E [0,0:) U (e' - 0:, e'].

The map /. equals fo on the boundary av. = (V' x to) U (V' x (to + e)) and it folds along two hypersurfaces which are C1-close to V~ = V' x to c V. This folding is nearly normal for small e -+ 0, and a small perturbation of /. near the two folds satisfies on V. the requirements of the stretching lemma.

We obtain the stretching on all ofVby first subdividing the interval [eo, 1 - eo] C

[0,1] into e-subintervals, then by applying the above stretching on each e-subinterval [to, to + e] of this subdivision and finally by letting e -+ 0 and eo -+ O. Q.E.D.


Exercises and Generalizations. (a) Prove that Lipschitz' maps are almost everywhere differentiable and then show that no map yn --+ Wn- 1 is isometric. (a') Let V be an n-dimensional manifold with a C2-smooth non flat Riemannian metric and let f: V --+ ~n be an arbitrary isometric map. Find a point w E ~n whose pullback f- 1(w) c V contains a non empty perfect subset and thus show that f is never T1-isometric [see (A') in 2.4.10].

(b) Let V and W be pseudo-Riemannian manifolds of the same type (n+, n_) for n+ + n_ = n = dim V = dim W, such that min(n+, n_) ~ 1. Show that every continuous map V --+ Wadmits a fine CO-approximation by isometric maps, which by definition are Lipschitz'maps f: V --+ W preserving the pseudo-Riemannian length of all rectifiable curves in V.

Hint. Define and use maps with normal folds and cusps (compare 1.3.1).

P-Convexity and c-1,1-Solutions of Partial Differential Relations. A relation fIl c x(r) is called P-convex (P for principal) ifthe intersection of fIl with every principal subspace R ~ ~q in x(r) is a convex subset in R. Denote by Convpfll c x(r), now for an arbitrary fIl c x(r), the intersection of all P-convex subsets in x(r) which contain fIl.

Let a relation fIl c x(r) admit an (in)approximation by some open relations fIlj c x(r), i = 0, 1, ... , such that:

(i) fIlj c Conv p fIli+1 for all i = 0, 1, .... (ii) The relations fIlj are uniformly bounded in the fibers of the fibration P;-1: x(r) --+

x(r-1). That is the union U~Oflljn(p;-1)-1(y) is a relatively compact subset in x(r) for all compact subsets Y c x(r-1).

(iii) If a sequence aj E fIlj converges as i --+ 00 to some point x E x(r), then x E fIl.

Theorem. Let fo: V --+ X be a C-section for which J;o(V) c fIlo. Then fo admits a fine C-1-approximation by C-1-maps f: V --+ X which almost everywhere satisfy the relation fIl in the following sense. The jet J;-1: V --+ x(r-1) is an almost everywhere differentiable Lipschitz map and J;(v)EfIl for almost all VE V.

Idea of the Proof. The solutions f are obtained as limits of piecewise Cr-smooth solutions of the relations fIl j. The basic ingredients of the proof are similar to (in fact, much simplier than) those in 2.4.6. The key step is a piecewise smooth version of the C~-approximation lemma (see 2.4.1). The actual proof is left to the reader.

Example. If fIl is the isometric immersion relation then one takes the strict shortness relation for fIlo and let fIl j , i = 1, ... , correspond to those strictly short maps V --+ W for which the induced Riemannian metric on V is 8j-c1ose, 8j --+ 0 for i --+ 00 to the given metric on V. Then the theorem delivers a Lipschitz map f: V --+ W, for dim W ~ dim V, whose differential Df : T,,(v) --+ Tw(W), w = f(v) is isometric for almost all v E V. Such a map f, however, may fail to be isometric. In fact, f may collapse an arbitrary submanifold of positive codimension in V to a single point in W Probably, there is a refinement of the above general theorem which produces "stronger" solutions, like actual isometric maps for fIl = fIlisom '

2.4.12 The Regularity Problem and Related Questions in the Convex Integration

If f!ll c x(r) is a generic CS-smooth submanifold whose intersections with the principal subspaces R c x(r), R ~ IRq, have positive dimension (this corresponds to underdetermined systems of P.D.E. with CS-smooth coefficients), then one expects this f!ll to admit many CS-smooth holonomic sections V ~ f!ll that correspond to e+1_ solution V ~ X of f!ll. The convex integration only delivers e-solutions and no regularity assumption on f!ll helps to smooth these solutions. One might try to apply the convex integration to the lift f!llS C x(r+s) that corresponds to the differentiation s times the corresponding P.D.E. system (see 1.1.1). Unfortunately, the differentiated system is linear in the top derivatives (of order r + s) and so the convex hull extensions Conv N f!llS do not enlarge f!llS for s ~ 1. Thus the convex integration fails to produce holonomic sections V ~ f!lls that would correspond to e+1-solutions V ~ X of f!ll. Observe that quasilinear relations like f!llS lie on the very boundary of the convex integration domain as small perturbation make them ample. One hopes that "nonintegrability" of generic quasilinear relations may substitute "nonlinearity" in the convex integration scheme [compare 2.3.8 and (B) in 2.4.5], but one does not know how to make this idea work.

A more realistic goal is that of obtaining solutions V ~ X of f!ll c x(r) in Holder classes e+a for some 0 < rx < 1. Some results in this direction are known for the isometric immersion problem. Borisov (1965) announced the existence of isometric C1+a-immersions vn ~ IRn+1 for all rx < (n 2 + n + 1)-1, and he also indicated some obstruction to isometric C1+a-immersions for rx > ~. Furthermore, Kallen (1978) proved that every n-dimensional manifold V with a CP-smooth metric, 0 < /3 < 2, admits an isometric C1+a-embedding V ~ IRq, q = 3(n + l)(n2 + n + 2) + 2n, for every rx < /3/2.

Convex Integration with Elliptic Operators. The success of the convex integration depends on our complete understanding of the (elliptic) operator dr/dtr on maps IR ~ IRq (see the C-L-approximation lemma in 2.4.1). The question is whether there exists a convex integration type theory which relies upon another elliptic operator, such as a or the Dirac operator.

Integrodifferential Inequalities. The convex integration works in those cases when an inequality like Ilf(b) - f(a)11 ::;; (b - a)S~ II f'(t) II dt, for f: [a,b] ~ IRq, fails. For example, if a map V ~ IRq is short, then it can be approximated by isometric maps. One asks himself if a similar role may be plaid by more interesting inequalities, such as the isoperimetric inequality,

(t»lfl n/(n-l)Y-1)/n ::;; constn tn Ilgradfil

for C1-functions f: IRn ~ IR with compact support. Unfortunately, one does not know all inequalities like (*) that are linear inequalities between the integrals SIR» Ilfllillgradfll j and their powers. The problem amounts to understanding the class of measures on 1R2 which are the push-forwards of the Lebesgue's measure on IRn under the maps XH(f(X), Ilgradfll(x)) for all C1-functions f: IRn ~ IR with compact supports. Observe that measures on IRq which are push-forwards of smooth


measures under smooth maps F: V -+ IRq are rather special even for those F which are free of any additional (holonomy) condition. For example, let V be a closed connected manifold with a positive COO-measure dJl. on V, such that JvdJl. = 1 (such a measure is unique up to a COO-diffeomorphism of V). Functions J;: V -+ IR, i =

1, ... , q, are called independent if they are independent as random variables on the probability space (V, dJl.). That is the push-forward measure F*(dJl.) on IRq for F =

(fl"" ,h) equals the product of the measures (J;)* (dJl.) on IR, i = 1, ... , q. One has with this definition the following unsolved

Problem ofV.Eidlin. Find a topological characterization of those manifolds V which admits nonconstant independent real analytic functions J;: V -+ IR, i = 1, ... , q.

Exercises. (a) Show that V admits independent nonconstant COO-functions fl' ... , h if and only if q :;;; dim V.

(b) Show that Cartesian products V = Vl X ... x Yq, dim V; > 0, i = 1, ... , q, admit q nonconstant independent Can-functions V -+ IR.

(c) Let F = (fl"" ,h): V -+ IRq be a Coo-map whose components J;, i = 1, ... , q, are independent. Show that a point x = (Xl, ... , Xq) E IRq is a noncritical value of F if and only if Xi E IR is a noncritical value of J; for i = 1, ... , q.

(d) Denote by M (f) c V for f: V -+ IR the union of those points v E V at which the function f assumes local maximum or local minimum. Let fl' ... ,fq be independent COO-functions, such that the function J;, i = 2, ... , q, is nonconstant in an arbitrarily small neighborhood lPjZ(v) c V of a given point v E M(fd c v. Show that the intersection M(fd n lPjZ(v) goes under the map (f2"" ,h): V -+ IRq-l onto a subset in IRq-l of dimension q - 1 (i.e. the interior is nonempty).

Hint. Apply the above (c).

(e) Let f: V -+ IR be a non-constant real analytic function such that every connected component of the subset M(f) c V has dimension n - 1 for n = dim V. Show that the fundamental group 11:1 (v) admits a surjective homomorphism either on 71. or on 71.2 * 71.2,

Hint. The map f admits a (unique) factorization,

J

~ V~X~IR,

where X is a one-dimensional polyhedron and such that the pullback fl-l(X) c V is connected for all X EX. Study the map fl and use the fact that every analytic subset MeV (of dimension m = n - 1) is a 71.2-cycle (of dimension m).

(e') Prove the converse to (e): If 11:1 (V) surjects onto 71. or 71.2 * 71.2 then there is a Can-function f: V -+ IR for which the subset M(f) c V is a Can-hypersurface in V.

Hint. Use harmonic maps V -+ Sl for a double covering V -+ V.

(f) Let V admit n nonconstant independent COO-functions for n = dim V. Prove with (e) and (d) the existence of a surjective homomorphism of 11: 1 (V) either onto 71. or onto 71.2 * 71.2,

Part 3. Isometric Coo-Immersions

The theory of topological sheaves (see 2.2) and the implicit function theorem (see 2.3) provide a general framework for the isometric immersion problem which consists of the construction and classification of isometric Coo-maps f: (V,g)-+ (w, h), for given forms g on V and h on W (compare 2.4.9). However, the direct application of 2.2 and 2.3 does not lead to geometrically significant results unless specific geometrical features of the forms in question are taken into account.

3.1 Isometric Immersions of Riemannian Manifolds

Let V = (V,g) be a Coo-smooth Riemannian manifold of dimension n. We want to give an upper bound to the minimal dimension q for which there exists an isometric Coo-immersion V -+ IRq. A similar question is studied for an arbitrary Riemannian manifold W = (w, h), dim W = q, in place of IRq.

3.1.1 Nash's Twist and Approximate Immersions; Isometric Imbeddings into IRq

Let S,:-l c IRq denote the e-sphere in IRq, let fo: V -+ S,:-l be a C1-map and let go = fo*(h) be the induced quadratic differential form on V for h = 2:[=1 dxf on IRq. If cp: V -+ IR is any C1-function, then the form gl = f1*(h) for f1 = cpfo: V -+ IRq, satisfies the following

Fundamental Formula.

Proof The equation (fo,fo) = e2 implies (fo, 8fo) = 0 for all vector fields 8 on V. Therefore,

gl(8,8) = (8f1,8f1) = <8(cpfo),8(cpfo)

= cp2(8fo,8fo) + 2cp8cp(fo,8fo) + (8cp)2(fo,fo)

= cp2go(8, 8) + e2(8cp)2.

Let us apply (*) to the map fo given by (YE' z,): V -+ S,l c 1R2, for y, = e sin(e-1 x),

222 3. Isometric Coo-Immersions

Z, = e COS(e-1 x), where x: V --+ [R is an arbitrary C1-function. Then we obtain, with go = (dy,)2 + (dZ,)2 = (dX)2,

Nash's Formula.

(**)

Exercise. Letthe manifold (V, g) admit an isometricimmersionf: (V, g) --+ Sq-1 c [Rq. Construct with (*) an isometric immersion

(V, cp2 g) --+ ( [Rq.!, - dX~+1 + it dxl ).

for all cp: V --+ [R.

Prove the existence of an isometric Coo-immersion (S2,g) --+ [R3,1 for all Coo_ metrics g on S2 by using a conformal diffeomorphism (S2,g) --+ S2 C [R3.

A Decomposition g = I cpl dxl· Let g be a continuous Riemannian metric on V. Clearly. there exist Coo-functions Xi: V --+ [R, i = 1, ... , 1, for some 1, such that the form gl Tv(V) is contained in the interior of the convex hull of the forms (dxYI T,,(v) for each tangent space T,,(v), v E V.

Exercise. Prove the existence of such Xi' i = 1, ... ,1, for 1 = [n(n + 1)/2J + n. Next, if the form g is C"-smooth for some IY. :5: 00, then there are (use a partition

of unity) C"-smooth functions CPi on V, such that

I

(+ ) g = I cpldxl· i=1

(A) Corollary: Nash's Approximation Theorem. An arbitrary Riemannian C"-metric g on V has a fine C"-approximation by some C"-metrics g' on V which admit isometric C"-immersions 1': (V,g') --+ [R21 for some 1 = l(n) < 00, n = dim V.

Proof According to (**) each form cpl(dxY admits an e-approximation by the form (dYi,,)2 + (dzi . .)2. If V is compact, then the sum g' = g; = Il=1 [(dYi.,)2 + (dzi,,)2J is e-close to g'. As this g' is induced by the map (Yi,., Zi,,): V --+ [R21 the proof is accomplished for compact manifolds V. If V is non compact, then in addition we require every function Xi' i = 1, ... , 1, to have its support in a disjoint union of compact subsets, say in U j Uij C V. With such a function Xi we use in (**) a function e = e(V), v E V (instead of a constant e > 0), such that e(V) is constant on every subset Uij' j = 1, 2, : .. , and such that the function e(V) is small in the fine COO-topology. Then the error g - g; is small in the fine C"-topology. Q.E.D.

(A') Remark. If 0( is an integer, then g admits a C"-approximation by some coo_ metrics, and so the above g' and l' can be taken Coo-smooth. If 0( is not an integer, then we have a CP-approximation of g by Coo-smooth metrics g', induced by imbeddings into [R21 for all f3 < 0(, such that (V,g') admits a Coo-immersion into [RI.

3.1 Isometric Immersions of Riemannian Manifolds 223

Exercise. Let (V, g) be a complete simply connected surface of non-positive curvature K(g) ~ O. Construct C"-maps f.: V -+ 1R3, e -+ 0, such that the induced metrics 1.* (h), for h = L~=1 dx;, C"-converge to g on compact subsets in V.

Hint. Use coordinates (U1,U2) on V for which g = dui + qJ(U1,U2)du~.

Remark. If K < 0, then every bounded domain U c V admits an actual (not only approximate) isometric immersion (U, g) -+ 1R3. But if K(g) ~ - X2 < 0, then there is no global isometric C2-immersion (V, g) -+ 1R3 [see the survey by Poznyak (1973)]. This is the famous theorem of Hilbert-Efimov.

Isometric Imbeddings V -+ IRq. Start with a free Coo-imbedding fo: V -+ IRqo which is strictly short relative to a given C"-metric g on V. Then the form g - fo*(ho), ho = L{g1 dx;, is positive and so there exists a C"-immersion /': V -+ 1R21 which is isometric relative to some C"-metric g', such that the "error" 0 = g - fo*(ho) - g' is arbitrarily small in the fine C"-topology. If ex> 2, then there exists a small perturbation of fo, say f1: V -+ IRqo of class C", such that f1*(ho) = fo*(ho) + 0 (see 2.3.2). Thus, we obtain an isometric C"-imbedding (f1'/'): (V, g) -+ IRqo+21 as g =

fo*(ho) + 0 + g'. [Compare Moore-Schlafly (1980).] We shall reduce in 3.1.7 the number 21 to q1 = (n + 2)(n + 3)/2 and then we shall

obtain the following

Imbedding Theorem. Every Riemannian C"-manifold, 2 < ex ~ 00, admits a free isometric C"-imbedding f: V -+ IRq for q = n2 + lOn + 3.

Remarks. If ex > 4, then the dimension of the ambient space can be reduced to q = (n + 2)(n + 3)/2 (see 3.1.7), but no such improvement is known for smaller ex.

Isometric C"-imbeddings V -+ IRq for ex = 3,4, ... ,00 and q = (n + 1) Gn(n + 1) + 4n] were obtained by Nash (1956). He starts with imbeddings of compact manifolds into IRq for q = !n(n + 1) + 4n, and then he reduces the non-compact case to the compact one with the following

(B) Compact Decomposition. Let V be a manifold without boundary, let V;, i = 1, 2, ... , be closed manifolds, dim V; = n = dim V, and let K V -+ V; be Coo-maps with the following two properties:

(i) The supports of the differentials D];: T(V) -+ T(V;) form a locally finite family: for every compact subset U c Vall maps"/;, i ~ io = io(U) have D];I U = O.

(ii) There exists an open cover U~1 Ui = V, such that the subset Ui c V is mapped difJeomorphically by ]; onto an open subset (j c V; for all i = 1, ....

Then, as the reader will agree, every C"-metric g on V admits a decomposition

g = L];*(?fJ, i

for some Riemannian C"-metrics ?Ii on V;.


(B') Corollary. Let Ui Vi = V be an arbitrary locally finite cover of V by open subsets Vi c V. Then every CIZ-metric g on V decomposes as

where Xj: V --+ [R are CIZ-functions on V such that the support of Xj is contained in some subset Vi for i = i(j) and i(j) --+ 00 for j --+ 00 (i.e. almost all functions Xj vanish on Vi for every fixed i)

Proof We may assume (refine the cover if necessary), that the subsets Vi satisfy (ii) for some maps 1: V --+ V; ~ sn. Then g = 2>K*Uli) and we obtain the required decomposition with isometric CIZ-immersions (sn, gi) --+ [Rq for all i = 1, ....

Exercise. Assume, given n, q and iX, there exists an isometric CIZ-immersion (sn,g)--+ [Rq for all CIZ-metrics g on sn. Show that an arbitrary n-dimensional Riemannian CIZ-manifold V admits an isometric CIZ-immersion into [R(n+1)q. Prove the existence of such an immersion V --+ [R2q, provided V is diffeomorphic to sn minus finitely many points, for example V ~ [Rn.

3.1.2 Isometric Immersions vn --+ W q for q ~ (n + 2) (n + 5)/2

The following proposition is proven on the basis of Lemmas (D) up to (F) at the end of this Sect. 3.1.2.

(A) Extension Lemma. Let (V, g) and (w, h) be Riemannian Coo-manifolds and let f: (V, g) --+ (W,h) be a free isometric Coo-immersion. Consider the cylinder (V x [Rl,

g EEl dt2 ) and let V c V = V x 0 c: V X 1R1 be a compact subset in V. If the subset V is contractible in V and if q 2: (n + 1)(n + 4)/2, then the map fl V: V --+ Wextends to a free isometric Coo-map 0--+ W for some neighborhood a c V x [Rl of V.

(A') Corollary. Consider the product (V x [R2, g EB dti EB dt~) and let V c V =

V x 0 c V X 1R2 be a subset as in (At. If q 2: (n + 2)(n + 5)/2 then t1.!.e map fl V extends to a free isometric Coo-map 0--+ W for some neighborhood a c V x [R2

ofv.

Proof First extend fl V to a and then apply (A) to a in place of V.

One may equivalently express (A') by saying that fl V extends to a free isometric Coo-map V' x D; --+ W for some neighborhood V' c V of V and for some e-ball D; c [R2, e > O. This leads to the following

(B) Cylinder Lemma. If q 2: (n + 2)(n + 5)/2, then the map fl V extends to a free isometric Coo-maps V' x [Rl --+ W

Proof Use the above with any isometric Coo-map tjJ: [Rl --+ DE2, tjJ(O) = o.


(B') Adding dx2• Let x = V --+ 1R1 be a COO-function with support in U. If q > (n + 2)(n + 5)/2, then there exists a free isometric Coo-immersion f1: (V, g + (dx)2)--+ (w, h) which equals the immersion f: V --+ Woutside the subset U.

Proof The graph of the function x is an isometric immersion

I'x: (V, g + (dx)2) --+ (V x 1R1, g EEl dt2).

Hence, the composition of the map I'xl U' with the above map U' x 1R1 --+ W is a free isometric map (U', g + (dx)2) --+ W which agrees with f outside U. Q.E.D.

(B") Remark. One can make the map f1 arbitrarily CO-close to f by taking a sufficiently small ball D; c 1R2.

(C) Theorem. Let (V, g) and (w, h) be Riemannian Coo-manifolds and let fo: V --+ W be a strictly short map. If q ;?: (n + 2)(n + 5)/2, then fo admits a fine CO-approximation by free isometric Coo-maps f: (V, g) --+ (w, h).

Proof Assume the map fo to be COO-smooth and free (see 1.1.4) and decompose the metric g - fo*(h) [see (B') in 3.1.1]

g - fo*(h) = 'LdxJ, j = 1,2, ... , j

for some locally finite sequence of Coo-functions Xj on V. Then the above lemma (B') provides free isometric Coo-maps

k

for gk = fo*(h) + 'L dxJ, k = 1,2, ... , j=1

which converge for k --+ 00 (in fact, the maps h stabilize on compact subsets in V) to the required isometric map f: (V, g) --+ (W,h).

(D) The Cauchy Problem for Cylinders. We study Coo-maps f: V x IR --+ (w, h) and use the local coordinates U1, ... , Un' t in V x IR. We denote by Vi' i = 1, ... , n, Vt, Vij = Vi Vj etc., the covariant derivatives in(W, h) relative to the vector fields ai = a/aUi' and at = a/at. Thus, the derivatives VJ = D,(ai), Vijf etc., are vectors fields in W along the mapped manifold f: V x IR --+ W, which are called, for brevity, fields along V x IR. Such a field X is called binormal to V x to C V x IR for a given to E IR if it is normal to the osculating spaces of the map fl V x to at all points (v, to) E V x to. Recall that the osculating space, called 'rv7toUI V x to) c Tw(W), w = f(v, to), is spanned by the covariant derivatives VJ and Vij!, i, j = 1, ... , n, at w, thus, the binormal fields X satisfy

(X,VJ) =0, (X, Vijf) = 0,

for (X, Y) ~ h(X, Y).

(D') Lemma. A Coo-map f: V x IR --+ (W, h) induces the cylinder metric g EEl dt2 on V x IR for the metric g = f*(h)1 V x 0, if and only if the field Vtf is binormal to V x t for all tE IR and (Vtf, Vtf) == 1.

226 3. Isometric CD-Immersions

Proof The isometric immersion equations for the metric g EB dt2 are

(1)

(2)

(3)

def (VJ, VJ> = gij = g(Oi' OJ),

(VJ,Vtf> =0,

(Vtf,Vtf> = 1.

i,j = 1, ... , n,

Since gij does not depend on t for the cylinder metric,

(1')

Then we differentiate (2) and alternate i +-+ j,

(2') (Vijf, Vtf> + (VJ, Vtif> = ° (VjJ, Vtf> + (Vjf, Vljf> = 0.

As Vij = Vji and Vit = Vii' we conclude with (1') and (2') to

(4)

The Eqs. (2), (3) and (4) show Vtf to be a binormal field of norm 1. This is the "only if" claim of the lemma and we obtain "if" by reversing the above calculation.

Janet's Equations. Differentiate (4) in t and write

(4')

for X = Vtf, Xi = VJ Differentiate (3) in uj and then in Ui,

(3')

(3")

(Vtf, VtJ> = 0,

(Va/,VaI> + (X,ViVtXj> = O.

The Eqs. (4'), (3") and the derivatives of (2) and (3) in t result in the following system of (n + l)(n + 2)/2 P.D.E. of second order

(Vllf, Vtf> = 0,

(5) (Vllf, VJ> = - (Vtf, Vitf>,

(Vllf, Vijf> = (Vi/f, Vjtf> + (Vtf, R(Vtf, Vif, VJ),

where R denotes the curvature tensor of (w, h).

(D") Lemma. Every solution f: V x IR ~ W of (5) which satisfies on V x ° c V x IR the initial conditions

(6)

(VJ, Vjf> = gij

(Vtf,Vtf> = 1

(Vtf,VJ> = 0

(Vtf, Vij!> = ° is an isometric map f: (V x IR, g EB dt2 ) ~ (w, h).


Proof By reversing the above calculation we obtain with (5)

Vt(VJ, VJ) = 0,

Vt(VJ, VJ) = 0,

Vt(VJ, vijf) = 0,

and so the field VJ satisfies the assumptions of (D') on V x IR. Q.E.D.

227

Thus, the extension of an isometric map fl V x 0: (V x 0, g) ~ (w, h) is reduced to the solution of the Cauchy problem for the system (5) with the initial data (6), where the equations (6) say that the map f is isometric on V x 0, and that the field X = VJ along V x 0 is binormal to V x 0 and has norm one.

(E) Solution of (6). Fix an isometric Coo-map fl V x 0: (V x 0, g) ~ (w, h) and decide whether there exists a unit vector field X along V x 0 which is binormal to V x O. If the map f: V x 0 ~ W is free, then we can form the binormal bundle BN ~ V x 0 whose fiber BNv = Tw(W) e T,,z, v = (V,O)E V X 0, W = f(v,O), is the orthogonal complement to the osculating space T,,z = T,,~o(fl V x 0). This is possible because the dimension of T,,2 is independent of v E V X 0 for free maps fl V x 0, namely dim T,,2 = [n(n + 1)/2] + n, n = dim V. The fields X in question are unit sections V x 0 ~ BN, and so such an X exists in the following two cases,

(i) dim BN > n, that is dim W = q > [n(n + 1)/2] + 2n (ii) BN is a trivial bundle of dimension> O. For example, if the manifold V is

contractible and q > [n(n + 1)/2] + n.

Exercise. Find a (non-free!) Coo-immersion IR ~ 1R3 which admits no continuous non-vanishing binormal field.

(E') Solution of (5). Let the metric h on W be real analytic and let fo: V ~ W be a free Can_map which admits a unit binormal field X. Then fo extends to an isometric Can_map f: ((9jZ V, q EB dt2 ) ~ (w, h) for a small neighborhood (9jZ V c V x IR, V =

V x 0, and for g = fo*(h).

Proof The system (5) is linear in Vttf and so it can be can-resolved in Vttffor linearly independent fields VJ, Vijf and VJ along V = V x O. This means the existence of a can-field Yalong V x 0 which satisfies (5) when substituted for VtJ. Then the system (5) can be can-resolved in Vttf on a small neighborhood (9jZV c V x IR. This means the existence of a real analytic vector function cP on (9jZ V with the entries v, t, W, Vi' Vt, Vii' Vit , such that cP(v, t,f(v, t), VJ(v, t), ... ) identically satisfies (5) (for all f) when substituted for Vttf To obtain such a cP on (9jZ V (not only within a given coordinate neighborhood) we observe that the system (5) is invariant under coordinate changes in V and so (5) defines a differential relation fJl in the pertinent jet space X(2) over V x IR which splits as a Whitney sum, X(2) = Xv EB X.l (where the summand Xv corresponds to the derivative Vtt compare 2.4.1,2.4.7). The initial data (fo, X) on V = V x 0 c V x IR define a section into X.l, say X: V ~ X.l, while the


field Y, which resolves (5) over V, lifts this section to [lJl c X(2) -+ X-L. Call this lift Y: V -+ [lJl for V = X(V) c X-L and extend Y to a section ifj: lDfi(V) -+ [lJl for some neighborhood lDfi(V) c X-L. Such a section ifj gives us the required cP as well as the correct global definition of cP. Notice t~at the sections Yand ifj (as well as Yand cP) are not unique unless q = [n(n + 1)/2J + n + 1.

Now, the Cauchy-Kovalevskaya theorem gives us a unique can-solution f of the system Vttf = cP( v, t,f, Vlf), ... ) on lDfi V, which satisfies a given initial condition. Q.E.D.

Remark. It is unclear whether a similar extension theorem holds true for C OO _

immersions, except for V = IR and W = 1R3 , where such an extension clearly exists.

(F) Free Cylinders V x IR -+ W. Let fo: V -+ W be a free map. A binormal field X: V -+ EN is called regular if the fields VJo, Vijfo, X, ViX are independent on V. This is equivalent to the regularity of the section X: V -+ EN for the connection in the bundle EN -+ V induced from the Riemannian connection of (w, h) (see 2.2.6).

Example. If X = Vtfforafreemapf: V x IR -+ W,flV x 0 = fo, thenXisaregular field.

(F') Lemma. If the manifold V is parallelizable and if EN -+ V is a trivial bundle of dimension ;::: n + 2, n = dim V, then there exists a unitary regular field V -+ EN. For example, a regular field exists if V is a contractible manifold and dim W = q ;::: (n + l)(n + 4)/2.

This is an immediate consequence of the h-principle for regular fields (see 2.2.6).

Fix a free map fo: V -+ Wand a regular field X along V. Then a field Yalong V is called free (relative to fo and X) if the fields VJo, Vijfo, X, ViX, Yare independent on V.

Example. Iff: V x IR -+ W is a free map, then the field Vttfl V x 0 is free relative to fo = fl V x 0 and X = Vtfl V x O. Conversely, if fl V x 0 is a free map, the field Vtfl V x 0 is regular and Vttfl V x 0 is free, then the map f is free on some neighborhood lDfi(V x 0) c V x IR.

Letfo: V -+ Wbe a free Coo-map and Xbe a regular Coo-field. Denote by 1'2 -+ V the subbundle of the bundle fo*(T(W)) -+ V, whose fiber 1'/, v E V, is spanned by the osculating space T.,2(V) c Tw(W), W = fo(v), and the vectors X and ViX at w. Observe that dim 1;,2 = dim Tv2 + n + 1 = [(n + l)(n + 4)/2J - 1, and that a field Y is free if and only if it is nowhere contained in the subbundle 1'2. A free field exists, for example, if the orthogonal complement fo*(T(W)) e T2 is a trivial bundle of positive dimension. This is always the case for contractible manifolds V, provided q ;::: (n + l)(n + 4)/2.

(F") Proposition. Let fo: V -+ W be a free COO-map, let X be a unit regular binormal coo lield along Vand let Yo be a free field with respect to fo and X. Then the map fo

extends to a free COO-map f: V x IR ~ W for V = V x 0Jor which the induced metric in a small neighborhood mft V c V x IR is cylindrical,

f*(h)lmftv = fo*(h) EB dt2 ,

and VJIV= X.

Proof First let fo and Xo be real analytic. Then the existence of f without the freedom condition is established in (E') with the aid of a field Yalong V which serves for Vttfl v: To make f free we have to make Y free. To do this we first deform the given free field Yo to a free can-field Y~ which is orthogonal to 1'2, i.e. Y~: V ~ fo*(T(V)) e 1'2. Then the field Y + Y~ satisfies (5) when substituted for Vttf, since Y~ is normal to X = VJ, to V;/ and to Vijf We multiply Y~ by a large can-function A = A(V), which makes the field Y + AY~ free, and then we solve (5) with this field Y + AY~ in place of Y. The new solution f has VJIV = X and VttflV = Y + AY~, and so this map f: V x IR ~ W is free on mftv c V x IR.

Now, if fo and X are COO-smooth, then the formal part of the CauchyKovalevskaya theorem provides a free COO -map f: V x IR ~ W, for which fl V x ° = fo, VJIV x 0= X and VttflV x 0= Y + AY~, and which is irifinitesimally isometric along V = V x 0, i.e. J,jl V x ° = ° for (j = f*(h) - (fo*(h) EB dt2 ) and for all r = 0, 1, .... Hence, a small COO-perturbation of f is isometric on mft V c V x IR by the implicit function theorem (see 2.3.6). Q.E.D.

The Proof of Lemma (A). Lemma (F') provides a regular binormal field along a small neighborhood mftu c v: As q ~ (n + 1)(n + 4)/2 there is a free field along mftu as well. Hence (F") applies to fo = flmftu. Q.E.D.

Ca.-Immersions for 4 < (J( < 00. Let (w, h) be a Coo-manifold and let fo: V ~ W be a free Ca.-immersion. The map fo does not, in general, admit an extension to a Ca.-map f: V x IR ~ W which is isometric for the cylinder metric fo*(h) EB dt2 , because the last equation in (6) relates the derivative VJ to Vijfo and so isometric cylinders f: (V x lR,fo*(h) EB dt2 ) ~ W which extend fo are at most Ca.-1-smooth for generic Ca.-mapsfo·

Observe however that the metrics go = fo*(h) are somewhat better than just Ca.-1-smooth for Ca.-maps fo. Indeed, the curvature tensor RgO is Ca.-2-smooth by the Gauss theorema egregium, and so go is CP-smooth for all P < (J( in appropriate (harmonic) local coordinates on V according to Jost-Karcher (1982).

Question. Does some neighborhood U c V x IR admit an isometric Ca.-immersion (U,fo*(h) EB dt2 ) ~ (w, h) for q ~ (n + 1)(n + 4)/2, for all free Ca.-maps fo: V ~ W and for a given (J( > 2?

There is a similar regularity problem for bendings of a given free Ca.-map fo: V ~ IRq. Does there exists a Ca.-continuous deformation.ft: V ~ IRq, t E [0,1], for which .ft*(h) = fo*(h) for all t (where h = L1=1 dxf), such that the maps.ft for t > ° are not congruent to fo by isometries of IRq? (The implicit function theorem of 2.3.2 yields Ca.-1-bendings for 3 < (J( < 00.)

Let us generalize Theorem (C) to Ca.-manifolds (V, g) for 4 < (J( < 00.

C"-ExtensionLemma [compare (A)]. Letf: (V, g) ~ (w, h) be a free isometric C"-map for which the metric 9 = f*(h) is C"-smooth and let U be a (closed) n-dimensional ball in V, n = dim V. If 0( > 4 and q ;;::: (n + l)(n + 4)/2, then there exists a C"-immersion j: V x IR ~ W with the following three properties:

(i) There is a neighborhood a c V x IR of U c V = V x 0 c V x IR on which the map j is free and the induced metric is cylindrical,

j*(h)1 a = gliB dt2,

(ii) j*(h)IV x 0 = g,

(iii) The map jl V x 0 is as C2-close to f as one wishes.

Proof. Start with a fine Cil-approximation of f by a Coo-map fo for some P in the interval 4 < P < 0(. Then we formally solve the systems (5) and (6) near U and thus we obtain a Coo-map io: V x IR ~ W whose CIl-l-norm is controlled (i.e. this norm is bounded on every compact subset K c V x IR by a constant C = C(K,f)) and such that the metric io*(h)1V x 0 is CIl-l-close to 9 and io*(h)l{9fiU is CIl-2-close to 9 liB dt2 for a small neighborhood {9fiU c V x IR. The implicit function theorem (see 2.3.2) now yields a CIl-2-small perturbation of io to the required C"-map j Q.E.D.

The Proof of Theorem (C) for C"-Manifolds (V, g). Let fo: V ~ W be a strictly short map. Application of the implicit function theorem allows one to approximate fo by a free C"-maps, say fo: V ~ W, such that the metric 9 - fo(h) is Coo-smooth and is therefore a sum

9 - fo(h) = "LdxJ j

for some COO-functions Xj with small supports. Then the proof of(C) goes along with the above extension lemma.

C1-Approximation. The proof of Theorem (C) shows that the isometric Coo-immersionsf: (V, g) ~ (W,h), which CO-approximate the strictly short mapfo: V ~ W, are C1-close to fo, provided the map fo is C1-smooth and the metric fo*(h) is CO-close to g. This equally applies to C"-immersions, 4 < 0( < 00, and so we have the following

(G) Theorem. Let the metric h be COO-smooth and let 9 be C"-smooth, 4 < 0( < 00.

Suppose fo: (V, g) ~ (W, h) is an isometric C1-map which admits a fine C1-approximation by strictly short C1-maps. If q ;;::: (n + 2)(n + 5)/2, then fo admits a fine C1 _

approximation by isometric C"-maps (V, g) ~ (W, h).

Exercise. Let fo: V ~ W be a C2-immersion such that every point w = fo(v)e w, ve V, admits a normal vector v E Tw(W) e T,,(V) for which the second fundamental form II. T,,(V) is positive definite. Obtain a fine C2-approximation of fo by strictly short maps [for the metric fo*(h)]. Find a sharper infinitesimal criterion for such short approximations.

3.1 lsometric Immersions of Riemannian Manifolds 231

(H) The Parametric h-Principle. The proof of (C) applies to families of maps and then yields the parametric h-principle for free isometric maps relative to strictly short maps.

Exercises. Prove the parametric h-principle for free isometric C"-map f: V -+ IRq, for q ~ (n + 2)(n + 5)/2, n = dim V, for all C"-smooth Riemannian manifolds V and for IX> 4.

Let i\, ... , Ok be COO -smooth linearly independent vector fields on V. Construct a Coo-map f: V -+ IRq, q = (k + 2)(k + 5)/2, such that <oJ, oJ> = bij where 6ij = 0 for i -=1= j and 6ij = 1 for i, j = 1, ... , k.

Hint. Study the maps V -+ IRq which are isometric and free (in the obvious sense) on the subbundle Span {oJ c T(V).

3.1.3 Convex Cones in the Space of Metrics

Let (V, g) be a Riemannian Coo-manifold of dimension n without boundary.

Proposition. Let a Riemannian Coo-metric 6 on V satisfy 100n6 < 9 which means the positive definiteness of 9 - 100n6. Then there exist Coo-diffeomorphisms N V -+ V, i = 1, ... , m = n + 8, such that Dn=l pr(g) = mg + 6.

Proof Let (w, h) be the Cartesian product of m copies of (V, g) and let fo: V -+ W be the diagonal imbedding for which fo*(h) = mg. Since mn ~ (n + 2)(n + 5)/2, there is a fine CO-approximation of fo by a Coo-map f, such that f*(h) = mg + 6. An inspection of the proof of Theorem (C) in 3.1.2. shows that for 6 < 100-n g the map f can be assumed so C1-close to Io, that the projections Pi of I to the factors Vof Ware immersions and, hence. Coo-difTeomorphisms V -+ V. Q.E.D.

(A)Theorem. There is no (non-trivial) Diff-invariant convex cone in the space ~': of Riemannian Coo-metrics on an arbitrary closed manifold V, where Diff stands for the group of co -diffeomorphisms of V.

Proof Let C c ~': be a non-empty cone of metrics such that 9 E C implies 9 + 6 E C for any 6 E ~': for which b < f.ng with a given fixed positive f. n > O. Then, obviously, C = ~': for compact manifolds V. The above proposition then applies to give the theorem.

Exercises. Extend the above theorem to CO-metrics on V and also to Can-metrics. Let V be an open connected manifold without boundary of dimension n ~ 2.

Show every Diff-invariant convex cone C c ~':(V) to be stable, i.e.

C+g c C for all 9 E ~,:.

Complete metrics and also metrics of infinite volume provide interesting examples of such cones.

Consider a complete COO-metric g = gs on ~2 which is Sl-symmetric around a fixed point Vo E ~2 and such that the length of the circle of radius R about Vo equals RS for a given S E ~ and for all R ~ 1. Let Cs denote the intersection of all Diffinvariant convex cones in <§:f which contain gs. Show that the cone Cs properly contains Cs, for all Sl > S :::::;; ° and that Cs = Cs, for all Sl ~ S > 0. Prove that Cs

contains no (non-trivial) convex Diff-invariant subcones for any s > 0, and so Cs is a minimal cone in <§:f(~2). Show that Cs, s > 0, is a unique minimal (convex, Difl'-invariant) cone in <§:f(~2). Find further examples of convex Diff-invariant cones in <§~(~2) and in <§:f(V) for manifolds V of dimension> 2.

3.1.4 Inducing Forms of Degree d> 2

Let t/J be a symmetric form of degree d on ~q, for example, t/J = L1=1 xt. Consider a smooth map f: V ~ ~q and let {a1 , ... , all} be a frame of tangent vector fields on V. Then the induced differential form on V satisfies

f*(t/J)(ai" ai2 , .. ·, ai) = t/J(ai,f, ... , aiJ)

for all d-tuples of indices i1, ... , id = 1, ... , n. For example, if d = 3, then

i, j, k = 1, ... , n.

Let us write down the linearization L = Lf(y) of the differential operator ff-+ f*(t/J) (compare 2.3.1). To simplify the notation, we assume d = 3 and we put J; = ad and Yi = aiy. Then

Lf(y)(ai, aj, ak) = t/J(Yi,fj,fIJ + t/J(J;, Yj'f,.) + t/J(J;,fj, Yk)'

In order to invert the operator Yf-+Liy) we consider to maps y: V ~ IRq which are t/J-normal to fj ® f,. as follows

(1) t/J(y, fj, f,.) = 0, j, k = 1, ... , n.

Differentiate (1) and get with J;j = aAf

(1')

Hence, the solution of the P.D.E. system Lf(y) = g is reduced to the solution of (1) and the following system

(2) i,j, k = 1, ... , n,

where L denotes the sum over the permutations of the indices i,j, and k. The systems

(1) and (2) constitute (n ; 1) + (n ; 2) linear algebraic equations in y, where

( a) a! b = b!(a - b)!'

The map f is called t/J-free if it is C2-smooth and if the total rank of the systems

(n + 1) (n + 2). . (1) and (2) equals . 2 + 3 at every pomt v E V. More generally, If t/J is a

form of any degree d ;::0: 2, then we associate to the map f: V --+ IRq the system

(3) ~(y,jf;2, ... ,jf;d) == 0

where L denotes the sum over all permutations of the given indices. Here we have

d ( n + d - 2) (n + d - 1) l' .. d 11 r == r(n, ) == d _ 1 + d mear equatIOns m y an we ca the map

. (n + d - 2) (n + d - 1) . ~-free If the system (3) has rank r == d _ 1 + d at all pomts v E V.

This definition of freedom is independent of the choice of the frame a1 , ... , an on V. It is now clear that the differential operator f ~ f*( ~) is infinitesimally invertible

at ~-free maps f: V ~ (IRq, ~) for all forms ~ of any degree d ;::0: 2 (compare 2.3.1). Again, let ~ == L[=l xt and let E c X(2), X == V x IRq --+ V, denote the subset of

the 2-jets of e 2-maps f: V --+ IRq which are not ~-free. A straightforward calculation gives codimE;::O: q - r(n, d), and so generic e 2-maps f: V --+ IRq are ~lree for q ;::0:

n + r(n, d) by Thorn's transversality theorem. Denote by Fro the space of ~-free ero-mapsf: V --+ IRq with the fine ero-topology

and let <gro denote the space of symmetric ero -forms on V of degree d. By the implicit function theorem (see 2.3.2) the differential operator f ~ f*( ~) sends Fro onto some open subset <gro(q) c <gro which is non-empty for q ;::0: n + r(n, d).

Positive Forms of Degree d == 2k. A symmetric form 9 on IRn of degree 2k defines a quadratic form on the symmetric power (IRn)k by

g(x 1 ® X 2 ® ... ® Xk,Yl ® Y2 ® ... ® yd == g(Xt>Yl,···' Xk,Yk)·

If this quadratic form is positive definite, then the form 9 on IRn is also called positive. Positive forms 9 on IRn constitute a unique (up to sign) minimal convex open cone (in the space of forms on IRn) which is invariant under the linear group GLn [see (B) in 2.4.9]. A symmetric differential form 9 of degree 2k on V is called positive if gl T,;(V) is positive for all v E V.

Theorem [compare (B') in 3.1.1]. An arbitrary positive ero-form 9 on V admits a decomposition

where each x/ V --+ IR is a ero-function on V with support in some ball ~ c V, and such that every compact subset in V meets at most finitely many balls ~.

Proof. Since the form 9 is positive, there exist ero-functions Zi on V, i == 1, ... , I for some I == l(n, K) such that the form 9 I T,;(V) is contained in the interior of the convex hull of the forms (dzykl T,;(V) for all tangent spaces T,;(v), v E V [compare 3.1.1 and (B) in 2.4.9]. Thus we can obtain a decomposition

I

9 == L cpr dz1\ i=l


for some COO -functions ((J;, where the support of ((J; is the disjoint union of arbitrarily small subsets in V.

Let x, = B((J sin B- 1 z and y, = B((J cos B- 1 z. Then (see 3.1.1)

for B ~ o. Next, we use the identity [compare (4) in 2.4.9] (a2 + b2)k = I,J~l (Aja + J.lj b)2\ where

[ 2k J-1/2k

Aj = (sin 2n/j) j~ (sin 2n/j)2k

and

J.lj = (COs2n/j)[~ (COs2n/j)2kT1/2k.

For example, if k = 2, then

Thus we obtain

(a2 + b2)2 = 1(a4 + b4) + -H(a + b)4 + (a - b)4].

2k I, [d(Ajx, + J.ljy,)]2k-;::::-t ((J 2k dz2k. j=l

This approximation together with the openness of C§oo(q) makes the argument of 3.1.1 work for forms of arbitrary degree 2k ~ 2. Q.E.D.

Exercises. Show that every positive C~-form g of degree 2k on V, admits for IX > 2, a representation

q

g = I, (d/;)2k ;=1

for some C~-functions J;: V ~ IR and for

(n + 2k - 2) (n + 2k - 1) q = n + 2k _ 1 + (2k + 1) 2k .

Show that g = I,:=1 g~ for some positive quadratic forms g; on V and for

( n + 2k - 1) I = 2k ' and thus reduce the above results on 2k-forms to the case k = 1.

Let g be a C~-form of odd degree 2k + 1. Find an algebraic formula which makes

q

g = I, (d/;)2k+1, ;=1

for some C~-functions J; on V, for

[ (n + 2k - 1) (n + 2k)] q = (2k + 1) n + 2k + 2k + 1

and for all IX ~ 1 and k ~ 1. Then use the implicit function theorem to obtain

3.1 Isometric ImmersIOns 01 Riemannian Manifolds

= 2 [n (n + 2k - 1) (n + 2k)] q + 2k + 2k + 1 '

provided ex > 2.

Hint. See 2.3.7.

3.1.5 Immersions with a Prescribed Curvature

Consider a 4-form lP on IW,

Xi E [Rn, i = 1, ... , 4,

which is symmetric under the following permutations of the entries

Xl ~X2' X3 ~X4 and (Xl,X2)~(X3,X4)

235

This means lP is a symmetric bilinear form on the symmetric square ([Rn)2, and so

. . n(n+1)( n(n+1)) the dImenSIOn of the space of all such forms lP equals 4 1 + 2 .

There is a canonical splitting of lP into the sum lP = lP + + lP -, where lP + is the symmetric 4-form on [Rn obtained by the complete symmetrization of lP, and where lP - = lP - lP + satisfies

The form lP- has the symmetry type of curvature tensors. These constitute a space of dimension

n(n + 1) (1 n(n + 1)) _ n(n + l)(n + 2)(n + 3) = n2 (n 2 - 1). 4 + 2 24 12

Now, let f: V ~ W be an isometric C2-immersion between Riemannian COO _

manifolds V = (V, g) and (w, h) of dimension nand q respectively. The map f then induces the form lP of the above type on every tangent space T,,(V), v E V Namely, take local coordinates Ui in V which are geodesic at v E V and define lP = lPJ by

lP(aiA, 13k, 131) = <Vijf, Vkd),

for ai = af(v)jaui and for the covariant derivatives Vij and their scalar products in W The part lP - of lP = lPJ depends only on the curvature tensor R of the induced metric g = f*(h) by the Gauss theorema egregium

R(g) = lPJ-.

The remarkable feature of this formula is the absence of third derivatives of f in the expression for R(f*(h)), where h H f*(h) is a first order differential operator and g H R(g) is a second order operator. However, the composition of the two is a second (not third!) order operator.

The symmetric part lP/ also has a simple geometric interpretation. Let y be

a geodesic in V in the direction of some unit vector a E T,,(v). Then the value rfJ/ (a, a, a, a) equals the (curvature)2 of the curve f(y) c Wat w = f(v) E W

Observe that the form rfJf can be identified with the quadratic form on the symmetric square (T(V))2 which is induced from h by the second differential DJ. This D} maps (T(V))2 to the normal bundle Nf -+ V by sending a; ® OJ to PN(Vijf), for all bivectors a; ® aj E(T(V))2 and for the normal projection PN : T(W)I V -+ Nf .

Exercises. Show that the form rfJ/ is positive (in the sense of 3.1.6.) if and only if the map f is free.

Let V be a connected manifold and let /;: V -+ W, i = 1, ... , be isometric C2_

immersions for which rfJf~ :S; rfJt [i.e. rfJo - rfJf~ I T,,(V) lies in the closure of the cone of the positive forms for all v E V] and for which the sequence /;(vo) E W has an accumulation point in W for a given point Vo E V. Show that some subsequence Ca-converges for all a < 2 to an isometric map f: V -+ W which is twice differentiable almost everywhere and which a.e. satisfies rfJ/ :S; rfJo.

(A) Theorem. Let fo: V -+ W be a free isometric Coo-immersion and let rfJ+ be a continuous symmetric form on V such that rfJ+ - rfJf: is a positive form. If q ~ (n + 2)(n + 5)/2, then the map fo admits a fine C1-approximation by free isometric C2-immersions f: V -+ W for which rfJ/ = rfJ +.

Example. If V = IRn and W = IRq then the theorem implies the existence of C2 _

solutions to the following system of [n(n + 1)/2] + [n(n + 1)(n + 2)(n + 3)/24] P.D. equations in the unknown map f: IRn -+ IRq.

<aJ,ajf) = b;j' 1 :S; i:S;j:S; n,

< aijf, akzf) = rfJ;lk/' 1 :S; i :S; j :S; k :S; l,

where b;j = 0 for i =I j and bu = 1, and where rfJ;lk/ are arbitrary continuous functions on IRn for which the form rfJ(a;, OJ, Ok' 0/) = rfJilk/ is positive at all points v E IRn [compare (B) in 2.4.9].

Proof [compare Nash (1954)]. Let rfJt ~ 0 be a Coo-form on V which equals (dx)4 for some Coo-function x: V -+ IR with the support in a ball V c V. Let us construct a family of free isometric Coo-maps I.: V -+ W for small e > 0, with the following four properties.

(i) I. equals fo outside V for all e > 0; (ii) the maps I. C1-converge to fo for e -+ 0;

(iii) the C2-distance between I. and fo is controlled by the CO-norm of rfJt, with some universal constant const = const(q),

III. - fol12 :S; const IlrfJt 110'

for some fixed norms in the (linearized) space of maps close to fo and in the space of 4-forms on V;

(iv) the forms rfJf: CO-converge to rfJf: + rfJt for e -+ o.

First, we obtain by (A') in 3.1.2 with the implicit function theorem (see 2.3.6) a family of Coo-maps 1.: V x 1R2 -> W with the following three properties

(a) J:IV\U = folV\U for V = V x 0 c V X 1R2 and for all £ > 0; (b) the family 1. is Coo-continuous in £ ~ 0 and 101 V = fo. (c) the induced metric 1.* (h) satisfies on some neighborhood -0 c V X 1R2 of U,

1.* (h) 1 -0 = gE ® dti ® dt~,

for gE = g - £4 dX2 and for all (small) £ > O. Then we put

h(v) = J:(v, £2 sin £-1 x, £-2(1 - cos £-1 x)).

These maps I. are isometric and they obviously satisfy (i)-(iii). We check (iv) by evaluating wI>n the geodesic coordinates u1 , ••• , Un at some point VE V. We have for Vi = VUi and ai = a/aui,

ViI. = VJE + £ai x [(cos £-1 X)VtIJ: - (sin £-1 X)Vt 2 J:],

and

Vijh = vijJ: + (aix)(ajx)z + 0(£),

for Z = -(sin£-l x)VtIJ: - (COS£-lX)Vt 2 J:. Theform WI, at the point v is

<Vijh, Vijh) = <VijJ:, VijJ:) + (aix)(ajx)(akX)(a1X) + 0(£),

as <Z,Z) = 1 and <VijJ:,Z) = 0 by (D") in 3.2.1. Hence, WI: -> WI: + dX4. Next, we use the decomposition wD" = Ljdxj for positive Coo-forms wo+ on V

(see 3.1.4) and thus we obtain maps I. with the properties (i)-(iv) for an arbitrary positive Coo-form wo+ > O. Finally, we break the form ,10 = w+ - WI: into the sum

,10 = wD" + ,11'

where wD" is a positive Coo-form and ,11 is a small positive continuous form. We take the map hI with a small £1 > 0 for a new map fl' For this f1 the error rp - rpJ~ is close to ,11' Then we pass to f2' f3, ... , for which the error W + - rpI: -> 0 for i -> 00, and the property (iii) allows us to go to the limit f = limi~oo.t; which has w/ = w+. Q.E.D.

COO-Immersions with Given Curvature. The isometric immersion relation for maps f: (v, g) -> IRq prescribes the scalar products

(1)

for all vectorfields a1 and a2 in V. This implies

<a1 azf,azf) = ta1 g(a2,a2 )

and so

(1')

Since commuting fields satisfy


the Eqs. (1) express via (1') the scalar products

(2)

by combinations of derivatives of g. Next, for commuting fields 0;, i = 1, ... , 4,

and so we express (3) by combinations of second derivatives of g. This amounts to Gauss' formula tPf- = R(g).

Now, let the curvature tP/ also be given. Then tP+ and tP- determine all of tP = tPf and so all scalar products

(4)

are expressed by tP + and by derivatives of g. Abbreviate this expression to (12,34). Then

(5) 01 (11,22) - 02(11,12) + tal (12,12) = (off,oif).

The scalar products (5) determine with (*) and with a similar expression for 0102 03 (by a combination of cubes of linear combinations of 01 , O2 and ( 3 ) all scalar products

(6)

Furthermore, the derivatives of (2) give us the scalar products

(7)

and so the forms g = f*(h) (for h = L1=1 dx'f in IRq) and tP/ uniquely determine the normal projections of the third derivatives of f to the (second) osculating space T,,2(f) c IRq for all v E V. In particular, if dim T,,2(f) = q for all v E V, then the third derivatives of f become functions of the first and second derivatives and so the system

(8) f*(h) = g, tP; = tP+

is completely integrable (see 2.3.8) in this case. In particular, (8) admits a COO-solution if and only if the forms g and tP + satisfy the compatibility condition which expresses the symmetry

(9)

in terms of g and tP +. This compatibility condition is called the Codazzi equation. Furthermore, Frobenius' theorem [see (F) in 2.3.8J implies the following rigidity of Coo-maps V -+ ~q,

(B). If fl*(h) = f2*(h) and tPf: = tPf:, and if the osculating space of the map fl has dim T,,2(fl) = q for all v E V, then f2 is congruent to fl by a rigid motion of ~q, provided the manifold V is connected.

The condition q = dim T"z(fl ) implies q ~ n(n + 3)/2. But the system (8) contains p equations for

l1(n + 1) n(11 + l)(n + 2)(n + 3) p= 2 + 24 '

and so one may expect a similar rigidity theorem for generic Coo-maps V ~ ~q for q < p. On the other hand, the Theorem (A) shows the complete breakdown of the rigidity for C2-maps f: V ~ ~q, q ;::-: (n + 2)(n + 5)/2.

Now let q ~ qo = (3 - )3)(n2 - 4n)/6. Then the dimension d of the space of quadratic form on (~n)2 = ~n(n+1)/2 of rank r ~ q - n satisfies

d = (q - n)[(n + 1)2 - q] < n2 (n 2 - 1) 2 - 12 .

This suggests that the tensor rl>J of a generic map f: V ~ ~q is uniquely determined by rl>J-' as rl>J depends (at every point v E V) on d parameters, while rl>J- may control n2 (n2 - 1)/12 parameters corresponding to the curvature tensor R(f*(h)). One knows (see Kobayashi-Nomizu) that the curvature tensor R(f*(h)) uniquely determines rl>J for generic maps f: V ~ ~q for q ~ n + (n/3) and so these maps are uniquely determined (up to an isometry of ~q) by the metric f*(h). On the other hand, if d < n2 (n2 - 1)/12, then a generic curvature tensor on V does not come from any map V ~ ~q and so generic COO-manifolds (V, g) admit no isometric C2-immersions into ~q for q < qo [see E. Berger (1981) and Berger-Bryant-Griffiths (1983) for deeper relations between rl>J and R(f*(h))].

Exercises. Show that the above Proposition (B) fails in general to be true for C2-maps.

Give an example of a COO -metric g on V ~ S3, for which the space of isometric COO-immersions f: (V, g) ~ ~4, normalized by f(vo) = 0 and by DJ T"o(V) = Do for a given isometry Do: T"o(V) ~ ~4, is homeomorphic to the Cantor set. Show that no can-metric on V possesses a similar property.

Construct a Coo-metric g on V ~ S3 which admits no C2-isometric immersion into ~4, but such that every unit ball in (V, g) admits an isometric COO-imbedding into ~4. Show that no Can-metric on S3 has this property.

Show that every COO-immersion fo: V ~ ~q, q ~ n + 2, admits a fine Co_ approximation by a Coo-immersionf: V ~ IRq, which admits an infinite dimensional family of COO-deformations h: V ~ ~q, t E T, such that h*(h) = f*(h) and

for all t E T.

Isometric Immersions of Order k;::-: 1 (Allendoerfer 1937; Spivak 1979). Let f: V ~ ~q be a Coo-map. Denote by T"k(f) c Tw(lRq), W = f(v), the kth osculating space which by definition is the span of the derivatives 01 02 ••• oJ(v) for alII ~ k and for all I-tuples of vector fields 01 , ... , 0, in V. The kth order differential DJ maps the symmetric power (T,,(V))k for all VE V to the orthogonal complement T"k(f) e T"k-l(f) by sending every k-vector 01 ® ... ® OkE(T,,(V))k to the normal projection

of the derivative a1 a2 ... ad(v) to T"k(f) e T"k-l(f). Thus, the Euclidean metric in IRq induces a certain quadratic form on (T(V))k, that is a 2k-linear form, called Gk(f) on T(V). For example, G2(f) is our old curvature form (/>J. Denote by gk(f) the symmetrization of Gk(f). Thus,

and

Exercises. Express the form Gk(f) by gk(f) and by derivatives (of order < k) of the forms G1(f) for I < k. Thus the forms gl(f), I = 1, ... , k, uniquely determine the forms G1(f), I = 1, ... , k.

Let dim T"k(f) = q for all v E V. Show that f is uniquely determined (up to isometries of IRq) by the forms g 1 (f), ... , gk(f) [compare (B)].

Now, let gl' I = 1, ... , k, be arbitrary symmetric differential 2/-forms on V. The isometric immersion problem for (V, g 1, ... ,gk) is that of finding a C~-maps

f: V ~ IRq, for a given IX ~ k, such that gl(f) = gl for I = 1, ... , k. If such an f exists, then gl ~ ° for I = 1, ... , k, and if gl > 0, then the map f is necessarily kth-order free.

Exercises. Show that the differential operator (of order k) ft---+(g1 (f), ... , gk(f)) is infinitesimally invertible (see 2.3.1) at all (k + l)1h order free maps f: V ~ IRq and apply the implicit function theorem to these maps f

3.1.6 Extension of Isometric Immersions

We prove in this section the h-principle for extensions of free isometric immersions from a given submanifold Vo c V to a small neighborhood {!}fz Vo c V. First, let Vo be a single point Vo E V and let h' hj' ... denote the covariant derivatives VJ, VFJ, ... of a map f: V ~ (w, h) for fixed local coordinates U i , i = 1, ... , n, in V. We write the isometry condition for f: (V, g) ~ (w, h) as

(1)

and then we differentiate,

(2)

where ak = a/aUk. The Eqs. (2) are equivalent to

(2')

for

We differentiate (2'),

(3)

and then obtain

(4)

( 1"1")_ ijk Ji,Jjk - A ,

( I" 1") _ I" I" _ ijkl ijkl Jil>Jjk (Jik,Jjl) - B + R ,

for

and

Rijkl = <h,(R(fk,fz),fj),

where R denotes the curvature tensor in (w, h).

241

The Eqs. (3) are linear in the third derivatives fjkl and the Eqs. (4) give a complete consistency condition for (3) because the vectors h are linearly independent. Hence, the system (3) is solvable in fjkl(VO) E Two(W), wo = f(vo), if the vectors hj(vo) E

Two(W) satisfy (4). Moreover, the space of solutions of the system (1) + (2) + (3) at Vo E V has the same homotopy type as the space of solutions of (l) + (2) + (4).

The Eqs. (4) are non-linear and their space of solutions (in a fixed tangent space Two(W)) may be quite complicated. However, the space of free solutions, that are defined as p-tuples, p = n + [n(n + 1)/2J, of linearly independent vectors hand hj in Two(W) which satisfy (1) + (2) + (4), has a fairly simple structure.

(A) Lemma. The space of free solutions of (1) + (2) + (4) at every point wo = f(vo) is homotopy equivalent to the Stiefel manifold Stp(Two(W)) = Stp IRq. Hence, the space of free solutions of (1) + (2) + (3) is also homotopy equivalent to Stp IRq.

Proof The equations in question are invariant under coordinate changes in V and so one may use geodesic coordinates at Vo for which Aijk(VO) = O. Then the vectors hand hj become orthogonal in Two(W) and we must only determine [n(n + 1)/2]tuples of independent vectors hj E IRq-n which satisfy (4). These tuples are given by the injective linear maps F of the symmetric square (IRnf = lR[n(n+1)/21 into IRq-n such that the 1/>- -part of the induced form I/>F on IRn is given by Bijkl + Rijkl (see 3.1.5). If I/> = I/>F is positive definite (as a quadratic form) on (IRn)2, then the space of isometric maps [(lRn)2,I/>J --+ W equals St p- n IRq-n. On the other hand, those forms if> which are positive definite on (IRnf and which satisfy 1/>- = R for a given R, form a nonempty convex subset in the Euclidean space of all 4-forms if>. This subset is nonempty, since 1/>- + AI/>+ is positive on (IRn)2 for every positive symmetric 4-form 1/>+ on IRn and for all sufficiently large A ::::: Ao = Ao(I/>-). Hence, the space of solutions of (4) is homotopy equivalent to Stp- n IRq-n and the lemma follows.

(A') Corollary. The space offree isometric Coo-immersion lDjl(vo) --+ W(compare 1.1.5) is weakly homotopy equivalent to the Stiefel bundle Stp(W) of p-frames Stp T,,(W), VE V, for p = n + [n(n + 1)/2].

Proof If a smooth map f: lDjl(vo) --+ W satisfies (1) + (2) + (3) at Vo E V, then f is infinitesimally isometric of second order at Vo, that is Jj*(h)(VO) = J;(vo). By the implicit function theorem (see 2.3.2) a small perturbation of f is isometric on lDjl(vo). Q.E.D.

Now, let e: Vo 4 V be an arbitrary submanifold of codimension ko in V and let f: Vo --+ W be a free isometric immersion. Let Ne --+ Vo denote the normal bundle

T(V) e T(Vo) and let T(W)I VO stand for the induced bundle f*(T(W)) ~ Vo. We study homomorphisms X: Ne ~ T(W)I VO over small (coordinate) neighborhoods Uo c Vo by taking frames of independent fields ai: Uo ~ Ne , i = 1, ... , ko, and then by expressing XIUo = {Xi} for the fields Xi = X(a;) along Uo (compare 3.1.2). Consider a smooth map j: (!)It Vo ~ W which extends fo = jl Vo and let X = Djl Vo. If the differential Dj: T(V) ~ T(W) is isometric on T(V)I Vo then X is an isometric homomorphism of Ne to the normal bundle Nf = T(W) e T(Vo). This is expressed by the equations

(5)

(6)

where f/l denote the first covariant derivatives r/lf = 0(a/au/l) in (w, h) for some local coordinates u/l in Vo, jJ. = 1, ... , n - ko. If the map f is infinitesimally isometric of the first order along Vo [i.e. J} 1 Vo = Ji 1 Vo for 9 = j*(h)], then the Eqs. (2') above imply

(7)

for 1 ::;; jJ., v ::;; n - ko, i = 1, ... , ko. The Eqs. (7) admit the following invariant description. The bundle T(W)I Vo splits into the orthogonal sum,

T(W)I Vo = T(Vo) EB (T(VO))2 EB BNf ,

where the tangent bundle T(Vo) is isometrically imbedded into T(W)I Vo by the differential Df and the symmetric square (T(VO))2 is imbedded into T(W)I V by the second differential DJ, while BNf denotes the binormal bundle of f: V ~ W (compare 3.1.2). Denote by pl and p2 the normal projections of T(W)I Vo on the subbundles T(Vo) and (T(VO))2 respectively. Then the Eqs. (6) say

(6')

and the Eqs. (7) become

(7')

p10X = 0

p2 0 X = A,

for the homomorphism A: Ne ~ (T(VO))2 which is defined by Ai/lV, such that <Aai,f/lv) = Ai/lv. Consider the adjoint homomorphism A': (T(VO))2 ~ Ne , for which

<Aa,(5) = <a,A'(5)

for the scalar products induced by the imbeddings DJ: (T(Vo)f ~ T(W)lVo and X = DjlNe ~ T(W)I Vo from the scalar product in (w, h). Since the covariant derivatives in V equal normal projections to V of covariant derivatives in W, the homomorphism A' equals the second differential D;: (T(VoW ~ Ne of the imbedding e: Vo c. V. The homomorphism A' also equals the normal projection of (T(Vo)f c

T(W)I Vo to Df T(Vo) c T(W)I Vo, and so A' is a short homomorphism. This means the positive semi definiteness ofthe form cPf - cPe which is viewed here as a quadratic form on (T(V))2 (compare 3.1.5). Hence, the inequality cPf ~ cPe between the curvatures of the maps f and e gives us a necessary condition for the existence of an isometric extension j: V ~ W. Furthermore, if the map j is free, then the subbundles T(V)

and T2(f) = T(Vo) EB (T(VO))2 are linearly independent in T(W) lVo, and so the homomorphism A' is strictly short, which amounts to the strict inequality (/Jf > (/Je'

The fields Xi = VJ, for infinitesimally isometric maps j: {!}ft Vo -+ W, also satisfy the following d!fferential equations on Vo

(8)

as Eqs. (2') show. These Eqs. (8) say that the connection induced by the homomorphism X: Ne -+ T(W)I Vo equals the normal connection of the subbundle Ne C

T(V)I Yo. Observe that for a free map j the homomorphism X is regular (compare 2.2.6 and 3.1.2). This amounts to linear independence of the vectors p 3 (J7JtX), 1 :-:;; fJ :-:;; n - ko, 1 :-:;; i :-:;; ko where p 3 denotes the normal projection of T(W)I Vo onto the binormal bundle ENf C T(W)I Yo. The regularity of X is clearly sufficient for the strict inequality (/Jf > (/Je'

(B) Lemma. If q z mo + [mo(mo + 1)/2] + ko(mo + 2) for mo = n - ko = dim Yo, then regular homomorphisms X: Ne -+ T(W)lVo, which satisfy (5) + (6) + (7) + (8), abide by the h-principle.

Proof If the submanifold V is totally geodesic then the Eqs. (7) become

<J"v,Xi ) = AiJtv = 0,

and so the homomorphisms X in question are regular isometric homomorphisms Ne -+ ENf which induce a given connection in Ne • The h-principle for these is proven in 2.2.6. In order to apply the argument in 2.2.6 to Ai/lv =I- 0, we must check the microflexibility of the sheaf of regular solutions of (5) + (6) + (7) + (8). We assume as in 2.2.6 the fields Vi: Vo -+ Ne to be orthonormal and then we bring together the equations which relate the field X, to Xi for i < 1 and for a given 1 :-:;; ko,

(9)

<X"X;) = 0

\Xe,X,) = 1

<X"i/l> = 0

<X,J/lv) = Ai/lv

<X" V/lX;) = A'/li

<V/lX" X;) = Ai/ll

All these equations but the last one are algebraic in X" Furthermore,

<V/lX"X;) = V/l<X"X;) - <X"V/lXi) = <X,V/lX;),

and so the last equation is redundant. If by induction we assume the homomorphism X to be regular on Span{v1, ... ,V,} eNe, then the vector Xi' f/l' f/lv, V/lXi, for 1 :-:;; i :-:;; 1 - 1, fJ = 1, ... , n - ko, are linearly independent and so the solutions X, of (9) for which the homomorphism X is regular on Span {Xl' ... ,X,} form a ddimensional sphere over every point v E Vo for d = q - 1 - mo - [mo(mo + 1)/2] -(1 - 1)mo. The solutions X, now become regular sections of the resulting sphere bundle and the argument of 2.2.6 applies.


(B') Corollary. If fP, > fPe and q :2: mo + [mo(mo + 1)/2] + ko(mo + 2), then the space of regular soutions X = {Xl'"'' Xk} of (5) + (6) + (7) + (8) is weakly homotopy equivalent to the space of all injective homomorphisms of the bundle Ne $ Ne ® T(Vo) to the binormal bundle BN,.

Proof As fP, > fPe' the homomorphism A': (T(VO»2 --+ Ne is strictly short and so the adjoint homomorphism A: Ne --+ (T(VO»2 c T(W)I Vo is also strictly short, which means the positive definiteness of the form g' = g - A *(h) on Ne, where A *(h) is the form induced by A.

Since p l 0 X = 0 and p2 0 X = A by (6') + (7'), the homomorphism X is uniquely determined by Y = p3 0 X: Ne --+ BN,. The Eqs. (5) are equivalent to

(10)

and the Eqs. (8) become

(11)

(Yi, lJ) = gij,

where the following notations are adopted. The derivative V' is the covariant derivative in the bundle BN" which means V~ lJ = p3 0 VlllJ, and so

(Yi,V~lJ) = (Yi,VlllJ),

where Yi stands for Y(Oi) for a given frame of sections 0i: Vo --+ Ne , i = 1, ... , ko. Then we have with X? = A(Oi) = Xi - Yi,

Cil\i = Aillj - (X?,VIlXjO) + (VIlX?, lJ) - (VIlXjO, Yi).

It is now clear (compare 2.2.6) that the space of the 1-jets of regular homomorphisms Ne --+ BN, which satisfy (10) + (11) is fiberwise homotopy equivalent to the Stiefel bundle Str(BNf ) for r = ko(mo + 1). Q.E.D.

Second Derivatives Xij = Vij! Let {UIl,Ui}, 1::::;; Jl.::::;; mo = n - ko, 1::::;; i::::;; ko, be local coordinates in V, such that ull extend local coordinates in Va' Ui are constant on Vo and the fields 0i = (O/OUi) I Vo are normal to Va. If the extension j of f is infinitesimally isometric of second order along Va' then the fields Xij = Vill Vo satisfy the following algebraic equations according to formulae (2') and (4).

(12)

(13)

( X .. J. ) = Allij IJ' Il

for 1 ::::;; i,j, k ::::;; ko, and 1 ::::;; Jl. ::::;; mo = n - ko and

(14)

(15)

(16)

(XijJIlV) = DVllij + (VvXi' VIlXj)

(Xij' VIlXk) - (Xjk, VIlXi) = Dijllk

(Xij,Xkl ) - (Xi/,Xkj ) = Dik/j,

where 1 ::::;; i,j, k, I ::::;; ko, 1 ::::;; Jl., v ::::;; mo. and Dabcd stands for nabcd + Rabcd•

Let the map f be free and the homomorphism X = {Xi} be regular. Then the fields f ll• fllv, Xi and VIlXi a span certain subbundle T2(J,X) c T(w)1 Vo of dimension mo + [mo(mo + 1)/2] + ko + moko. Denote by BN(J, X) the orthogonal com-

plement T(W) e T2(f, X) and orthogonally split Xij = X;j + X;j for X;j: Vo ~ T 2(f,X) and X;j: Vo ~ BN(f, X). If we substitute X;j + X;j for Xij in the Eqs. (12)-(15), then we obtain similar equations with X;j in place of Xij'

(12')

(15')

<x. J. ) = Allij '1' JJ

while the Eqs. (16) become

(16') <X~ X'; > - <X~' X".) = Eiklj lJ' kl ,I, kJ '

where

Eiklj = Diklj + <X~ X'.) - <X~. X' > ,I, kJ lJ' kl •

The Eqs. (12')-(15') form a non-singular system of linear equations in X;j and so the solutions X;j of these equations are just sections of an affine subbundle in T 2(f,X).

The Eqs. (16') are quadratic in X;j. However, free solutions X;j of (16') have very simple structure [compare Lemma (A)], where the freedom means linear independence of X;j which is equivalent to the freedom of the map j Namely, for every point VE VO, and for every E(v) = Eiklj(V), where Eiklj(V) is a function if fiv), fIlJv), Xi(V) and VIlXi(V), the space of free solutions of (16') in the fiber BN(f, X)I V is a smooth manifold, say G = G(E(v)), which is homotopy equivalent to the Stiefel manifold of [ko(ko + 1)/2]-tuples of independent vectors in BN(f, X)lv. Furthermore, this manifold continuously depends upon v and E(v) such that the union U G(E(v)) over all v E Vo and all possible values of E(v) form a fibration over the space of pairs (v, E(v)). This analysis and Lemma (B) give us the following description of systems of vector fields Xi and Xij along V, where Xi satisfy (5) + (6) + (7) + (8) and the regularity condition, while the fields Xij satisfy (12) + (13) + (14) + (15) + (16) and the freedom condition (i.e. the vectors X;j are independent).

(C) Lemma. The above systems of fields satisfy the h-principle. This h-principle shows that the space of these systems (Xi' Xij) is weakly homotopy equivalent to the space of injective homomorphisms

Third derivatives X ijk = VJij' An extension] of f is infinitesimally isometric of second order along Vo if and only if the first derivatives VJ = Xi satisfy the Eqs. (5)-(8), the second derivatives Vij] = Xij satisfy (12)-(16) and the third derivatives Xijk = VkVij] satisfy [see (3) and the following discussion]

<Xijk,fll) = Ok Allij - <Xij' VIlXk ) (17)

_;'l lij <Xijk,XI ) - UkA - <Xij,Xkl ),

for 1 ~ i,j, k, I ~ ko, 1 ~ J1 ~ mo = n - ko. Since these equations are linear in Xijk and since the Eqs. (14)-(16) give us the complete consistency condition for (17), these equations (17) are solvable in some vector fields X ijk along Vo and the space of these solutions is contractible. Hence, Lemma (C) yields the following


(C') Corollary. Free extensions j: {r}fe Vo -+ W, for jl Vo = J, which are infinitesimally isometric of second order along Vo satisfy the h-principle. The space of these extensions has the weak homotopy type of the space of the above injective homomorphisms f

Now, we apply the implicit function theorem [compare (A')] and obtain the following solution to the free isometric extension problem.

(D) Theorem. Let V and W be Riemannian Coo-manifolds, let e: Vo 4 V be a c oo _ submanifold and let f: Vo -+ W be a free isometric Coo -immersion. Then free isometric Coo-immersions {r}feVo -+ W (for an "infinitely small" neighborhood (r}feVo c V, see 1.4.1) which extend f = jl Vo satisfy the h-principle. Such an extension j exists if and only if the following two conditions are satisfied.

(i) the (relative) curvature of the immersion f is strictly greater than the curvature of e: Vo 4 V, that is the form f/Jf - f/Je is positive definite on the symmetric square (T(Vo)f;

(ii) there exists an injective homomorphism

J: Ne EB (Ne ® T(Vo» EB N; -+ BNf ,

where Ne is the normal bundle of e and BNf is the binormal bundle of f

(D') Remark. The above considerations show that the restriction map jl-+ f = jl Vo is a Serre fibration of the space of free isometric Coo-immersion over the space of those free COO-immersions f: Vo -+ W which satisfy the condition (i).

Exercises. Extend the Theorem (D) to Can-immersions. Let the manifolds V and W be parallelizable and let Vo c V be a totally geodesic

submanifold. Show that every free isometric Coo-immersion f: Vo -+ V extends to a free isometric COO -immersion j: {r}fe Vo -+ W, for q = dim W ;;::: n(n + 3)/2, n = dim V.

Let V and W be Riemannian can-manifolds, let e: Vo 4 V be a l-codimensional Can-submanifold and let f: Vo -+ W be a free isometric Can-immersion for which CPf > CPe· Show that f extends to a (not necessarily free) isometric can-immersion j: {r}fe Vo -+ W if and only if there exists an injective homomorphism Ne -+ BNf .

Hint. Use the Cauchy-Kovalevskaya theorem in place of the implicit function theorem.

The h-Principle for Isometric Immersions {r}feVo -+ W The results in 2.1.2 and the proof of the Theorem (A) of 3.1.5 show that an arbitrary strictly short map fo: Vo -+ W admits a CO-approximation by isometric Coo-immersions f: Vo -+ W with an arbitrarily large curvature form f/J;, provided dim W = q ~ (mo + 2)(mo + 5)/2 for mo = n - ko = dim Yo' If the form f/J; is large, then the quadratic form f/Jf on (T(V»2 also is large and for f/Jf> f/Je the Theorem (D) allows one to extend the immersion f to {r}fe Vo c V. Moreover, we obtain with the Remark (D') the hprinciple for free isometric Coo -immersions {r}fe Vo -+ W relative to strictly short maps for q ~ (mo + 2)(mo + 5)/2. This inequality is automatically satisfied for ko = codim Vo ~ 2 and so we have the following

(E) Theorem. Let Vo c V be a C'X) -submanifold of co dimension ko :?: 2 in an ndimensional Riemannian Coo-manifold V and let fo: Vo --+ W be a strictly short map, where W is a Riemannian Coo-manifold of an arbitrary dimension q. Then the following condition (i) is necessary and sufficient for the existence of a free isometric Coo-map f: (!Jfz Vo --+ W whose restriction fl Vo is arbitrarily CO-close to fo.

(i) There exists an injective homomorphism of the bundle T(V) EB (T(V))ZlVo to the induced bundle fo*(T(W)).

(E') Corollary. If the manifold V is parallelizable, then some neighborhood U c Vof an arbitrary Coo-submanifold Vo c V of codim Vo = ko :?: 2 admits a free isometric Coo-immersion f: U --+ [Rq for q = n(n + 3)/2 where n = dim V.

Observe that no neighborhood U c V admits a free map U --+ [Rq for q < n(n + 3)/2.

Exercises. Extend (E) and (E') to the case ko = 1 and q :?: (n + I)(n + 4)/2. Let the manifolds V, Wand e: Vo c; V be real analytic and parallelizable, let

ko = codim Vo :?: 3 and let the normal bundle Ne --+ Vo admit a I-dimensional sub bundle (e.g. ko > mo = dim Yo). Show for q :?: n(n + 1)/2 the existence of an isometric (possibly non free) can-immersion of some neighborhood U c V of Vo into W Then assume ko = 2 and prove the existence of such an immersion U --+ W for q :?: n(n + 3)/2.

3.1.7 Isometric Immersions vn --+ W q for q ~ (n + 2)(n + 3)/2

Let V = (V, g) and W = (W,h) be Riemannian Coo-manifolds. We use special coordinates u1 , Uz, ... , Un in V, where Uz , u3 , ... , Un are global cyclic coordinates of some Coo-imbedded normally oriented torus T n- 1 c; V and where U 1 is the normal coordinate in some tubular neighborhood U c V of T n - 1, such that U 1 = 0 is the equation for T n - 1 cUe V. Then we use coordinates {u 1 , ... , Un' t, B} in V x [R2

for the Euclidean coordinates {t, B} in [Rz.

Let F: V x [Rz --+ Wbe a Coo-immersion and let x = x(u 1 ) be a Coo-function with compact support in the real variable u1 • Then we consider the functions t(u 1 ) = 8 sin 8-1 x and B(u 1 ) = 8(1 - cos 8-1 x) for 8> 0 and we define the following Coo-map f = Ie: V --+ W for all 8 > 0,

for VE U,

and

f(v) = F(v,O) for VE V\U.

A straightforward computation gives the following formulae for the covariant derivative of f in the Riemannian manifold W = (W, h).

(1)

for 2:s;; i,j :S;;n;


(2)

where x' = dX(Ul}/du1 and yl = cVtF + sVoF, for c = cose-1x and s = sine-1x;

(3)

for 2 ~ i ~ n;

(4)

where

y2 = -sVtF + cVoF

and

y3 = 2(cVltF + sVloF) + x'(c2VttF + 2csVtoF + s2VooF).

Here Vlt = VIII Vt and so on. If the map

F: (V x ~2, g EEl dt2 EEl t02) -+ (w, h)

is infinitesimally isometric of infinite order along V = V x 0 c V X ~2, then the induced metric J.*(h) Coo-converges to g + (dX)2 for e -+ O. Moreover the jet of the difference satisfies for e -+ 0

for all k, r = 1,2, .... Indeed, the map xl-+(esine-1x,e(1 - cose-1x) isometrically sends (~, dx2) into (~2, dt2 + d02), and so the argument of 3.1.2 applies.

Now, we test the freedom of the maps f = I., e -+ 0, for possibly non free maps F. We assume the restriction FI V to be a free map V -+ Wand we denote by BN c T(W)I V the binormal bundle of the map FI V. Put

Y(a) = (cosa)VtF + (sina)VoF,

for all a E [0, 2n)], and let Zi(a), i = 2, ... , n, denote the normal projections of the covariant derivatives Vi Y(a) I U to BN for the coordinates domain U c V.

(A) Lemma. If the map F is infinitesimally isometric of order r ~ 1 along U = U x o c V X ~2, and if the fields Zi = Zi(a), i = 2, ... , n, are linearly independent for every a E [0, 2n], then the map f = I.: V -+ W is free for all sufficiently small e > O.

Proof The Eqs. (1)-(4) in 3.1.2 show that the fields Y1IU and Y21U are mutually orthogonal, have unit norm and that they are orthogonal to the fields ViFI U, VijFI U and Vk yl for 1 ~ i,j ~ nand 2 ~ k ~ n. Since the fields ViFI V and VijFI V are linearly independent, since the binormal projections Zk of Vk yl also are independent and since the field y3 remains bounded for e -+ 0, the following n + [n(n + 1)/2] fields also are independent along U for all small e > 0,

ViF, VijF, vij + X'ViYl'

V1F + x'y\ V11 F + x"yl + e-1(x')2y2 + x'y3 for 2 ~ i,j ~ n,

and the lemma follows.


Remark. The above argument shows the maps I. to be uniformly free for e -+ 0, which means the uniform bound from below for the norm of the Hessian of the derivatives ViI. and VijJ., 1 :$; i, j :$; n.

This remark and the relation (*) allows one to apply the maps I. the implicite function theorem for operators of polynomial growth (see 2.3.6) and thus to obtain the following generalization of the adding dx2 lemma [see (B') in 3.1.2].

(B) Lemma. Let the map F be infinitesimally isometric of infinite order along V and let the fields Zi' i = 2, ... , n, be independent. Then there exists a small Coo -perturbation fl of the map I., for any sufficiently smalle > 0, such that j1(h) = 9 + (dx)2.

Let us show that the assumptions of this lemma can be met for q ~ (n + 2)(n + 3)/2. Consider a free isometric Coo-immersion fo: (V, g) -+ (w, h) and let the special coordinate domain T n- l x IR ~ U c: V lie in some topological ball in V.

(C) Lemma. If q ~ (n + 2)(n + 3)/2, then there exists a Coo-maps F: V x 1R2 -+ W which satisfies the assumptions of (B) and for which FI V = fo.

Proof First we construct an orthonormal 2-frame of binormal fields Xl and X2 along U c: V -+ W, such that

(i) (ViXk,X1) = 0,1 :$; i:$; n, I:$; k,l:$; 2, (ii) the field Xl is regular,

(iii) the frame (Xl' X2) is semi regular (see 2.2.6) along the hypersurface Ul = const in U for all const E IR. This means the linear independence of the orthogonal projections of the derivatives Vi(CXl + sX2), i = 2, ... , n, to the binormal bundleBN = BNfo c: T(W) IV for every pair of constants (c, s), c2 + S2 = 1. The existence of these fields Xl and X2, which are sections U -+ BN, follows for dimBN = q - [n(n + 3)/2] ~ n + 3, from the h-principle for semi regular bundle homomorphisms (see 2.2.6).

Next we construct (compare 3.1.2) a free infinitesimally isometric (along U) map Fl: (U x lR,g EB dt2) -+ W, for which Fli U = fol U and VtFll U = Xl' such that the field VttFll U is normal to X2.

Finally, we extend Fl to the desired infinitesimally isometric map F: (U x 1R2,g EB dt2 EB dt2 EB dlP) -+ W with VeFI U = X2. Q.E.D.

Now, we are able to produce isometric immersions V -+ W with the best known lower bound on q = dim W.

(D) Theorem. Let fo: V -+ W be a strictly short map between Riemannian coo_ manifolds of dimensions nand q such that q ~ (n + 2)(n + 3)/2. Then fo admits a fine CO-approximation by free isometric Coo-maps f: V -+ W.

Proof We only need to show (compare 3.1.2) that every Riemannian Coo-metric on V decomposes into a locally finite sum Li dx], where every function Xj is of the form

Xj = xiud for some special coordinate system in a small domain U = ~ ~ T n- 1 x IR in V. If V is compact, then the existence of such a (finite) decomposition directly follows from Theorem (A) in 3.1.3. The compact decomposition [see (B) in 3.1.1] reduces the non-compact case to the compact one. Q.E.D.

Exercises. Extend the C 1-approximation theorem (G) in 3.1.2, the parametric hprinciple (H) in 3.1.2 and the curvature theorem (A) in 3.1.5 to the dimensions q in the interval (n + 2)(n + 3)/2 :s;; q < (n + 2)(n + 5)/2.

Generalize the above to eX-immersions of eX-manifolds for a ~ 100.

3.1.8 Isometric Cylinders vn x IR --+ W q for q ~ (n + 2) (n + 3)/2

If q ~ (n + 3)(n + 4)/2, then the above Theorem (D) gives us a complete hold of free isometric immersions (V x IR, g EB dt2 ) --+ (w, h). We shall see presently how the proof of that theorem yields such an immersion for q ~ (n + 2)(n + 3)/2for compact (possibly with boundary) manifolds V.

Consider a COO-immersion F: (V x 1R2,g + dt2 + d(}2) --+ (w, h) which is infinitesimally isometric of first order along V and whose restriction FI V: V --+ W is free. Then the differential DF maps the normal bundle N = V X 1R2 --+ V of V =

V x 0 c V X 1R2 into the binormal bundle EN c T(W)I V of the map FI V by sending the fields a/at and alae in V x 1R2 to the orthonormal fields VtF and VeF along V.

Consider the circle S. c 1R2 given by t2 + (}2 = 82 and concentrate on the map FlY x S.: V x S. --+ W.

(A) Lemma. If the manifold V is compact and if the homomorphism DF : N --> EN is semi regular (see 2.2.6), then the map FI V x S. is free for all sufficiently small 8> O. Moreover the maps FI V x S. are uniformly free for 8 --+ O.

Proof The semi regularity assumption amounts to linear independence of the binormal components ofthe vectors Vu,(cVtF + sVoF), i = 1, ... , n, along V for every pair (c, s) E 1R2, c2 + S2 #- 0 and for every system oflocal coordinates u1 , ••• , Un in V. We parametrize S. c 1R2 by t = 8 sin 8-1 x and () = 8 cos 8-1 x for x E IR and we have the following formulae for the covariant derivatives of the map!. = FI V x s.

for 1 :s;; i,j :::;; n, and

Vi!. = ViF

Vij!. = VijF,

Vx!. = cVtF - sVeF

Vix!. = Vi(cVtF - sVoF)

Vxx!. = - 8-1 (sVtF + cVoF) + c2 VttF - 2csVto F + S2 Vee F,

where s = sin 8-1 x and c = cos 8-1 x. These derivatives are independent for 8 --+ 0 [compare (A) in 3.1.7] and the lemma is proven.


If the map F is infinitesimally isometric of infinite order along V, then there is a small perturbation of the map 1. which is a free isometric map

1.': (V x S;, 9 Ef) dx2 ) ~ (w, h)

by the implicit function theorem (compare 3.1.7). We can even find a perturbation, such thatJ.'1Y x So = FlY x 0, for a given point So E Sl. This is done by translating Se1 C 1R2 to the circle Se1(so) around So and then by perturbing the map FlY x Se1(so) without changing it on V x 0 c V X Se1(SO).

(A') Corollary. Let V be a compact parallelizable manifold and let f: V ~ W be a free isometric COO-immersion for which the induced bundle f*(T(W)) is trivial. If q ~ [(n + 2)(n + 3)/2] + 1, then there exists a free isometric COO-immersion 1.: V X

Se1 ~ W for aile > 0, such that 1.1 V x So = f for a given point So E Se1•

Proof The binormal bundle of f satisfies

BNJ = f*(T(W)) e [T(V) Ef) (T(V)f],

and so BNJ is trivial in our case. Furthermore

. n(n + 3) dIm BNJ = q - 2 ~ n + 4,

and so there is a semi regular homomorphism N = V X 1R2 ~ BNJ as in (B') of 2.2.6. This homomorphism extends to an infinitesimally isometric map F as in 3.1. 7. Thus, we obtain the required immersion V x S; ~ W for short circles Se1 and by taking cyclic coverings of V X Se1 we make the length of the circle any number we want. Q.E.D.

(A") Example. If (V, g) is isometric to a flat torus, then V x Sl is also a flat torus and so we obtain by applying (A') and the Theorem (D) of 3.1.7 a free isometric COO-immersion of the standard flat torus yn+l into any given Coo-manifold W, provided dim W ~ [(n + 2)(n + 3)/2] + 1. The universal covering of T n+1 gives us a free isometric immersion IRn+l ~ W.

(A"') Remark. If the manifolds V and Wand the map f are real analytic, then by the Cauchy-Kovalevskaya theorem there exists an isometric (non-free) map of a small neighborhood @ftV c V X 1R2 into W for all q ~ (n + 2)(n + 3)/2 [see (E') in 3.1.2] and thus we obtain (non-free) isometric Can-maps IRn+1 ~ W for dim W =

(n + 2)(n + 3)/2.

Exercise. Prove the above result for Coo-manifolds and maps.

Hint. See 2.3.6 and 3.1.9.

Cylinders Between Isometric Immersions. We consider two homotopic isometric COO-immersions fo and fl: V ~ Wand ask ourselves if there exists an isometric COO-cylinder between fo and fl, which by definition is an isometric Coo-map 1: (V x [0,1], 9 Ef) dt2 ) ~ W, for some 1 E (0, CIJ), such that 11 V x 0 = fo and 11 V x 1 =

fl. An obvious obstruction for transforming a given homotopy between fo and fl to a cylinder may come from the "longness" ofthe homotopy and so we assume the existence of a strictly short homotopy !t, t E [0, 1 J, between fo and fl for which the map!t is strictly short for ° < t < 1. Next we assume the maps fo and fl to be free. If q ~ (n + 2)(n + 3)/2, then the techniques of the previous section allow us to stretch a strictly short homotopy to a free isometric COO-homotopy. This is also denoted by f, but now the map f: V x [0, 1J -+ W is COO-smooth and !t = flY x t: V -+ W is a free isometric immersion for all t E [0,1]. Finally, let the manifold V be compact and parallelizable and assume the induced bundle fo*(T(W)) to be trivia1.

(B) Theorem. If q ~ [(n + 2)(n + 3)/2J + 1, then the maps fo and fl can be joined by a free isometric COO-cylinder V x [O,IJ -+ w. If q = (n + 2)(n + 3)/2, then a (possibly non-free) isometric COO-cylinder exists, pr.()vided the Riemannian manifolds are real analytic (with no analyticity assumption on the maps fo and fl).

Proof Take the product V x Sl for Sl = Sf C 1R2 given by t2 + (P = 1 and apply (an obvious generalization of) the proof of (A') to the free isometric maps !t, t E [0, 1]. Thus, we obtain a Coo-family of free isometric maps f,: V x Sl -+ W, such that io I V x ° = fo and it I V x ° = fl' where the circle Sl is parametrized by [0,2n]. Since free isometric maps V x Sl -+ Ware microflexible (see 1.4.2 and 2.3.2), there is a small perturbation it' of it such that

(i) the maps it': V x Sl-+ Ware free isometric COO-immersions for all tE[0,1J and it' = it for t = ° and t = 1;

(ii) there exist some points ° = to < ... < ti < ... < tk = 1 and some points Si and s; in Sl, such that

i = 1, ... , k,

for s~all neighborhoods (9p(V x Si) c V x Sl. Thus we obtain an isometric map f: V x S -+ W for the non-Hausdorff manifold S which is obtained from k + 1 copies of Sl by gluing together the small intervals (9P(Si-l) c Sl = Sl-l and (9p(si) c Sl = Sl, i = 0, ... , k. There is an obvious isometric Coo-map of some interval [O,IJ into Swhichjoins the points OES1 = SJ c SandOES1 = - - '" Sf c S, say cp: [0, IJ -+ S. Then the composed map f 0 (Id x cp): V x [0, IJ -+ W is the required cylinder.

The case q = (n + 2) (n + 3)/2. Suppose for a while that the maps fo and fl are real analytic. Then the maps !t can be made real analytic for all t E [0, 1 J and next they extend to a family of isometric Can-maps Fr: V x D2 -+ W, for a small disk D2 =

D; C 1R2 = {t, e} around the origin, such that the map Ft is free on V x D n (lRl x 0) for all tE [0,1]. Thus, one obtains a family of isometric maps f,: V x Sl -+ W, such that the map itl (9fi(V x 0) is free for all t E [0,1]. This partial freedom is sufficient for the above microflexibility argument to apply and the case of analytic maps 10 and fl is concluded. In fact, this argument provides a COO-cylinder 1: V x [O,/J -+ W, such that the restrictions 11 V x [0,6J and 11 V x [I - 6,/J for small 6 > ° are given free isometric Can-maps, say 10 and 1t: V x [0,6J -+ W, where the interval [I - 6,/J is identified with [0,6J. (The maps 10 and 1t are, of course, not


quite arbitrary. They are assumed to extend 10 and II respectively and they must be homotopic, which means the existence of a family of free isometric maps 1r: V x [0, e] ---+ W, t E [0, 1], between 10 and Jl')

Now, let the maps 10 and II be COO-smooth. Then they can be extended to free isometric Coo-maps Jo and It of V x [0, e] ---+ W Since the Riemannian manifolds V and Ware real analytic, one can make these extensions real analytic outside V = V x ° c V x [0, e] by the implicit function theorem (see 2.3.2 and 2.3.6). Then the Can-maps Jol V x [e/2, e] and It I V x [e/2, e] are joined by an isometric cylinder and thus the maps 10 and II are also joined by a cylinder. Q.E.D.

Example. Let II and 12 be free isometric immersions SI ---+ IRq. If q 2 6, then there is an isometric COO -cylinder between II and 12 which can be assumed free for q 2 7.

Question. Does an isometric cylinder between 10 and II exist for q = 4 and q = 5?

Exercises. Prove for can-manifolds V and W that every free isometric COO-cylinder between can-maps 10 and II: V ---+ W admits a COO -approximation by an isometric can-cylinder between 10 and II'

Hint. See 2.3.6.

Let V = (V, g) be a compact parallelizable Riemannian Coo-manifold and let 10: V x SI ---+ (w, h) be a strictly short map, such that the induced bundle Io*(T(W)) is trivial. Prove for dim W 2 [(n + 2)(n + 3)/2] + 1 the existence of an approximation of 10 by a free Coo-map I: V x SI ---+ (w, h), for which the induced metric I*(h) equals g EB A dx 2 , where the (large) positive constant A may depend on the required precision of the approximation.

Stretching a Cylinder. Let I: V x [0, I] ---+ W be an isometric cylinder. Suppose that the map II V x [a, b], for some subinterval [a, b] c [0, I], extends to an isometric map 1: V x [a, b] x [0, e] ---+ W for some e > 0. Show the existence of an arbitrary long isometric cylinder V x [0,/ 1 ] ---+ W for an 11 2 I which equals I near the ends of the cylinder. Study the freedom of this long cylinder. (An interesting case is when I is free but J may be non-free). Study extensions J which are only infinitesimally isometric (along V x [a,b]) and give a sufficient condition for the existence of a stretching for q 2 (n + 2)(n + 3)/2, n = dim V.

Folded Cylinders. A Iolded COO-cylinder is a continuous map I: V x [0, I] ---+ W with the following property. There is a subdivision of [0, I] into closed subintervals

[li-l,l;] for i = 0, ... ,k and 10 = 0, lk = I,

such that the map I is COO-smooth on each segment [Ii-I' 1;] and

I(v, Ii - b) = I(v, Ii + b),

for all v E V, i = 1, ... , k - 1, and for all sufficiently small 15 > 0. Let It: V x [0, e] ---+ W, t E [0, 1], be a Coo -family of free isometric Coo -maps.

Prove, in case V is compact, the existence of a folded COO-cylinder J between 10 IV x 0 and II: IV x 0, which is free at every segment [L1, I;] where J is coo_ smooth.

Urifolding the Folds. Let f: V x [0,1] -+ W be a folded COO-cylinder and suppose fl V x [Ii, Ii + 15] admits an extension to an isometric Coo-map

j: V x [Ii, Ii + 15] x [0,8] -+ W,

for all folding points Ii C [0,1], and for some positive 15 and 8. Prove the existence of an isometric COO-cylinder (without any folds) betweenflV x ° andflV x 1. Study the freedom of this smooth cylinder and analyse the possibilities of infinitesimally isometric extensions j Give an alternative proof of the Theorem (B) by using folded (and then unfolded) cylinders.

Let (T 2,g) be a torus with an arbitrary flat metric and let fo: yn -+ (w, h) be an arbitrary continuous map such that the induced bundle fo*(T(W)) is trivial. Prove for a can-manifold W of dimension q ;::: (n + 1)(n + 2)/2 the existence of a Coapproximation of fo by a Coo-map f: Tn -+ W for which the induced metric f*(h) equals Ag for some constant A = A(f) > 0. Show for q ;::: [(n + 1)(n + 2)/2] + 1 the existence of a similar free approximation f

Consider a manifold V = Vo x [0, 1] with an arbitrary (noncylindrical) Riemannian metric g on V. Observe that the metric g + A dt2 is "nearly cylindrical" for the projection t: Vo x [0,1] -+ [0,1] and for large constants A> 0. Combine this observation with the implicit function theorem and with the above discussion on cylindrical immersions and prove the following

Proposition. If a manifold V ~ Vo x [0,1] is compact and parallelizable and if a strictly short map fo: (V, g) -+ (w, h) induces the trivial bundle fo*(T(W)) on V, then for dim W;::: [(n + 1)(n + 2)/2] + 1, n = dim V, there is a free Coo-approximation f: V -+ W of fo for which the induced metric f*(h) equals g + Adt2 for some constant A = AU) > 0.

Generalize this proposition to nonsplit (compact or not) manifolds V by taking an arbitrary Morse function t: V -+ IR and thus show the existence of a free isometric COO-immersion of (V,g), (for all Riemannian metrics g on V) into the pseudoRiemannian manifold (W x IR, h EB (-dt 2 )), where W is an arbitrary Riemannian manifold of dimension q ;::: [(n + 1)(n + 2)/2] + 1. For example, every parallelizable Riemannian COO-manifold V of dimension n admits a free isometric immersion into the pseudo-Euclidean space IRq+! = IRq, 1 with the metric

q

L dXf - dX~+l' i=l

l' (n + 1)(n + 2) 1 lor q = 2 +.

Give examples of n-dimensional (non-parallelizable!) manifolds V which admit no free maps into IRq for q < n(n + 4)/2.

3.1.9 Non-free Isometric Maps

The study of non-free maps faces two major problems. First, the algebra-geometric structure of jet spaces of isometric maps becomes quite complicated unless some regularity (freedom like) condition is imposed on these maps. The second difficulty


arises from the failure of the implicit function theorem for non-free Coo-maps. This problem is less severe for Can-maps due to Cauchy-Kovalevskaya theorem. (Jacobowitz 1982; Bryant-Griffiths-Yang 1983).

A Weakened Freedom Condition. Take a hyperplane H c T,,( V), V E V, and let Vo c V be a hypersurface through the point v which has T,,(Vo) = H and which is geodesic at v. We call an isometric C2-map f: V --.. W free along H, if the map fl Vo is free in some neighborhood U c Vo of v E Yo. The map f is called weakly free if it is free along some hyperplane H c T,,(v) for all points v E V.

The structure of local isometric weakly free maps (9jt(vo) --.. W, Vo E V, is very simple in the Can-category. Indeed, every free isometric Can_map fo: Vo --.. Wextends near Vo E Vo c V to an isometric Can_map of a small neighborhood (9jt(vo) c V into W, provided the binormal space ENfo(Vo, Vo) c Two(W)' wo = fo(vo) has dimension b :?: 1. If b = 1, then the extension is unique with a choice of an orientation in ENfo ' If b :?: 2 (which amounts to dim W :?: [n(n + 1)/2] + 1), then the space of these extensions is homotopy equivalent to the sphere Sb-l. This follows from our analysis in 3.1.6.

Example. Consider a surface V = V 2 with a Riemannian Can-metric g on V. One obtains all weakly free isometric can-immersions (9jt(vo) --.. 1R3 by taking any geodesic Vo c V through Vo E V and then by construction a free isometric can_ immersion of a germ of Vo at Vo into 1R3. The extension to V is then unique up to a choice of a binormal to Vo in 1R3. If the Gauss curvature Kg does not vanish at v E V, then every isometric immersion (9jt(vo) --.. 1R3 is free on some geodesic Vo c V, and so the above description gives us all isometric Can-immersions (9jt(vo) --.. 1R3. This is not true anymore if Kg(vo) = O. Moreover, Hopf and Schilt (1937) found the following example of an isometric can-immersion f: (9jt(vo) --.. 1R3 which cannot be isometrically C2-deformed to any weakly free isometric immersion. To see an obstruction to such a deformation, we look at the tangential (Gauss) map Gf :

(9jt(vo) --.. S2, whose Jacobian equals the Gauss curvature K of the surface. If Vo E V is an isolated zero of K = K(v), then the map Gf is a covering of the punctured neighborhood {9jt(vo)\vo onto (9jt(so)\so for So = Gf(vo) C S2, (9jt(so) C S2. The local degree of this map deg(Gf , Vo) clearly is invariant under isometric deformations of the map f. If f is a weakly free map, then obviously, Ideg(Gf , vo)1 = 1, and so no analytic surface in 1R3 with Idegl > 1 can be isometrically deformed to a weakly free surface.

Exercises. (a) Show for K :?: 0, that deg(Gf,vo) = 1 for the above surfaces V. Then consider a smooth function ((1: 1R2 --.. IR with an isolated critical point Vo E 1R2 and prove Kronecker's inequality

index(grad ((1, Vo) ~ 1.

(b) Construct for a given integer -d ~ 1 a can-disk V c 1R3 with an isolated zero Vo c V of the Gauss curvature and with deg(G, Vo) = -d for the tangential map G: V --.. S2.


Let f: V -+ 1R3 be a Can-immersion for which the point Vo E V is an isolated zero of theGausscurvatureKg(v)oftheinducedmetricgandletldeg(G"vo)1 = d ~ 2. Then the map f is d-flat at Vo' Namely, the dth-osculating space T,,~(f) c Two(1R3), Wo =

f(vo), coincides with the tangent space T,,~(f) ;, D,(T(V)) C Two(1R3). Suppose that the map f is generic in the space of d-flat maps. Then T,,~+1(f) ;:/= T,,~(f) and small perturbations of J, preserving the d-flatness at Vo also have at Vo an isolated zero of the induced curvature, while the degree deg(G" vol is constant under such perturbations.

Theorem (Efimov 1949). The above generic d-flat Can-immersion f: V -+ 1R3 is rigid for d = B. Namely, f admits no Can-family of isometric maps j,: (v, g) -+ 1R3 , fo = J, for which D'tl T"o(V) = D,I T"o(V)'

Proof The basic algebraic ingredient of the theorem is

Efimov's Lemma. Let F = F(x, y) be a homogeneous polynomial (form) of degree d + 1 in two variables and let Fll , F12 and F22 denote the second derivatives of F. If d = 8 and the form F is generic, then the linear differential operator

tP ~ Fll tP22 - 2F12 tP12 + F22 tPll

is injective on the space of formal power series tP = tP(x, y) which start with terms of degree d + 1,

tP(x, y) = aox 9 + a1 x 8 y + ... + a9y9 + box 10 + .... To prove the lemma, one should find one single form F for which the above

operator is injective. Let us show this to be the case for the form F(x, y) = !x3 y6. The corresponding operator

tP~xy6tP22 - 6x2y5 tP12 + 5x2y4tPll

maps' the space {tPm} of forms of degree m into the space {tPm+4} and the matrix of the operator {tPm} -+ {tPm+4} is diagonal in the monomial basis. The entries on the diagonal are

P(k, I) = 1(1- 1) - 6kl + 5k(k - 1),

for k = 0, ... , m and I = m - k. We must show that the equation P(k, I) = 0 admits no integral solution k, I ~ 0 for k + I ~ 9. Observe that

P(k, I) = (1- k)(l- 5k) - 5k -I.

Put p = 1 - k and q = I - 5k. Then the equation P(k, 1) = 0 becomes

(*) 2pq = 5p - 3q.

The integral solutions of (*) are

5p' - 3q' p = 2q'

where

and 5p' - 3q'

q = 2p'

pi = ± 1, ± 3 and q' = ± 1, ± 5.

Since I + k = t(3p - q) ~ 9, we must have

(5p' - 3ql)(3p' - q') I I ~ 36,

pq

which is satisfied by no choice of the above pi and q'. Q.E.D.

We prove the theorem by studying the derivatives of the isometric immersion equations (2') and (4) of 3.1.6, which now become

(1) <I;,jjk) = AUk, 1 :s;; i,j, k :s;; 2,

and

(2) f11f22 - f/2 = B.

If we differentiate (1) m - 2 times, then we express the scalar products <I;, am f) by functions in (derivatives of) the metric 9 and in the derivatives akf of order k < m. Hence, the orthogonal projection of every derivative amf(vo) E Two(/R3) onto the tangent space DJ('T"o(V)) c Two(/R3) is uniquely determined by the induced metric 9 on V and by akf(vo) for k < m. Since the map f is d-flat, the normal projection of amf(vo) to Nvo(f) = Two(/R3) e DJ'T"o(V), called amf(vo), vanishes for m = 2, 3, ... ,

d, while the normal component iJd+1f(vo) of iJd+lf(vo) defines a form F = F(x,y) =

Lk+l=d+1 ak1xkyZ, where x and y correspond to the local coordinates U1 and U2 on

(d + 1)ak+f (Vo) . . . V and where akl = ~d ~-~T· If we differentlate the Eqs. (2) 2(d - 1) hmes

k u1 dU 1

and then take the normal projection of the derivatives off, then we see that all terms with amf(vo) for m > d + 1 appear in products with amf(vo) for m < d + 1. Therefore, all these terms vanish and so we left with some quadratic equations in akl . These equations can be compactly expressed by

(2')

where Bd is a function in (derivatives of) the metric 9 and in the derivatives amf(vo} for m :s;; d. Since f is a generic d-flat map, the form F is also generic and so for d = g there is no deformation of F which keeps the Hessian Fll F22 - Fl2 of F unchanged. Indeed, the linearized deformation equation is

F11 (/>22 - 2F12 cJ>12 + F22 (/>11 ,

and so Efimov's Lemma applies.

This form cJ>m for m > d + 1, unlike cj)d+1 = F, is not invariant under coordinate changes in V but it does satisfy the following equation in our fixed coordiwne system,

F11 (/>';2 - 2F12 (/>f2 + F22 (/>f1 = Bm ,

where Bm is a function in g and in anf for n < m. This is seen by differentiating (2) (m + d - 3) times and then by looking at the normal components of the derivatives. The key observation here (like for m = d + 1) is the vanishing of the terms which contain derivatives anf for n > m, as they appear in products with derivatives alf for I :s; d. Efimov's lemma implies the rigidity of the forms (/>m for all m = d + 2, d + 3, and thus the rigidity of f [compare (G) in 2.3.8]. Q.E.D.

Remarks and Exercises. Efimov's lemma (and hence, the theorem) is likely to be true for all d ~ 3. But the lemma obviously is false for d = 2, and generic 2-flat immersions may be non-rigid.

Questions. Let Vbe a surface with a can-metric g on V. What is the number N = N(g) of connected components in the space of isometric can-maps «(l)jt(vo),g) ~ 1R3,

Vo E V? Efimov's theorem gives examples of N(g) ~ 2 (we identify immersions which are congruent by reflections of 1R3 ). Do any metrics have N(g) = 3? Is there a universal upper bound N(g) :s; No independent of g?

Exercises. Take a generic form F = F(x, y) of degree d ~ 4 and show that the differential operator

(/> f--+ F11 (/>11 + F12 (/>12 + F22 (/>11

is injective on the space of forms (/> of degree m for all m ~ 4. Consider a quadratic differential operator D on functions f: IRn ~ IR,

n

D: ff--+ L aij(a;/)(ajf), i,j=l

where aij are generic Can-functions on IRn. Take a generic function f in the space of the can-functions f on IRn which are d-flat at the origin, that is JJ(O) = 0, and show for d ~ (100)n this f admits no can-deformation ft for which D(ft) = D(f).

Generalize this rigidity to systems of homogeneous polynomial differential operators of arbitrary order and degree ~ 2.

Questions. Take a generic Can-immersion f in the space of immersions vn ~ IRq which are d-flat at a given point Vo E Vn. Is f can-rigid for q :s; n(n + 1)/2 and for d large, say for d ~ (100)q? This rigidigy means non-existence of non-trivial isometric Can-deformations in the space of d-flat immersions V ~ IRq. Unfortunately, there is no generalization (?) of the theorem of Hopf-Schilt to codimension ~ 2 which would guarantee d-flatness of all isometric deformations.

If q < n(n + 1)/2, then the rigidity is expected with no flatness assumption (see 2.3.8), but for q = n(n + 1)/2 the flatness plays a crucial role in making the isometric immersion equation overdetermined. This is seen by differentiating m times the Eqs. (2') of 3.1.6 and by differentiating m + d - 1 times the Eq. (4) of 3.1.6. The resulting algebraic equations for anf(vo) contain no terms with n > m + 2, as those derivatives are coupled with the vanishing derivatives akf(vo), k = 2, ... , d, and so the equations in anf(vo), n = 2, ... , m + 2, are formally overdetermined for large d.

The local rigidity of a can-immersion V ~ IRq at some point Vo E V obviously implies the global Can-rigidity. This idea is used by Efimov (1949) to prove the Can-rigidigy of the standard round torus T Z c 1R3.

Exercise. Let the first Stiefel-Whitney class of a connected manifold V = vn satisfy (Wl)3 i= O. Show that every can-immersion f: V ~ IRn +1 is rigid. (This is only interesting for open manifolds V as all closed hypersurfaces V c IRn+1 are rigid for n ~ 3.)

Hint. If the Gauss map G/ V ~ pn IR has rank Gf Z 3 at some point Vo E V then f is rigid near Vo.

Question. Let the first rational Pontryagin class of V satisfy (Pl)3 i= O. Then is every can-immersion f: V ~ IRn+Z rigid?

Making Non-free Maps Free. The Euclidean space IRq obviously admits large families of isometric Can-deformations in the ambient spaces IRN :=J IRq. This allows one to make some maps f: vn ~ IRq free by deformating in a space IRN of sufficiently high dimension N.

Exercises. Show that every isometric C'Xl-immersionfo: (V, g) -+ IRq C IRN admits an isometric deformation to a free isometric immersion f: V ~ IRN for N z q + [n(n + 3)/2].

Hint. Apply Thorn's transversality theorem to deformations of V which come from isometric deformations of IRq in IRN.

Let V be a compact Riemannian manifold and let fo: V ~ IRq C IRN be an isometric immersion. Construct for N ~ q + [(n + 1)(n + 4)/2] an isometric C)cylinder f: V x IR ~ IRN, such that fl V x 0 = fo and such that the map f is free near some submanifold V x to c V x IR, to E IR. Then prove that any two isometric COO-immersions fo and fl: V ~ IRq can be joined by an isometric cylinder V x [0,1] -4 IRN :=J IRq for N z q + [(n + l)(n + 4)/2] (compare 3.1.8).

Show that every isometric COO-immersion fo: V ~ IRq C IRN admits an isometric deformation to an isometric COO-imbedding f: V -+ IRN , provided

N :s:; min(2q + n + 1, q + 3n + 1).

(This estimate for N, probably, can be improved.)

3.2 Isometric Immersions in Low Codimension

There is no general theory of isometric immersions vn ~ W q for q < (n + 2)(n + 3)/2 but various facts are known for special manifolds. What follows is a brief (and not at all complete) discussion on some results. For further information we refer to the surveys by Gromov-Rochlin (1970), Posnjak (1973), Posnjak-Sokolov (1977), Aminov (1982) and to the book by Burago-Zalgaller (1980).


3.2.1 Parabolic Immersions

Consider a smooth immersion f of V into a Riemannian manifold (w, h). A point v E V is called parabolic for f if the induced metric g = f*(h) has the same sectional curvatures at v as the metric h at w = f(V)E W:

Rg(X, Y; X, Y) = Rh(X, Y; X, Y),

for all vectors X and Yin T,,(v) c Tw(W), This is equivalent by the Gauss theorema egregium to the symmetry of the 4-form cPfl T,,(V) (see 3.1.5), which means

cPj = cPf - cP; = o. We call an immersion f parabolic if all points v E V are parabolic. For example,

isometric immersions between manifolds of constant sectional curvature ,,= K(V,g) = K(W,h) are parabolic

Flat Directions. A tangent vector (direction) X E T,,(V) c T(V) c T(W)I V is called flat for f if V xT E T,,(V) for the covariant derivative V x in W of an arbitrary vectorfield T tangent to V. This is equivalent to the inclusion X E Ker cPfl T,,(V), where KercP consists, by definition, of the vectors X, such that cP(X,Xl ,X2 ,X3 ) = 0 for all triples of vectors (Xl' X2 , X3 ) in T,,(V).

If W is a complete simply connected manifold of constant curvature then the space of flat directions, denoted by Fl = Ker cPf c T(V), clearly equals the kernel of the differential of the obvious tangential map Gf : V ~ Gr" W, where Gr" W stands for the space of the (complete connected) totally geodesic submanifolds X of dimension n in W Furthermore, if W = IRq, then the space Gr"lRq canonically projects onto the Grassmann manifold, n: Gr"lRq ~ Gr"lRq with the fibers ~ IRq-", and the composition no Gf equals the ordinary tangential map Gf : V ~ Gr" ~q. It is clear that the differential of n is injective on the image DGf(T(V» c T(Gr"lRq). Therefore,

Fl = Ker DGf = Ker DGf .

We need a formula for the curvature of the natural Euclidean connection V (see 2.2.6) in the canonical n-dimensional bundle H ~ Gr"lRq. We regard the points P E Gr" IRq as orthogonal projectors P: IRq ~ IRq of rank n and we view the vectors X E Hp c H as vectors X E IRq for which P(X) = X. If X = X(u l , ... , u,,(q_"» is a section of H expressed in some local coordinates U;, i = 1, ... , n(q - n) = dim Gr" IRq, then

(V;X)(P) = P(o;X)

by the definition of V. The curvature operator Q of V is defined in the coordinates U; by the operators Qij = - Qj; = (V; Vj - VF;): H ~ H.

(A) Lemma. The operators Qij satisfy

Q;iX) = (P;~ - ~P;)X,

where Pi: IRq ~ IRq denotes the partial derivative o;P of the projector P = P(u l , ... , u"(q_,,»: IRq ~ IRq.

3.2 Isometric Immersions in Low Codimension

Proof Since pZ = P and PX = X, we have

ViVjX = PoiPojX = 0i(POjX) - PiPojX = Pi(OjX - PojX) + POijX

= Pi(OjX - 0iP X) + ~X) + POijX = (Pi~ + POij)X,

and the lemma follows.

261

Take an r-dimensional linear subspace I c Tp(GrnlRq) and choose local coordinates Ui such that the vectors od Tp(GrnlRq ) lie in I for i = 1, ... , r. We call I an D-isotropic subspace if Dij = 0 for i, j = 1, ... , r. This can be expressed in the invariant language by saying that the (operator valued) curvature form vanishes on I, that is DII = o.

(A') Lemma. The dimension of every D-isotropic subspace I abides

r = dim I :s; q - n.

Proof Let S c Hom(lRq -4 IRq) be the span of the operators Pi = OiP and let A =

P(lRq) c IRq. Then the following three conditions are satisfied.

(i) The operators s E S are symmetric. (ii) The vector s(a) is normal to A for all s E S and a E A. (iii) The operators s E S commute on A,

slsZ(a) = sZsl(a) foralls1,szES and aEA.

Indeed, the operators s are symmetric as linear combinations of derivatives of a symmetric operator. Furthermore, the identity pZ = P differentiates to P Pi + PiP = Pi which yields (ii). Finally, the condition DII = 0 is equivalent to (iii).

Now, the proof is immediate with the following

Sublemma. Let linear subspaces A c IRq and S c Hom(lRq -4 IRq) satisfy (i)-(iii). Then dimA + dimS s q.

Proof Take a non-zero vector ao E A and let So c S be the subspace of those operators s for which s(ao) = O. Then the linear subspace S(ao) = {s(ao)lsE S} c IRq has dim S(ao) = dim S - dim So. Furthermore, all So E So satisfy

<so(x),s(ao) = <x,sos(ao) = <x,sso(ao) = 0,

for all x E IRq and so the images of the operators So E So are normal to S(ao). Take the orthogonal complement Ro c IRq to the span of S(ao) and ao and let Ao =

An Ro c Ro. Since qo = dim Ro < q, we may assume by induction the inequality dim Ao + dim So :s; qo which implies what we need,

(dim A - 1) + (dim S - dim S(ao)) :s; q - dim S(ao) - 1.

(A") Example. The curvature D on the oriented Grassmannian GrzlRq is an ordinary (real valued) 2-form which is non-singular (symplectic) by (A'). Hence the Euler class (of the canonicaI2-bundle) [D] E HZ(GrzlRq; IR) satisfies [D]q-Z i= O.

Exercise. Show that the orthogonal projection Cq -+ ~q c Cq induces a diffeomorphism ofthe quadric L[=l zr = ° in Cpq-l onto Gr 2 (~q), such that the simplectic Kahler form on the quadric goes to the form Q.

(Alii) Corollary (Tompkins 1939). If v E V is a parabolic point of an immersion f: V -+ W, then the subspace Flv = Flv(f) c T.,(V) of flat vectors has dim Flv ~ 2n - q.

Proof The form <Pfl Tv(V) for T.,(V) c Tw(W), w = f(v), is invariant under the map exp-l: U -+ Tw(W), where exp: Tw(W) -+ W is the (geodesic) exponential map and U c W is a small neighborhood of WE W Therefore, one may assume W = ~q =

Tw(W) and use the map Gf : V -+ Grn~q. The parabolicity of the point VE V now is equivalent to the vanishing of the induced curvature,

GJ(Q)I T.,(V) = 0,

and so the image DGf(T.,(V)) c TAGrn~q), x = Gf(v), is Q-isotropic. Hence,

dim Flv = n - rank DGfl T.,(V) ~ 2n - q.

Remark. Let <P be an arbitrary symmetric 4-form on ~n, such that the associated quadratic form (p on the symmetric square (~n)2 is positive semidefinite. The following inequality is an algebraic equivalence of (Alii), .

deC -rank <P = n - dim Ker <P ~ rank <P.

Examples. If dim W = dim V + 1, then the inequality dim Flv ~ n - 1, obviously, is equivalent to the parabolicity of the point v. In particular, parabolic immersions f: V -+ ~n+1 are characterized by the inequality rank v Gf ~ 1 for all v E V, which (obviously) is invariant under projective transformations of ~n+l. If we take, for instance, an isometric immersion f of the disk (B2 = {x2 + y2 ~ 1}, go = dx2 + dy2) into ~3 and then compose this f with some projective map p: ~3 -+ ~3 [which must be regular on f(B) c ~3], we obtain a new flat metric g = (p 0 f)*(h) on B2 (where h = Lr=l dxr)· Thus, the metric go generates an interesting (?) class of flat metrics g on B2 as f and p vary.

Fix a projective map p of the open unit ball Bn+1 c ~n+1 onto the hyperbolic space H n+1 (of constant curvature K = -1). Then parabolicimmersionsf: V -+ Bn+1

go to parabolic immersions po f: V -+ Hn+1 which induce in V = vn metrics of constant curvature K = -1. This gives many examples of complete parabolic C"'-hypersurfaces in H n+1 which correspond to proper parabolic imbeddings f: V -+ Bn+l.

Exercise. Classify for all n ~ 2, all complete parabolic hypersurfaces in H n+1 with free non-Abelian fundamental groups.

(B) Integrability ofFI c T(V). Denote by io = io(f), for a given immersionf: V -+ W, the minimum of dim Flv over all v E V and let Iio c V be the subset where this minimum is assumed. This Iio is an open subset in V and the spaces FlV' vEIio, form an io-dimensional vector bundle Flio -+ Iio. If the manifold W has constant curvature, then the bundle Flio is integrable because FI = Ker DGf . Furthermore, if

W is complete and simply connected, then the integral leaf 2v c tio through v E tio is the path connected component of the point v in tio n Gjl(X) for x = Gf(V)E Grn W The points v E tio, for which c = Gf(v) lie in the image Gf(V\tiO ) c Grn W, have zero measure in tio, because rank Gfl V\tio < rank Gfltio = n - io and Sard's theorem applies. Hence, generic leaves 2v are smooth ii-dimensional properly imbedded submanifolds in V.

Lemma. Let X and Y be smooth vector fields in FI c T(V) c T(W)I v. If the manifold W has constant curvature, then the covariant derivative Z = V x Y (in W) also lies in Fl.

Proof. To show Vzr E T(V) for all fields r E T(V) we observe that the Lie bracket [Z, r] = Vzr - V,Z always lies in T(V) (for any two fields tangent to V) and so we must prove that V,Z = V,Vz Y lies in T(V). Since W has constant curvature, the operator V,V x - V xV, = R(X, r) preserves the tangent bundle T(V) c T(W)I v. Hence the field

where r' = V, Y E T(V) and r" = [r, X] E T(V), lies in T(V). Q.E.D.

Corollary (O'Neil 1962). Let W be a complete simply connected manifold of constant curvature and let dim FlvU) > ° for all v E V. Then there exists a complete totally geodesic submanifold 2 c V of dimension io > 0, such that the map f is flat on 2, [i.e. f(2) c W is contained in some io-dimensional totally geodesic submanifold Xio c W] and the map Gf has rank io at all points v E 2.

Proof. The generic leaves 2 = 2v discussed above are totally geodesic and fl2 is flat by the lemma.

Theorem (O'Neil 1962). Every isometric map f: [Rn -+ [R2n-l is flat on a straight line and so the image f([Rn) c [R2n-l is unbounded.

Proof. We have, by (Alii), dim Flv > 0, and the map f is isometrically flat on some affine subspace 2 c [Rn. Q.E.D.

Theorem (Ferus 1975). Every isometric Coo-immersion f: sn -+ s2n-l is flat, if both spheres are assumed to have constant curvature K = 1.

Proof. By our earlier considerations there is an open subset U c tio which is foliated by great spheres 2v cUe sn of dimension io such that the map f is flat on these spheres. Since great spheres are invariant under the central involution s r--.. - s of sn, the subset U c sn is also invariant and the map fl U: U -+ s2n-l is symmetric:

f( -u) = -f(u), for all UE U.

Since the map fl U is isometric as well as symmetric, every great circle Sl c U goes by f onto a great circle in s2n-l. Since U is open, the map f is flat on U and so io = n. Q.E.D.


Exercises. (a) Let f: S" -+ sq be a short (for example, isometric) map. Show that f is flat isometric under either of the following two conditions,

(i) There are points Si E sn, i = 1, ... , n, in general position (i.e. contained in no equator sn-l C sn), such that f( - s;) = - f(s;), for i = 1, ... , n.

(ii) There is a single point s E sn for which f( - s) = - f(s) and such that the differential Dj : T.(sn) -+ 1[(s)(sq) exists and it is isometric (for example, f is Cl-isometric near SEsn).

(a') Let U c sn be an open subset whose complement C = sn\ U is a smooth k-dimensional submanifold which contains no pairs of opposite points. Show that every isometric Coo-immersion f: U -+ sq is flat in the following two cases,

(i) n ~ k + 2 and q :s; 2n - 2; (ii) n ~ 2k + 2 and q :s; 2n - l.

(b) Let f: IRn -+ IRq be a short map which isometrically and bijectively sends the ray {Xl ~ O,Xi = O,fori = 2, ... ,n} in IRn onto the ray {Xl ~ O,xi = Ofori = 2, ... ,q} in IRq. Show that the half-space {Xi ~ O} c IRn goes into {Xl ~ O} c IRq.

(b') Assume that the above short map f is flat isometric on the line {Xi = 0 for i = 2, ... , n} c IRn and show that the map f splits in the following sense. Every hyperplane {Xl = const} c IRn goes into the hyperplane {Xl = const} c IRq and every normal line to {Xl = const} c IRn isometrically and bijectively goes onto a normal line to {Xl = const} c IRq. Use this and obtain an orthogonal splitting of an arbitrary isometric Coo-immersion f: IRn -+ 1R2n- 1 [compare Stiel (1965)].

(c) Let V be a complete manifold of non-negative sectional curvature and let f: V -+ IRq be a short map which isometrically and bijectively sends some geodesic 'Y E V onto the line {Xi = 0 for i = 2, ... , q} c IRq. Show that the manifold V splits isometrically, V = V' x IR, where 'Y = v~ x IR for some v~ E V', and the map f also splits: The hypersurfaces V' x reV go into hyperplanes {Xl = const} c IRq, and the geodesics Vi x IR go onto normal lines to these hyperplanes [compare Hartman (1970)].

(c') Generalize (c) to manifolds V of non-negative Ricci curvature by applying the splitting theorem of Cheeger-Gromoll (1971).

(d) Study short maps of an open hemisphere S,!- c sn into sq which are flat isometric on some geodesic arc of length n in S,!- .

(C) Remark. In fact, all leaves .Pv, vEIio, (not only generic ones) are properly imbedded into V. Indeed, consider the normal bundle L.l -+.P of a given leaf .P = .Pvo c Vand let hv: L~ -+ L;, VE.P, denote the holonomymap ofthefoliation, which is

where X = GAv) = Gj(vo). Since the foliation is totally geodesic in V; the bundle L.l continuously extends to the closure il c V of .P in V and the holonomy maps hv continuously extend to all v E ii. Then, by continuity, Dvo = Dvhv for all v E ii, which implies

and so il = .P. Q.E.D.


(D) Geodesic Laminations. Consider an isometric C1-immersionf: V --+ W between manifolds of constant sectional curvature K = K(V) = K(W) and let us define a partition of V into totally geodesic leaves 2'v c V for all v E V We first assume that W is the standard (i.e. complete simply connected) manifold, W = Mq[K] of curvature K, and that V is an B-ball in such a manifold, V = Bn(B) c Mn[K], where we assume 4KB2 ::; rc2. Then we define 2'v c V to be the subset of those points v' E V for which dist(f(v'),J(v)) = dist(v', v). Since K(V) = K(W) and since the map fis smooth, the subset 2'v c V is geodesically convex for all v E V and so 2'v' = 2'v for all v' E 2'v. Hence, V is partitioned into closed convex subsets called leaves, 2'v c V If the map f is COO-smooth, then this partition agrees on the subset Iio c V with the above foliation Iio = {2'v}' Furthermore, let a point v E V lie in the interior of the subset Ii c V where the rank of the map Gf equals i ~ io. The interior of Ii is foliated, like Iio, into i-dimensional leaves which are subsets of the convex leaves 2'v. Since the union U?=io Int Ii is dense in V, every convex leaf 2'v is locally a limit of i-dimensional leaves of (local) geodesic foliations and so 2'v satisfies the following three properties

(i) dim2'v ~ io ~ 2n - q for all VE V (ii) If io ~ 1, then the (convex!) leaf 2'v c V has no extreme points inside V = Bn(B),

and so 2'v equals the convex hull Conv(2'v n aV). (iii) If io ~ r + 1 for r ~ 0, then (ii) holds true for the intersection of 2'v with every

complete totally geodesic subspace x n- r c Mn[K] ~ V of codimension r,

2'v n x n - r = Conv(2'v n x n - r n aV).

(D') Corollary. The intersection 2'v n xn- r have

Diam(2'v n x n- r ) ~ dist(v, av), for all totally geodesic subspaces x n- r c M n [K] through v and for all v E V

Exercises. (a) Let Vo be the center of the ball V = Bn(B) c Mn[KJ. Prove that

Diam 2'vo = Diam V = 2B, for 2io > n,

and that

for io ~ 1,

where An is a regular geodesic n-simplex in V with the vertices on the boundary av (b) Let V be the unit ball Bn(1) c IRn. Prove that every isometric Can-immersion

f: V --+ 1R2n - 1 has Diamf(V) = 2. Construct for given numbers n = 2,3, ... , and b > ° an isometric Coo-immersion

f: V --+ 1R2n - 1 for which Diamf(V) ::; J3 + b. (c) Let V be a closed hemisphere in sn = Mn[1] and let Vo E V be the center

of V = Bnm c sn. Show that the leaf 2'vo c V equals V, provided io ~ 1, and hence every isometric Coo-immersion V --+ s2n-l is flat. Find non-flat isometric Coo -immersions of the interior Int V into sn+1 for all n = 2, 3, ....

Denote by Q the quotient space of the partition of the ball V = Bn(B) c Mn[K] into leaves 2'v. If V is a closed ball, then Q is a compact space.


(D") Lemma. The topological dimensi.on of Q satisfies

dimQ::; n - io.

Proof There exists, by (D'), a countable collection of geodesic subspaces Xrio c M"[KJ,j = 1,2, ... , such that the images of the balls Brio = vnXrio under the quotient map V -+ Q cover the space Q and so the problem is reduced to the following

Sublemma. Let rx: B"-io -+ Q be a continuous map for which the pullback rx-1(X)EB"-io

is a convex subset in Bn-io for all x E Q. Then

dim rx(B"-io) ::; n - io.

Proof We assume by induction

dim rx(B') ::; n - io - 1,

for all convex subsets B' c Bn-io of dimensional n - io - 1. Then for an arbitrary neighborhood U c Q of a given point x E Q we take a neighborhood U' c rx-1(U) c B"-io of the subset rx-1(x) c B"-io, such that the boundary au' c Bn-io is a finite union of convex subsets B' of dimension n - io - 1. Since the pullbacks f- 1 (x') are convex and hence, connected for all x' E Q, there exists a neighborhood U" c U of x, such that the boundary au' is contained in the image rx(oU'). As dim rx(oU') ::; n - io - 1, the sublemma follows from the very definition of dim Q.

Now let V and W be arbitrary manifolds of constant curvature K and let f: V -+ W be an isometric Coo-immersion. Then every small ball in V is partitioned into convex leaves .!l'v and these local partitions define a global partition, called a lamination [compare Thurston (1978)J, of V into locally (possibly noncomplete) geodesic submanifolds .!l'v c V. This lamination has codimension n - io by (D") and the map f is isometrically flat on every leaf .!l'v c V. If the manifold V is compact (possibly with boundary) and if the receiving space W is simply connected, then all leaves are closed subsets in V and the quotient space Q is a compact Hausdorff space of dimension n - io.

(E) The Holonomy Dimension Hd(V). Let V be an arbitrary n-dimensional manifold of constant curvature K = K(V). The holonomy covering of V is a unique minimal Galois' covering ii: V -+ V, such that V admits an isometric immersion d: V -+

Mn[KJ, called the developing map. Denote by rthe Galois group of the covering ii, let K(r, 1) be the corresponding Eilenberg-Mac Lane space and let H: V -+ K(r, 1) be the classifying map which isomorphically maps the Galois group r = 1t1 (V)/ 1t1 (V) onto 1t1 (K(r, 1)) ~ r. Denote by Hd(V) the minimal integer k for which there exists a k-dimensional polyhedron K and continuous maps F: V -+ K and G: K-+ K(r, 1), such that the composed map Go F: V -+ K(r, 1) is homotopic to H.

Examples. If V is a closed manifold, then Hd(V) = dim V unless V is the sphere S" of curvature K > O.

Let U be an open subset in a complete manifold V of dimension n. If the inclusion homomorphism on the k-dimensional homology, Hk(U) -+ Hk(V) does not vanish, then Hd(U) ~ k, unless U = V = S", n = k.

Remark. The developing map d: V ~ Mn[KJ isomorphically maps F onto a subgroup d*(F) in the isometry group Is(Mn[KJ) such that the map d is F-equivariant.

Exercise. Show for n ~ 3, that every finitely presented subgroup in Is(Mn[KJ) is the image d*(F) for some compact manifold (with a boundary!) of constant curvature K.

(E/) Theorem. Let V be an n-dimensional manifold of constant curvature K, such that the group d*(F) acts freely on the image d(V) c Mn[K]. If V admits an isometric C'XJ-immersionf: V ~ Mq[KJ, then

q ~ dim V + Hd(V).

Proof Since the map f is flat isometric on every leaf 2v c V and since the group d*(F) is free on the image d(2v), the holonomy covering V ~ V is trivial over every leaf 2v c V. Therefore, the classifying map H: V ~ K(F, 1) extends to the mapping cylinder C of the quotient map ot:: V ~ Q (which collapses each leaf to a point in Q) and so

Hd(V):::;; dimQ:::;; n - io:::;; q - n.

(F) Exercises and Open Questions. (a) Find, for given nand q satisfying n :::;; q < 2n, a compact manifold V = vn of constant curvature K = -1, which admits an isometric Coo-immersion V ~ Mq[KJ (here Mq[KJ is the hyperbolic space) but has no such immersion into Mq-l [K].

(a/) Do similar manifolds V exist for K = 1? What happens to q = 2n for K = ± 1? (b) Let K be a k-dimensional piecewise smooth subpolyhedron in an n

dimensional manifold V of constant curvature K. What is the minimal q = q(n, k, K), such that some small neighborhood U c V of K admits an isometric Coo-immersion U ~ Mq[KJ for all V and K? No estimate better than n + k :::;; q :::;; (n + 2)(n + 3)/2 is known for k ~ 1.

(b/) Show that q(n, 1, K) = n + 1. (b") Let U. be the e-neighborhood of a closed geodesic in an n-dimensional

manifold of constant curvature K. Prove for small e > 0 the existence of an isometric can-immersion U. ~ M n+1 [K]. Study the boundary au. for n = 3 and show that every complete flat 2-dimensional manifold admits an isometric immersion into M4(K) for all K. For example the 2-torus with an arbitrary flat metric (as well as every flat Klein bottle) admits an isometric Can-immersion into ~4.

(b"') Observe the following lackobowitz [1976J that every flat 2-torus admits a flat isometric immersion into a flat split 3-torus which is an isometric product of

. n(n + 1) three circles. [In general, every flat n-torus goes to a spItt 2 torus, see

lackobowitz (1976).J Use this to show that every flat 3-torus admits an isometric Coo-immersion into ~8.

(c) Let yn be the flat torus which is obtained by identification or opposite faces ofthe cube {Ix;! :::;; 1, i = 1, ... , n} c ~n, and let B(r) c Tn be the ball {Li'=l xf :::;; r2} for some r < 1. Show that the complement U; = Tn\B(r) admits no isometric coo_

immersion into ~2n-l for 2(vIn - r) ~ 1I:r. Construct such an immersion u,z ~ ~3 for r = 0.9.

(c') Let V(e) denote the complement to the square {Ixil ~ e, i = 1, 2} in T2. Show that V(e) admits an isometric Coo-immersion into 1R3 if and only if e > t. Show that V(0.9) admits an isometric Can-immersion into 1R3 but V(0.51) has no such immersion.

(c") Find a flat metric on S2 minus three open disks which admits no isometric Coo-immersion into 1R3. {No such metric is known on the cylinder Sl x [0,1], but many are known on the Mobius band, see Halpern-Weaver (1977).}

(d) Show that the compact disk B2 with an arbitrary flat metric admits an isometric Coo-imbedding into 1R3. [No nontrivial imbedding result is known for flat metrics on B" for n ~ 3.]

(d') (Michel Katz, unpublished). Construct infinitely many pairwise topologically non-isotopic isometric Coo-imbeddings ofthe flat cylinder Sl x [0, e] into 1R3 for length (Sl) = 1 and for e = 10-6•

(d") Find a complete (nonflat!) Coo-surface V which admits uncountably many pairwise non-isotopic proper isometric Coo-imbedding V -+ 1R3.

(e) Let U c IR" be an arbitrary open subset. Define Radk(U c IR") to be the lower bound of those numbers R ~ 0 for which there exists a continuous map oc: U -+ IR" with the following two properties

(i) the topological dimension of the image satisfies

dim!(U) ~ k,

(ii) dist(u,f(u» ~ R for all UE U.

See Appendix 1 in Gromov (1983) for basic properties of Radk •

Examples. If U contains a (k + I)-dimensional cycle C cUe IR" which is not homologous to zero in the e-neighborhood U.(C) c IR", then Radk(U) ~ e.

If the complement IR"\ U is a union of disjoint compact subsets in IR", then Rad"_2(U) = 00.

If the complement IR"\ U can be covered by compact subsets K j c IR", j = 1, 2, ... , such that Diam Kj ~ const < 00 and such that no I subsets among K j

intersect, then Rad,,_I(U) = 00.

(e') Let U admit an isometric Coo-immersion into a ball B"+k(R) c 1R,,+k of radius R. Prove that

Radk(U) ~ JnR.

(e") Let K c V be a properly imbedded smooth submanifold of dimension k. What is the minimal q such that some small neighborhood U c IR" of K admits a bounded isometric Coo-immersion f: U -+ IRq? The above (e') implies the estimate q ~ n + k for "sufficiently spread" submanifolds K and the upper bound q ~ n + 1 is obvious for k = 1. Nothing beyond this is known.

(f) Let V be a complete n-dimensional manifold of constant curvature Ie ~ 0 which admits an isometric Coo-immersion!: V -+ Mq[Ie]. Show that Vis homotopy equivalent to a (q - n)-dimensional polyhedron.

Additional References. Dajczer-Gromoll (1984), Laroubi (1984), Zeghib (1984).

3.2.2 Hyperbolic Immersion

Consider a smooth immersion f: V ..... (w, h) and compare sectional curvatures of (w, h) and (V, g) for the induced metric g = f*(h). Namely, set

,1 = ,1((\,02) = R9(Ob02;01,02) - Rn(01,02;01,02),

for all pairs of orthonormal tangent fields 01 and O2 in T(V) c T(W)I V. The immersion f is called hyperbolic if ,1 ~ 0 and it is called strictly hyperbolic if ,1 < 0 for all orthonormal fields 01 and O2 on V. (The equality ,1 == 0 amounts to the parabolicity of f.)

Denote by IIv the second quadratic form (on V) for a normal vector v E Nv =

N c T(W) e T(V)I v. Recall that

IIv(ol (v), 02(V)) = <Val VaJ(v), v),

where the covariant derivatives are understood in (w, h). Let

be the discriminant of IIv' Then Gauss theorema egregium claims

for an arbitrary orthonormal basis (VI, •.. , Vq - n ) in Nv •

Denote by a: Gr2 V ..... V the Grassman bundle of tangent 2-planes in V and let L ..... Gr2 Vbe the canonical oriented 2-bundle. Take a plane Lx c Lover some point x E Gr2 T,,(v) c Gr2 V which is spanned by some orthonormal vectors 01 (v) and 02(V) in T,,(V). Then the discriminant of the form IIvLx equals the product of the eigenvalues a1 and a2 of the form IIvlLx. Let us define another form iiv = iivx on Lx as follows

iiv = (IIv - !(a1 + (2)g)ILx'

The form iiv clearly has zero trace and iiv = 0 if and only if a1 = a2. We call the immersion f: V ..... W umbilic on Lx if iivx = 0 for all normal vectors

v E Nv • The umbilicity of f on Lx obviously implies ,11Lx :::::: 0 and so strictly hyperbolic immersions are nowhere umbilic.

Denote by N ..... Gr2 V the lift of the normal bundle, N = a*(N), and let P ..... Gr2 V be the 2-bundle whose fiber L'2 consists of quadratic forms on Lx with zero trace. The correspondance v f--+ IIv is a homomorphism of bundles,

ii:N ..... P,

whose zeros x E Gr2 V correspond to umbilic planes Lx c T(V). Thus we conclude to the following

(A) Lemma. Iff: V ..... W is a strictly hyperbolic immersion, then the homomorphism ii: N ..... P does not vanish and so the Euler class X of the bundle Hom(N ..... £1) is zero.


(A') Corollary (Liber-Chern-Kuiper-Otsuki-Springer). There is no strictly hyperbolic immersionsf: V --+ W for

dim W = q ~ 2n - 2, n = dim V.

Proof The oriented 2-bundles Land P can be viewed as complex line bundles and they obviously satisfy the relation P = L -z, which implies

x(p) = -2X(L)

Then the restriction of the class X to the Grassman manifold

GrzlRn = Grz T,,(V) c Grz V,

where the bundle N is trivial, satisfies for all v E V

xl GrzlRn = (-2X(L))kIGrz lRn,

for k = dim N = q - n. The class (X(L))k E HZk( Gr zlRn; IR) does not vanish for 2k ~ dim GrzlRn = 2(n - 2) by (A") of 3.2.1 and the Lemma is proven.

(A") Corollary. Let some normal Pontryagin class Pl(N) E H41(v, IR) be non zero while Pi(N) = 0 for i =I 1. Then the hyperbolicity of the immersion f: V --+ W implies q > 2n + 21- 2.

Proof Since Pi = 0 for i =I 1 the Euler class X of Hom(N --+ p) is expressed in terms of the induced class Pl(N) = rx*(pl(N)) by the well-known formula

X = ( - 2X((L)trx*(Pl(N)),

for m + 21 = dim N = q - n. To conclude the proof we must show that

for all P E H*(V; IR).

The class (X(L))n-zEHzn-Z(Grz V) is given by a (2n - 2)-form w whose restriction to GrT,,(V) = GrzlRn is a normalized volume form [see (A") in 3.2.1]. Hence, the integrals of the forms n on V over cycles C c V satisfy

t n = Ie W 1\ rx*(n)

for the pull-back C = rx-1(C) c Grz V. If P =I 0, then P is given by a form n such that Ie n =I 0 for some cycle C in V and the above implication follows. Q.E.D.

Example. Let us construct for given nand 1 ~ n/4 a manifold V = Vn, such that the normal bundle N of every immersion f: V --+ IRq has Pl(N) =I 0 and Pi(N) = 0 for i =I 1. It is enough to consider the case n = 41 and q > 2n. Take a (q - n)-dimensional vector bundle M --+ S41 for which Pl(M) =I O. As q - n > n, the total space of the bundle M has the same proper rational homotopy type as the trivial bundle IRq-n x S4 (this is a standard corollary of Serre's theorem on finiteness of the stable homotopy groups of spheres) and so there is a proper map cp: IRq-n x S41 --+ M of non-zero degree. Make this cp smooth and transversal to the zero section S4 c M and then take the pullback V = cp-l(S41) C IRq-n x S41.

Corollary. There is no strictly hyperbolic immersion V x IRk --+ IRq+2k for all k = 0, 1, ... ,for q ::; ~n - 2 and for the above manifold V of dimension n = 41.

Exercises. (a) Fill in the details in the above argument. (b) Express the condition X # 0 in terms of characteristic classes of the bundles

T(V) and N over V (without any simplifying assumptions like Pi = ° for i # I).

Asymptotic Directions. A non-zero tangent vector (direction) X E T.,(V) is called asymptotic for an immersion f: V --+ W if IIv(X, X) = ° for all normal vectors v E Nv. This is equivalent to the system of the following q - n homogeneous quadratic equations

IIv(X, X) = 0, J

j = 1, ... , q - n,

where {v 1, ... , vq - n } is some basis in Nv •

(B) Lemma (T. Springer and T. Otsuki 1953). If q ::; 2n - 1 and iff is hyperbolic, then there is an asymptotic direction X in every tangent space T.,(V). Furthermore, if f is strictly hyperbolic, then q = 2n - 1 [compare (A')] and there are exactly 2n

distinct hyperbolic directions in T.,(v), which, moreover, continuously depend on v E V.

Proof If n > q - n, then the system (*) has, according to Bezout's theorem, a non-zero complex solution Z = X + Y J=1, for X, Y E T.,(V). The equations IIv(Z, Z) = ° amounts to the system

J

IIv(X, X) = IIv(Y, Y) J J

IIv(X, Y) = 0. J

Since f is hyperbolic,

L1(X, Y) = L IIv(X, X)IIv(Y, Y) - (IIv(X, y))2 ::; ° • J J J

J

for all X and Yin T.,(V). Therefore, the solutions X and Y of (**) also satisfy (*) and at least one of them is non-zero, which is the required asymptotic direction.

If f is strictly hyperbolic, then L1(X, Y) < ° for all linearly independent vectors X and Y in T.,(V). It follows that all non-zero solutions X of (*) are simple and isolated (in the real projective space p n- 1 ). Indeed, if X + aX' is an infinitesimal deformation of a solution X of (*), then IIv(X, X') = 0, and the inequality L1(X, X') ::; ° implies X' = A.X, A.E IR. For the same'reason all solutions of (**) are real, Z = (a + bJ=1)X, and so all complex solutions of(*) are simple and isolated (in Cpn-l) as well as real. It follows, with Besout's theorem, that q - n ~ n - 1 and the number of solutions (in pn-l) equals 2n. As these solutions are simple, they are continuous (in fact, smooth) in v E V by the implicit function theorem. Q.E.D.

Remark. If f is a parabolic immersion, then every asymptotic direction X is flat since L1(X, y) = 0, Y E T.,(V). Thus we obtain another proof of (A"') in 3.2.1 for q = 2n - 1. In fact, the general case q ::; 2n - 1 also follows from the above lemma.

(B') Corollary. (Borisenko 1977). Letf: vn --.. wq be a strictly hyperbolic immersion for q = 2n - 1. If yn is a closed manifold, then the Euler characteristic X(Vn) = O.

Proof Indeed every asymptotic direction X E T.,(v), IIXII = 1, extends to a unique normal asymptotic vector field on some finite covering of V".

Remark. If vn and wq have constant sectional curvature, then a finite covering of V is parallelizable, since there are n out of 2n asymptotic directions which are linearly independent on V (see Moor 1972; Borisenko 1977). However, one does not yet know whether there are (isometric) hyperbolic immersions between round spheres, sn --.. s2n-1 for n = 3, 7.

Let us slightly generalize (A'). Take an arbitrary immersion f: V --.. Wand consider a unit normal vector v E Nv at some point v E V.

(B") Lemma. If the form IIv is positive definite on somek-dimensional subspace L c T.,(v) and if

q-n=dimW-dimV<k,

then the immersionfis not hyperbolic. Moreover, there are orthonormal vectors X and Yin Lfor which

LI(X, Y) ~ Div.(X, Y) = IIv(X,X)IIv(Y, Y) - (IIv(X, y»2 > O. J

Proof Take an orthonormal basis {Vj} in Nv,j = 1, ... , q - n with V1 = v. The proof of (A') yields orthonormal vectors X and Yin L such that L1:i Divj(X, Y) ~ 0 and so

q-n

LI(X, Y) = L Div.(X, Y) ~ Div(X, Y) > O. j=l J

Exercises. (a) Prove (B") with the argument of (B). (b) (Chern-Kuiper 1952; Jacobowitz 1973). Let f: vn --.. 1R211-1 be a smooth

immersion. Show that f cannot be hyperbolic if the manifold is closed. Assume, moreover, that the image f(V) lies in the ball of radius R, say Bx(R), x E IRq, and prove the existence of a 2-plane r in T.,o(V) for some Vo E V at which the sectional curvature K(r) ~ R-2•

Hint. Consider a maximum point VoE V of the function Vf--+ dist(v, x) and apply (B") to v = (f(v) - x)/llf(v) - xii.

(b') Prove the existence of r with K(r) ~ R-2 under the following weaker assumption: the orthogonal projection of f(V) C 1R2n- 1 into some hyperplane 1R2n-2 C 1R2n- 1 is contained in a ball of radius R. Derive the following relation betweenD = Diam VandK+ = sUPtK(r) for those Riemannian manifolds Vwhich admit isometric C2-immersions into 1R211-1,

K > D-22n -1 +- n-l'

Verify (*) for the real projective space p 3 of constant curvature and prove that there is no isometric C2-immersion p 3 --+ IRs. Generalize this by showing that no metric on pn whose sectional curvature is pinched betweeh 0.99 and 1 admits an isometric C2-immersion into IRn+2 for all n = 3,4, ....

(b") (A. Gray 1969). Let f: vn --+ wq be a proper C2-immersion where wq is a complete simply connected manifold of non-positive sectional curvature. Show V to be homotopy equivalent to a k-dimensional polyhedron for some k :::;; q - n, provided one of the following two conditions is satisfied.

(i) f is hyperbolic, (ii) the manifold wq has constant negative curvature and the induced metric in vn

has non-positive curvature.

Hint. Study the critical points of the function v --+ dist(wo, v) for some Wo E wq.

(C) Flat Directions of Hyperbolic Immersions. Let f: V --+ W be a hyperbolic immersion. Then the quadratic form II. has rank II. :::;; 2q - 2n for all normal vectors v E T,,(v), VET. This is an immediate corollary of (B"). In particular, the forms II. are singular for q < ~n. In order to exploit this for manifolds W of constant curvature we recall classical facts on the following

Geometric Legendre Transform. Let P = pq be the real projective space and let P* be the dual space. Points y E p* by definition are hyperplanes in P, called y* c P. Observe that P** = P and so points x E P correspond to hyperplanes x* c P*. Observe that x E y* <:0:> Y E x* and denote by Q c P x P* the set of those pairs (x, y) for which x E y*. Denote by nand n* the projections of Q to P and to P* respectively. Observe that the bundle n: Q --+ P is canonically isomorphic to the Grassmann bundle of hyperplanes in T(P) called Grq-1P --+ P, while the bundle n*: Q --+ P* is canonically isomorphic to Grq - 1 P* --+ P*.

The manifold Q carries a natural hyperplane field (a contact structure) {} c T(Q), where the hyperplane (}z c T,,(Q), Z = (x, Y) E Q, is defined as the span of the tangent spaces to the fibers n-1(x) c Q and (n*fl(y) c Qat Z E Q. The tangent spaces to the fibers, say T: and TI of dimension q - 1 in T,,(Q), have T: n TI = 0 and so they span a hyperplane in T,,(Q) for all Z E Q.

Take an arbitrary smooth submanifold V c P of positive codimension and denote by V c Q the subset of those pairs (x, y) E Q C P x P* for which x E V and the hyperplane y* c P is tangent to Vat the point x. For example, if V consists of a single point VEP, then V = n-1(v); if V is a hyperplane, V = y* for some YEP*, then V = (n*fl(y).

A submanifold Z c Q is called Legendre if dim Z = q - 1 and if the tangent space ~(Z) c ~(Q) lies in (}z for all Z E Z.

Lemma (Legendre). The above submanifold V c Q is Legendrefor all V c P. Furthermore, let Z c Q be Legendre, let the projection n I Z have constant rank = n and let the image V = n(Z) c P be a smooth submanifold in P. Then V::J Z. Moreover, if the intersection Z n n-1(x) is a closed manifold for all x E P, then V = z.


The (well known and easy) proof is left to the reader.

Exercise. Let P be an arbitrary smooth manifold of dimension q and let Q = Grq_1 P. Define (}z c T(Q), Z = (x, H), for x E P and H c T,,(P), to be the pull-back of the hyperplane H under the (differential of the) projection Q --+ P. The resulting hyperplane field Z f---+ (}z on Q is called the canonical contact structure on Q and the submanifolds Z = Zq-l C Q tangent to {} are called Legendre. Show this new field {} to be equal to the previous one for the projective spaces and extend Legendre's lemma to all smooth manifolds P.

Definition. Legendre's transform of a submanifold V c P is the image V* = n*(V) c P*.

Warning. This V*, in general, is not a submanifold in P* since the map n*IV may have non-constant rank.

Exercise. Let co dim V ~ 2 and let the map n* I V have constant rank. Show that V is flat, i.e. V lies in a projective subspace pn c pq for n = dim V.

The above Legendre's lemma immediately implies the famous Legendre duality theorem which claims the identity (V*)* = V for the submanifolds V of positive co dimension in P. Since V* may be singular, the identity (V*)* = V applies only to those points Z E V where the map n* I V has locally constant rank. Namely, if rank n* 10' == k for some small neighborhood 0' c V of Z E V, then the image n*(O') is a k-dimensional submanifold, say U* c P, such that O*:J 0'. Recall that dim 0* = dim 0' = q - l.

Denote by Ej* c V,j = 0, ... , q - 1, the subset of those Z E V where rank n* I V =

j. The (connected components of the) pull-backs of the map n*IV foliate the interior ofthe subset Et,j = 0, ... , q - 1, into (q - 1 - j)-dimensionalleaves 2z c

IntEt, through the points zElntEt Every such leaf projects under the map nl V: V --+ V C P onto a (q - 1 - j)-dimensional submanifold, called!lz c V, which is uniquely determined by the point

Z = (x,Y)ElntEt eVe Q c P x P*,

where x E!lz eVe P and where the hyperplane y* c P* is tangent to Vat x. Since the hyperplanes t* c P*, IE !lz, are tangent to V* c p* at Y E V* by Legendre duality, the submanifold !lz c P is flat for all Z E Int Ef, j = 0, ... , q - 1, and the hyperplane y* is tangent to Vat all points I c !lz. Furthermore, ifjo is the greatest intege! for which Ej~ "# 0, then the leaf !lz is a closed subset in V for a generic point zEn(Ej~) c V according to Sard's theorem [compare (B) in 3.2.1].

Finally, we observe that the kernel of the differential Dzn*IV isomorphicaUy projects onto Ker II. under (the differential of) the map n I V, where v E T,,(P) is the unit normal to the hyperplane y* c P for all Z = (x, y) E V. (Here we use the standard metric of constant curvature + 1 in the projective space P and we may take an arbitrary point Z E v.)

Exercise. Study the relations between the manifolds .Pz and the flat leaves 2v [see (B) in 3.2.1].

Since the above considerations are projectively invariant they apply to all manifolds W = wq of constant sectional curvature and so we have the following

Proposition. If an isometric immersion f: V --t W has rank II. ::; k < n = dim V for all normal vectors v, then there exists a totally geodesic submanifold 2 c V of dimension k which is (relatively) complete in V and such that the map f12: 2 --t V is isometric flat.

Corollary. If the manifolds V and Ware complete and if the map f is hyperbolic, then the image f(V) c W contains a complete totally geodesic submanifold of dimension ~3n - 2q.

This result is due to Borisenko (1977) where the reader is referred to for a further study of hyperbolic immersions.

(D) Examples of Hyperbolic Immersions. There are various constructions of complete hyperbolic surfaces in [R3 (see Rosendorn 1966). For example, let K c [R3 be a piecewise linear one-dimensional polyhedron (graph), such that every vertex ko c K lies in the interior of the convex hull of the neighbor vertices. Then the boundary of an appropriate small neighborhood of K is a hyperbolic surface which is complete, provided K contains no connected infinite chain of edges whose total length is finite.

Exercise. Make the above precise. Then construct a complete bounded hyperbolic immersion [R2 --t [R3. Observe that the direct product of hyperbolic immersions is hyperbolic and construct a complete bounded hyperbolic immersion [R2n --t [R3n,

n = 2, 3, ....

If wq is a manifold of constant sectional curvature, then the hyperbolicity is quite a restrictive condition for complete immersions vn --t wq for q ::; 2n - 2, and especially for q < !n. Yet, there is no geometric classification of these immersions. In fact, one does not even know which closed manifolds vn admit non-flat hyperbolic immersions in a given manifold wq for n + 2 ::; q < !n.

Clifford Torus. The product of n unit circles Sl c [R2 is called the Clifford torus Tn c s2n-1 C [R2n, where

s2n-1 = {XE[R2n lllx ll = In}. This torus is strictly hyperbolic in the sphere s2n-1.

The cone from the origin over the Clifford torus is a (non-strictly) hyperbolic immersion C: Tn X (0,00) --t [R2n, which is non-complete at the origin. Furthermore, the map (t, x) f---+ (C(t, x),J(x)) for f(x) = x-1 is a complete hyperbolic imbedding of

T" X (0, 00) into ~2"+1. Observe that the sectional curvature of the induced metric converges to zero for x -+ 00.

Question. Does the Klein bottle admit a (strictly) hyperbolic immersion into S3?

Isometric Immersions of Hyperbolic Spaces. The hyperbolic plane H2 (with the complete metric of constant curvature -1) admits no isometric C2-immersion f: H2 -+ ~3 by a famous theorem of Hilbert. In fact, no complete immersion V2 -+ ~3 is uniformly strictly hyperbolic which means K ~ -" < 0 for the Gauss curvature K of the induced metric (see Efimov 1964; T. Milnor 1972).

Question. How does the Hilbert-Efimov theorem generalize to manifolds of dimension ~ 2?

It is unknown whether the hyperbolic plane admits a Ck-immersion into ~4 for k ~ 2. However, there exist closed strictly (and hence, uniformly) hyperbolic surfaces in ~4 (see Rosendorn 1961). The universal covering of such a surface is a complete bounded immersion ~2 -+ ~4 which is strictly uniformly hyperbolic [see the survey by Poznjak (1973), for additional information].

Theorem (Rosendorn 1960). There exists an isometric Coo-immersion H2 -+ ~5.

Proof Take (horospherical) coordinates t and u in H2, -00 < t, U < 00, such that the hyperbolic metric equals (dt)2 + e2t(du)2. Decompose, e2t = (fJ!(t) + (fJ~(t), for some Coo-function (fJi(t), i = 1,2, such that the support of (fJi' i = 1,2, is a disjoint union of closed subintervals in ~. Let Bi(t) be a positive function which is locally constant on the support of (fJi for i = 1, 2. Set

ii = ii(t, u) = Bi(t)(fJi(t)sin(uBi1 (t»

and

fi' = Bi(t)(fJi(t)COS(UBi1(t»,

for i = 1,2. By Nash's formula (see 3.1.1)

L (dii)2 + (diiY = e2t(duf - <5(t)(dt)2, i=1,2

for <5(t) = j=~2 (Bj(t»2 (d~t(t) y. If the functions Bj are chosen sufficiently small, then 1 - <5(t) > O. Therefore, there

is a Coo-function a(t) such that (da(t)/dt)2 = 1 - <5(t) and the five functions ii, ii' and a, for i = 1,2, define the required isometric immersion H2 -+ ~5.

An Isometric Coo-Immersion H" -+ ~4"-3. The metric (of constant curvature -1) on the hyperbolic space is (dt)2 + e2t Lj:i(duj)2, where t, u1, ... , U"-1 are the horospherical coordinates in H". Define with the above (fJi and Bj the functions iij =

Bj(fJj sin ujBi1 and iii = Bi(fJi cos ujBi1 and observe that the map (iij, iii, ajn=1): H" -+ ~4"-3 for i = 1,2, andj = 1, ... , n - 1 is isometric for the above a = a(t).


An Isometric can-Immersion H~ ~ 1R2n-1 (F. Schur 1886). Denote by H~ c Hn the horoball, H~ = {t,u1, ... ,unlt::;; O}. Take fj = aet sin a-1uj andfj' = aetcosa-1uj for j = 1, ... , n - 1 and for a small constant a > 0, and let 13 = f3(t) satisfy for t ::;; 0,

df3(t) dt = 1 - (n - 1)a2e2t.

Then the map (fj,fj', 13): H~ ~ 1R2n- 1 is can-isometric.

Corollary. Every relatively compact open subset U c Hn admits an isometric can_ immersion into 1R2n- 1 .

Immersion H~ ~ M 2n- 1 [K]. Consider a Riemannian can-manifold U and let y c U be an infinite geodesic. Let the isometry group I s(U) be transitive on y and let the isotropy subgroup Iw , WE y, contain the torus T n- 1 as a subgroup. Then an obvious generalization of the above formulae provides an isometric Can-immersion H~ ~ U. In particular, there is an isometric can-immersion of H~ into the a-neighborhood of an arbitrary (infinite or periodic) geodesic y in any given space of dimension ;;::: 2n - 1 of constant curvature K for all K E IR and all a > 0.

Exercises. Consider the (Riemannian) product of (n - 1)-copies of H2 with the metric Ij:t(dtj)2 + e2t(duy. Show that the submanifold {tj' ujl t1 = t2 = ... = tn - 1} is isometric to H n with constant curvature -(n - 1).

Consider coordinates x and y in H2 in which the hyperbolic metric is dx2 + [ch(x)] 2 dy2 for ch(x) = (eX + e-X)j2. Prove with an appropriate splitting [ch(x)]2 - ! = cpr(x) + cpi(x) the following theorem of Blanusa (1955).

There exists a proper isometric Coo -imbedding H2 ~ 1R6 whose image is the graph of a COO-map 1R2 ~ 1R4. Furthermore, there exists a proper isometric Coo-imbedding Hn ~ 1R6n - 6 for all n ;;::: 2.

Find a small isometric COO-perturbation of the linear imbedding ~6 ~ ~10 which makes the composed isometric map H2 ~ 1R 10 free and then approximate this map by an isometric can-imbedding H2 ~ ~10. Prove a similar result for 1R8 instead of 1R10 and thus show that existence of a proper isometric Can-imbedding Hn ~ 1R8n-8.

(D') An Application of the Theory of Sheaves. The strict hyperbolicity condition for immersions V ~ W is clearly open and (DitT V)-invariant. Hence, the h-principle applies to open manifolds V (see 2.2.2), and so the construction of hyperbolic immersions reduces to the study of the corresponding space of jets.

Lemma. Choose some vectors ai in IRn, i = 1, ... , k, put

and let F: Id Et> f1 Et> ... Et> k IRn ~ IRn+k. If the vectors ai span a subspace of dimension ;;::: n - 1 in IRn and if the number A is sufficiently large, then the map F is strictly hyperbolic at the origin ° E IRn.

Proof Let h(x, y) be the symmetric forms corresponding to the (quadratic) function /;. Then, obviously,

k

LI(x, y) = L h(X, X)h(Y, y) - (h(X, y)f > 0 i=1

for all large A and for all orthonormal vectors x and y in IRn. Q.E.D.

Corollary. Consider an immersion cp of an open manifold V into a Riemannian manifold W. Let Vi E N", c T(W)I V,i = 1, ... , k, be mutually orthogonal normal (to "V) field and let IX: N", -+ T(V) be a homomorphism. If the tangent fields ai = IX(Vi) span a subspace of dimension ~ n - 1, at every point v E V, then the map f is homotopic to a strictly hyperbolic immersion.

Proof. Consider the above functions

h on IRn = T,,(v), VE V, for h = ia,(v),).(v),

where A(V) is a sufficiently large continuous function in VE V and define

k

Fv = Dvcp + L hVi: T,,(V) -+ Tw(W), w = cp(v). i=1

The maps Fv form a continuous in VE V family of maps T,,(V) -+ Tw(V) which are strictly hyperbolic near the zero section V -+ T(V). Hence, by the h-principle, the map cP can be deformed to a hyperbolic immersion.

Theorem. If q ~ 3n - 2 then an arbitrary continuous map cP: vn -+ wq is homotopic to a'hyperbolic immersion, provided vn is open.

Proof. We may assume cp to be an immersion. Construct normal fields Vi' i = 1, ... , q - n by induction in i as follows. Let I(i) c V be the subset where the fields V1, ... , Vi are independent. We construct Vi+1 by first taking a generic normal field Vi+1 on the complement V\I(i) such that < vi+1, Vj> = 0, j = 1, ... , i, and then by taking Vi+1 = t/lvi+1 where t/I is a smooth function on V whose zero set equals I(i). The subset Ik c V, where codim Span {Vi };=1, ... ,q-n ~ k clearly has codim Ik ~ kfor all k = 0, 1, ... , q - n. Therefore, the fields ai = IX(Vi) for a generic homomorphism IX: N", -+ T(V) satisfy the following inequality over a small neighborhood of the (n - I)-skeleton of some triangulation of V,

dim Span {ail ~ q - 2n + 1.

Hence, the above corollary applies for q ~ 3n - 2.

(E) Exercises and Open Questions. (a) Replace the above genericity argument by a homotopy theoretic consideration.

(b) Prove that an arbitrary immersion of an open manifold V into a Riemannian manifold W can be regularly homotoped to an immersion f: V -+ W which is e-parabolic, that is III.(X, Y)I ~ e for all unit normal vectors V, and for all unit tangent vectors X and Y, where e > 0 is any given number.

(b') Show that the existence of a smooth immersion V --+ [Rm implies the existence of a hyperbolic immersion V --+ [R2m-l for all open manifolds V.

(c) What is the minimal q = q(n) for which the above existence theorem holds true?

(d) Let V be an arbitrary (possibly closed) n-dimensional manifold and let W have positive sectional curvature. Prove the h-principle for strictly hyperbolic immersions V --+ W for dim W ~ (n + 2)(n + 3)/2 (compare 3.1.7).

(d' ) Find counterexamples to the above h-principle for dim W = 2n - 1. (No such example is known for dim W ~ 2n).

(e) An immersion V -+ W is called strictly elliptic if the discriminant J (defined at the beginning of this section) satisfies J(X, Y) > ° for all orthonormal tangent vectors X and Yin T(V). (If W = [Rq, this amounts to the positivity of the sectional curvature of the induced metric in V.) Prove that every strictly elliptic immersion admits a non-vanishing normal vector field. Prove the converse for open manifolds V: if an immersion f: V --+ W admits a non-vanishing normal field, then f can be regularly homotoped to a strictly elliptic immersion V --+ W

(e' ) Recall Whitney's theorem: no imbedding of the projective plane p 2 into [R4 admits a normal field. Hence, no imbedding p2 --+ [R4 is strictly elliptic.

(e") Does there exist an elliptic immersion p2 -+ [R4? (e"') Construct an elliptic imbedding of p3 ~ SO(3) into [R6.

3.2.3 Geometric Obstructions to Isometric C 2·Immersions V2 --+ [R3

In this section we discuss several inequalities between geometric invariants of surfaces V in [R3 which indicate a complete break-down of the h-principle for isometric C2-immersions V -+ [R3.

(A) A Lower Bound on the Curvature K(v) of a Compact Surface V. Let V(a) =

{VE VIK(v) ~ a2} for a ~ ° and assume the existence of an isometric C2-immersion f: V --+ [R3 such that Ilf(v)11 ::;; Ro for all VE V.

(i) If V has no boundary and if ° ::;; a ::;; R01, then

3a2 Area V(a) + r K(v)dv ~ 4n(1 - aRO)2. JV(a)

In particular, Sv(o)K(v)dv = Sv max(K(v),O) dv ~ 4n. (ii) Iffor some rx < 1 the boundary points of V satisfy Ilf(v)11 ::;; rxRofor all v E av, then

3a2 Area V(a) + r K(v)dv ~ (2n - 4 arc sin rx)(l - aRO)2. JV(a)

In particular,

Iv max(K(v),O)dv > 2n - Srx.

Proof Take the (interior) normal field v of the immersion fl V(a) for which the second fundamental form II. has non-negative eigenvalues 0::;; Al (v) ::;; A2 (v),


VE V(a), called the principal curvatures of f. Consider the (normal exponential) map e: V(a) x IR -+ 1R3, such that e(v, 0) = f(v), and which isometrically sends the line v x IR onto the straight line in 1R3 oriented by the normal vector v(v) for all v E V. The Jacobian of the map e is clearly J(v, t) = (1 - Al (v)t)(1 - A2(V)t).

Let W = {(v, t) E V(a) x IRIA2(V) ~ t ~ a-1} c V(a) x IR. Then the image e(W) c 1R3 contains the ball {llx II ~ Ro - a-1 } c 1R3. Indeed, each maximum point v = V(X)E V ofthe distance function v -+ dist(f(v), x) satisfies e(v(x), dist(f(v), x» = x. Therefore,

4n(Ro - a-1)3 ~ fw J(v,t)dvdt ~ fw (1 + Al(V)A2(V)t2)dvdt

= r (1 + K(v)t2)dvdt ~ a-1 Area V(a) + ta3 r K(v)dv, Jw JV(a)

which implies (i). If V has a boundary, then e(W) contains the subset Xv - {XE1R3111xll ~

min(Ro - a- 1 , Ilx - f(v) II - aRo} for all VE V. If Ilf(v)11 = Ro, then Vol Xv ~ t(2n - 4 arc sin a)(1 - aRof which implies (ii). Q.E.D.

(A') Exercises and Generalizations. (a) Prove (A) for a = ° by applying the GaussBonnet formula to the boundary of the convex hull Conv f(V) c 1R3.

(b) Consider a complete 3-dimensional Riemannian manifold W 3 whose sectional curvatures are pinched between + 1 and - 1 and whose injectivity radius is bounded from below by a constant Po > 0. Assume the existence of an isometric C2-immersion f: V -+ W 3 and find a lower bound on SV(a) K(v)dv in terms of Po, a and d = Diamf(V). Show in particular that

Iv max(K(v),O)dv ;;::: 4n - b

for closed surfaces V, where b = b(po, d) -+ ° for d -+ 0. (b') Generalize (b) by allowing the manifold W 3 to have a locally convex

boundary. (c) Consider an n-dimensional manifold V for n ~ 3 and denote by K+ (v) the

upper bound of sectional curvatures on the 2-planes in the tangent space T.,(v). Assume the existence of an isometric C2-immersion f: V -+ IRn+l and estimate from below the integral Sy(max(K+(v),0»n/2dv. Find a similar estimate for immersions f: V -+ W n+1•

Question. Does the existence of an isometric C2-immersion vn -+ IRq, n + 2 ~ q ~ 2n - 1, imply some integral inequality for the curvature of the (closed) manifold V n?

(B) An Upper Bound on Rad V. Assume the boundary av non-empty and set Rad V = sUPvEvdist(v,aV). Observe that manifolds of positive sectional curvature K(V) ~ ,,2 > ° have Rad V ~ n,,-l.

If K(V) > ° and if V admits an isometric C2-immersion into the ball B(Ro) = {llxll ~ Ro} c IRn+1, n = dim V, then Rad V ~ nRo.

Proof The image f(V) c IRn+l is a locally convex hypersurface. Take the (exterior) normal ray rv to f(V) at f(v) with respect to which the second fundamental form at v is negative definite and let s = s(v) be the intersection point of rv with the sphere aB(Ro). The resulting map V ~ aB(Ro) for v H s(v) is clearly infinitesimally enlarging (see 1.2.4), that is the induced metric of constant curvature Ro2 in V is greater than the original metric in V. Hence, Rad V ::;; nRo by the above remark.

Exercises. (a) Let Vo be a closed Riemannian manifold whose fundamental group is finite. Prove the existence of a constant Ro = Ro(Vo) < 00, such that no manifold V with Rad V ~ Ro and with dim V = dim Vo admits an infinitesimally enlarging (for example, isometric) map into Yo'

(b) Give an upper bound on Rad V for a locally convex hypersurface V in a Riemannian manifold W with a locally convex boundary aw.

(c) Construct, for a given I: > 0, a COO-metric 9 = g. on the 2-distD2 with the following three properties

(1) The curvature of 9 everywhere is positive ~ 1. (2) The manifold (D2, g) admits an isometric Sl-action with a fixed point

voED2, such that distg(vo,aD2) > 1. (3) The unit ballB in (D2, g) around Vo has JBK(v;g)dv::;;!n and DiamgaB <

dist(aB, aD2) + length aD2 ::;; 1:.

Prove·for I: < 1 that no isometric C2-immersion (D2, g) ~ 1R3 exists. Then consider an arbitrary closed Riemannian manifold W 3 and show that no isometric C2-immersion (D2, ag) ~ W3 exists for all small positive fJ = fJ(W3). Estimate fJ in terms of W.

(C) An Upper Bound on R(V) for Non-Elliptic Surfaces V. Let us construct a projective map P + of the unit ball B3 c 1R3 into the unit sphere S3 c 1R4 as follows. First move S3 to a new position in 1R4 where it is tangent to B3 at the center 0 E B3 and then radially project B3 to the moved sphere from the center of this sphere. In a similar way one obtains a projective map P_: B3 ~ H3 for the hyperbolic space H3 of curvature - 1.

Consider an immersion f: V ~ B3 which induces a metric 9 on V with the sectional curvature K = K(v). Denote by g± the metric induced by the map P ± 0 f and let K ± denote the curvature of 9 ± .

(C) Lemma. There exists a universal constant Co > 0 (which is, in fact, ::;; 100), such that

and

(**)

Proof The inequality (*) is obvious for a constant Co which majorizes the norms of the differentials DP ± and (DP ± rl. To prove (**) we observe that projective maps


send planes in 1R3 to (totally geodesic) planes in S3 (and in H 3 ). Hence, the discriminants of the second fundamental forms of the maps f and P ± 0 f satisfy with some constant Co > 0,

COl Discr II ~ Discr II ± ~ Co Discr II,

and (**) follows [compare (E') in 2.4.4].

(C") Corollary. If a surface V = (V, g) has curvature K ~ R(j2 for some Ro ~ ° and if V admits an isometric C2-immersion into the ball B(R) c 1R3 for R ~ Ro/~, then

RadV~nR~.

Proof We may assumeR = 1. ThenK+ ~ t by (**). Hence, Rad(V,g+) ~ nj2, and Rad(V,g) ~ n/~ by (*).

(C"') Exercises and Generalizations. (a) Let V be a complete non-compact surface, such that lim inf K(v) [dist(v, VO)]2 = ° for some Vo E V. Prove every isometric C2_

immersion V -+ 1R3 to be unbounded. (a') Let a compact surface V with a connected boundary satisfy

K(V) ~ _R(j2, for some Ro ~ 0,

2n - Iv max(K(v),O)dv ~ p > 0,

and

length av ~ RoP/30~.

(For example, V is flat.) Assume Rad V > nr,J2Ca for r = 30p-1 length av and show that no isometric C2-immersion V -+ 1R3 exists. Apply this to a closed surface with K ~ ° minus a small ball.

(b) Geralize (C') to C2-hypersurfaces V in the unit ball Bn+l c IRn+l, n ~ 3. Let the sectional curvature of V satisfy K (V) ~ - (2CO)-1 for the above Co and establish the following properties of V.

(b l ) Rad V ~ n~. (b2) If V is closed, then the homology group Hi(V; IR) = ° for 1 ~ i ~ n - 1.

Hint. Use an estimate by A. Weinstein (1970) on the curvature operator of manifolds vn c IRn+2 [see Aminov (1975), Baldin-Mercuri (1980) and Moore (1978) for further results].

(c) Let V be a complete non-compact surface with a finitely generated fundamental group. Let lim sup K(v) ~ ° and let V admit a bounded isometric C2_

v-+oo

immersion f: V -+ 1R3. Show the balls B(R) c V around a fixed point Vo E V to satisfy

lim infR-llogAreaB(R) ~ t: > ° R-+oo

under one of the following three conditions.

(i) Area V = 00, (ii) liminfK(v)?: -const > -00,

v"" 00

(iii) K(v) < ° outside a compact subset in V.

Hint. Use the map P_ to prove (i) and (ii). Consult Verner (1970) for (iii).

Questions. (a) Are the conditions (i)-(iii) essential? How does one generalize (*) to hypersurfaces vn c IRn+l for n ?: 3?

(b) Does (C") generalize to immersions V ~ W 3 , where the Riemannian manifold W 3 is not projectively flat? Is there a generalization of (C") to immersions vn ~ IRq for n + 2 ~ q ~ 2n - I? Under what conditions on (the dimensions of) vn and wq all isometric (or C2-neady isometric) C2-immersions f: vn ~ w q satisfy Diamf(Vn) ?: const = const(Vn, wq).

(D) I soperimetric Inequalities for Surfaces. Let V be a compact connected surface and let w+ = Iv max(K(v) + 1,0) dv. Then the area A = A(V) admits the following bound in terms of the length L = L(aV), of the Euler characteristic X = X(V) and of the integral w+.

Basic Inequality (Fiala 1941; Ionin 1969).

(*) L2/2 ?: A(AI2 - w+ + 2nx).

Proof Let V(t) = {VE Vldist(v,aV)?: t} c V. Then

(1) L(t) ~ L(aV(t)) = _ dA~(t)) .

If the boundary curve aV(t) is smooth, then by Gauss-Bonnet,

dLd(t) = r K(v)dv - 2nX(V(t)). t JV(t)

For non-smooth curves one still has the inequality

dL(t) ~ r K(v) dv - 2nX(V(t)). dt JV(t)

Since X(V(t))?: X(V) for t ?: 0, and since IV(t)K(v)dv ~ w+ - A(V(t)), we get

dL(t) (2) dt ~ w+ - A(V(t)) - 2nX·

Now, we obtain (*) by multiplying (1) with (2) and then by integrating in t over [0,00].

Exercise. Fill in the detail in this proof.

A Modification of (*). Let w_ = Ivmax(K(v), -1)dv and let I> = 2nX - w_IA. (a) If I> ?: 1, then L ?: A. (b) IfO ~ I> ~ 1, then L?: 2nX - w_.

Proof Since Iv(t) K(v) dv :<:;; OL + A - A(V(t)), we obtain with the above

iT dL(t) iT dA(V(t) - L(t)~ ~ (w_ - 2nx + A - A(t)) d .

o dt 0 t

We get (a) with T = 00 and we obtain (b) with T = To, such that A(To) = (1 - 6)A.

Now, let V admit an isometric C2-immersion f into the unit ball B3 c [R3. Then V satisfies the following (refinement of a special case of)

Inequality of Burago (1968). Let Wo = Iv max(K(v), 0) dv. Then

(**) U ~ A(C1A - C2WO + C3 X),

where Cl = C02, C2 = 2CJ and where C3 = C3(V;j) satisfies 4nCo2 :<:;; C3 :<:;; 4nC5 for the constant Co of Lemma (C').

Proof We get with (C') the inequality w+ (g_) :<:;; C5wo and we apply (*) to the length and the area of(V,g_),

L:/2 ~ A_ (A_/2 - C5Wo + 2nx).

Then (**) reduces to the following inequalities which are immediate with (C').

C5L2 ~ L: and COl A :<:;; A_ :<:;; CoA.

Corollary. If x(V) ~ 0 and K(V) :<:;; O. Then Area V:<:;; C1l/2 length av.

Exercises. (a) Let a surface V in the unit ball B3 c [R3 have X(V) ~ 0 and K(V) :<:;; (2Cofl. Show that Area V:<:;; C length av for some universal constant C > O. [Compare Jorge-Xavier (1981).]

(b) Find a sequence of smooth imbedded surfaces V; c B 3, i = 1, ... , of nonpositive curvature with the following two properties,

(1) The boundaries aV;, i = 1, ... , are equal to a fixed connected smooth curve a = aVl = aV2 = "',

(2) X(V;) --+ - 00 for i --+ 00 and Area V; ~ J - X(V;) for i = 1,2, ....

(c) Let V' = {VE VIK(v) ~ (2Cofl} and w' = Iv,K(v)dv. Prove (**) with w' in place of Wo (and with some constants C; instead of C;).

(d) Let the boundary av be connected, let Wo :<:;; n and let V admit an isometric C2-immersion V --+ [R3. Show that 10L4 ~ A(C1A - C2nU + C3 XL2) and find for all X < 0 examples of (non-immersible into [R3!) surfaces V which violate this inequality.

(E) Conformal Maps and Isoperimetric Inequalities. A Cl-map between two oriented Riemannian manifolds, f: vn --+ wn is called (orientation preserving) conformal if the Jacobian of f is related to the differential of f by the equality J(v) = IIDvlln for all v E V. A CO-map f is called quasiconformal if the (distribution) differential locally has I II Df II n dv < 00 and if J (v) ~ CII Dv II n for some constant C > 0 and for almost all v E V.


Consider a proper function x: V --+ IR+ and let V(t) = x-1 [0, t] C V. Such an increasing family of compact subsets V(t) c V is called an exhaustion of V. Any monotone change of the parameter t --+ e E IR+ gives another labelling of the exhaustion V(t). Let us assume V to carry a Riemannian metric and let us parametrize (label) a given exhaustion V(t) with the following

Conformal and Subconformal Measures in IR+. A measure de for a monotone increasing function e = e(t), t E IR+, is called subconformal [relative to V(t)] if for every two points tl and t2 > tl in IR+, and for an arbitrary continuous function h: x-1 [tl ,t2] --+ IR+ there exists a point t3 E [tl' t2] such that

where the function hn is integrated with respect to the (n-dimensional) Riemannian measure on V and hn - 1 is integrated over the (n - I)-dimensional measure on the fiber X- 1(t 3 ). Here and below we assume the fibers x-1(t) to be rectifiable (n - 1)dimensional subsets in V. Moreover, to avoid a trivial mess we assume the integral fx-l(t) F to be continuous in t for all continuous functions F on V.

Measures majorized by subconformal measures are subconformal. Therefore, there exists a unique maximal subconformal measure de which is called the conformal measure.

Example. Let V be complete and let V(t) be the ball of radius t around a fixed point in V.·Then the measure de = (VolaV(t))l/(l-n)dt obviously is conformal. (The boundary aV(t) may be properly contained in the sphere {v E Vldist(vo, v) = t}, but we assume this does not happen for our manifold V).

Exercises. (a) Assume the function x: V --+ IR+ to be smooth, put cp(t) =

(fx-'(t) Ildxll n- 1 )1/(1-n) and show that cp(t)dt is the conformal measure. (b) Prove the conformal measure to be invariant under conformal changes of

the Riemannian metric in V. (c) Let x: V --+ W be a smooth proper map, such that dimx-l(w) = m, WE W

Take a covector IE T':(w), pull back by Dr T':(W) --+ T,,*(V), v E x-1 (w), and set 11111 = (fx-'(w) IID]lllm)l/m. Prove that the (Finsler) metric in W defined by the dual norm on T(W) is a conformal invariant of V. Study similar metrics on submanifolds Z c W for xlx-l(Z): x-1(Z) --+ Z and use these to generalize the definition to non-smooth maps x.

Subcoriformal Exhaustions. An arbitrary exhaustion V(t) of V can be reparametrized bye = f de for some subconformal (e.g. the conformal) measure de. Then V(e) =

V(t( e)) is called a subcoriformal exhaustion and the range of e is denoted by (e_, e+) c IR.

Let W, dim W = dim V, be a manifold with a Riemannian metric g, whose (oriented) volume form is denoted by OJ, and let f: V --+ W be a conformal map. Set A(e) = Voig V(e) ~ fv(e)f*(OJ) and let L(e) = VolgflaV(e) which by definition is the (n - I)-dimensional volume of f(aV(e)) c W counted with the geometric multiplicity.

Lemma. If the exhaustion V( e) is subconformal, then

(+ ) 18+ (L(e)!A(e))n/(n-l)de ::;; const < 00,

8 0

Proof Let h(v) = IIDvfll. Since the map f is conformal,

A(e) = r hn and L(e) = f hn- 1. JV(8) av(8)

Since the exhaustion is subconformal,

and so

y(T) = f: (L(e))n/(n-l) de ::;; A(T)

for all T E (e _ , e +). Therefore,

18+ (L(e)/A(e))n/(n-l)de ::;; 18+ y-n/(n-l) :~ ::;; (n - l)(y(eo))l/(l-n) < 00.

8 0 8 0

Parabolic Exhaustions. An exhaustion is called parabolic if e+ = 00 for the conformal parameter of this exhaustion.

Example. If the concentric spheres sn-l(R) around some point in a complete manifold V have

Loo [VoISn-1(R)]1/(1-n) dR = 00,

then the exhaustion by the balls Bn(R) is parabolic.

Now the Lemma yields the following

Corollary. For parabolic exhaustions A(t),

(+ +) liminfL(t)/A(t) = o. t .... oo

The inequalities (+) and (+ +) show that the integral A(t) = JV(td*(w) is rather stable under reasonable changes of w. Namely, let w' be a form equivalent to w in the sense that the difference w - w' is a differential of a bounded measurable form on V, say w - w' = da. Then A'(t) = JV(td*(w') satisfies by Stokes' theorem

I(A(t) - A'(t))/A(t)1 ::;; IlaIIL(t)/A(t),

and so

liminfl(A(t) - A' (t))/A(t) I = 0 t .... oo

in the parabolic case. In particular, if w is equivalent to zero (i.e. w = du), then every conformal map f: V -+ W is constant, provided (some exhaustion of) V is parabolic.

Exercises. (a) Show, by making w = du, that the following manifolds W receive no non-constant conformal maps from parabolic manifolds without boundary (e.g. from IRft),

(i) W is compact connected and has a non-empty boundary. (ii) W is complete, the sectional curvature of W is everywhere negative ~ -1 and

the fundamental group n1 (W) is solvable. (iii) W is a (Riemannian) normal covering with a free non-Abelian Galois group of

an open subset U cWo, where Wo is a closed Riemannian manifold and the complement E = Wo \ U is a smooth simplicial subcomplex in Wo. For example, U equals S2 minus three points.

(b) Generalize ( + ), ( + +) and the above (a) to quasi-conformal maps.

Distribution of Values W = f(V)E W. The above bound on IA(t) - A'(t)1 may fail for forms w' = w + d if the (distribution) form u is not bounded. Consider, for example, the singular form (current) bw on W for some WE W which (viewed as a measure) consists of a single atom at W of mass one, fwbw = 1. Then the integral Nw(t) = fv(td*(bw)dv equals the number of solutions VE V(t) to the equation f(v) = w. Observe that fw Nw(t) dw = A(t) for all t. Assume Vol W < 00 and introduce the error (often called the defect) dw(t) = 1 - (Vol W)Nw(t)f A(t). If the map f misses w, then dw(t) = 1 and so IA - A'I = IA - Nwl = A.

Now, let dim W = dim V = 2. ·Put K(t) = fV(t) K(f(v))f*(w) for the curvature K(w) of W. In other words K(t) is the integral curvature of V(t) for the metric induced from W outside the singularities of the map f. If K(w) ~ -1, WE W, then the modified inequality (*) implies L(t) ~ min(A(t),2nX(t) - K(t)) for X(t) = X(V(t)) for all t. In fact, since the map is conformal, the singularities of the induced metric 9 in V correspond to the branching points of f (the curvature of the induced singular metric has the atomic mass - 2nm at every branch point where the differential Df vanishes with order m) and these singularities may be removed by a smooth approximation g' of g. Hence, the isoperimetric inequalities of (D) apply to metrics with isolated singularities (with an obvious definition of the curvature at the singular points). Now, we have with M2(t) = [max(O,2nX(t) - K(t)]2 the following relation for the conformal parameter e,

(+ + +)

We apply this inequality to the following very special class of metrics 9 in W: The metric 9 has constant negative curvature -1 outside a discrete subset {Wi}' i = 1, 2, ... , in Wand the singularity at Wi E W carries an atom of positive curvature, say Pibw for 0 < Pi < 2n. Every such metric is locally isometric to the sector of the angle 2n ~ P in the hyperbolic plane with the two sides identified. For example, the

boundary W of every convex polyhedron in the hyperbolic space H3 carries such a metric whose (intrinsic!) singularities are the vertices of the polyhedron.

If W is a closed surface with our special metric, then

Area W = L Pi - 2nx(W) i

and

K(t) = -A(t) + LPiNwi(t). i

Therefore, the inequality ( + + + ) provides a non-trivial relation between the errors (defects) dw,(t) for t --+ 00, in case the exhaustion V(t) is parabolic.

Finally, observe that the defects dw.(t) are rather unsensible [like A(t) see above] , to a choice of a (singular or non-singular) metric in Wand so the inequality ( + + + ) for singular metrics may yield defect relations for smooth ones. More precise statements and further results are in the following references and exercises.

References. Our brief exposition of conformal maps followed Ahlfors' geometric approach to Nevanlinna's value distribution theory. This theory and various generalizations can be found in Hayman (1964), Griffiths (1974), Cowen-Griffiths (1976), Rickman (1983), Mattila-Rickman (1979).

Exercises. (a) Let W be the torus T2 with a smooth metric and let V = 1R2, exhausted by concentric balls V(t) = {llvll :s; t}.Letd~(t) = max(O,dw(t) - e) and show that

100 (d~(t))2 dlog t < 00,

for all WE W, for all e > 0 and for all non-constant quasi-conformal maps f: V--+ W.

Hint. Use a metric on T2 whose curvature mass at the only singular point WE T2 equals 2n - e.

(a') Prove a similar defect relation for quasi-conformal maps 1R2 --+ S2 by using metrices on S2 with k > 2 = X(S2) singular points.

(b) Let V be an infinite cyclic covering of a closed surface Yo. Show that every non-constant quasi-conformal map f: V --+ S2 misses at most two points.

(b') Exhaust V by concentric balls V(t) (for the Riemannian metric induced from Yo), consider a quasi-conformal map f: V --+ T2, for which A(t)/t --+ 00, and show that Sf (d~(t)f dlog t < 00 for the above d~. Modify this for maps V --+ S2.

(c) Let V and W be complete Riemannian manifolds with the following properties.

(i) W is a regular (Riemannian) covering of a compact manifold Wo, such that the Galois group contains a free non-Abelian subgroup.

(ii) The unit ball Bv(1) c V satisfies Vol Bv(1) :s; C < 1 for all v E V.

Prove each quasiconformal map f: V --+ W to be uniformly continuous. Moreover, dist(f(vd,f(v2)) ~ const(Wo, C, d) for d = dist(v1 , v2).

(d) Let V be a surface with a parabolic exhaustion V(t) and let f: V --+ 1R3 be a conformal immersion, that is the induced metric g on V equals h2go for the original metric go on V and for some positive function h on V. Let A(t) = Areag V(t), D(t) =

Diamf(V(t)), X(t) = X(V(t)) and K+(t) = SV(t)max(O,Kg(v))(dv)g' Show that

limsupD2(t)(K+(t) + IX(t)l)jA(t) ~ e > 0

for some universal e > 10-8 .

3.2.4 Isometric Coo-Immersions V2 --+ IRq for 3::5 q ::5 6

Let us recall the classical result of Alexandrov (1944) and Pogorelov (1969) on

Elliptic Isometric Imbeddings S2 --+ M3[K]. If a COO(can)-metric on the sphere S2 has curvature K(s) > K then there exists an isometric COO(Can)-imbedding f: (S2,g)--+ M 3[K], where M3[K] is the standard (spherical for K > 0, Euclideanfor K = 0 and hyperbolic K < 0) 3-space of constant curvature K. (This imbedding is unique up to isometries of M3[K].) In particular, every metric g admits an isometric imbedding into some hyperbolic 3-space.

The proof (see Pogorelov 1969) is based on the continuity method, where the starting point is the following fact: the space of metrices g on S2 for which Kg > K

is connected for all K E IR.

Proof The claim is obvious for K < O. To handle K ~ 0 we use the conformal representation g = e2'Pgo for the standard metric go on S2. A straight-forward computation gives Kg = e- 2'P (l - A<p) for the Laplace operator A = Ag. Hence, the relation Kg > K reduces to Ke2'P < 1 - A<p. Since the function e2'P is convex, the space of solution <p of this inequality is a convex subset in the space of functions. Q.E.D.

Exercises. (a) Let V be an arbitrary (possibly non-compact) manifold of dimension n ~ 2 and let G(a, b), - 00 ~ a < b ~ 00, denote the space of complete metrics g on V whose sectional curvatures satisfy a < Kg < b. Show the space G(a, b) to be weakly contractible (i.e. non-empty, path connected, with zero homotopy groups 1ti

for i ~ 1) in case a < 0 < b. (b) Show the group Diff S2 to be weakly homotopy equivalent to the orthogonal

group 0(3).

Hint. Compare the spaces G'jDiffS2, where G' is the (contractible!) space of me tries g on S2 with no symmetries, and cP' jConf S2, where cP' is the space of functions <p on S2 for which e2'Pgo E G and Conf denotes the group of conformal automorphisms of S2.

(b') Determine the weak homotopy type of Diff V for an arbitrary surface V. Hint. Use the conformal representation g = e2'fJgo for all metrics g on V, where go is a metric of constant curvature.

Hyperbolic Immersions D2 ~ 1R3. Take an isometrically imbedded interval Y =

[y 1, Y 2] in a surface V and assume the normal exponential map exp: Y x IR ~ V to be a smooth imbedding on Y x [Xl' X2]. Then the metric g of V is expressed in the coordinates X and y by g = dX2 + qJ2 dy2 for some function qJ = qJ(x,y) (which is determined by (V, g) and Y c V). Let /.: D2 ~ 1R3, for D2 = exp(Y x [X 1 ,X2]), be defined by the functions x, aqJsina-1y and aqJcosa-1y. Then the induced metric ge on D2 is g + a2(dqJ)2 (see 3.1.1).

Example. Let V be a complete simply connected surface of non-positive curvature and let Y be a double infinite geodesic in V. Then the map exp: Y x IR ~ V is a diffeomorphism and so the metric g is globally dX2 + qJ2 dy2. Therefore, every compact disk D2 c V admits an a-approximate isometric immersion D2 ~ 1R3 for alIa> O.

It is unknown for general Coo -manifolds V whether the immersion/. for small a> 0 admits an approximation by isometric COO-immersions. However, such an approximation was obtained by Posnjak (1973), in the following two cases,

(a) the curvature K(V) is negative, (b) the surface (V, g) is real analytic.

Hence, in these two cases the disk D2 c V admits an isometric Coo(Can)-immersion into 1R3.

Counterexample. The above approximation result may fail for C2-metrics. Moreover, Pogorelov (1971) found a C2-metric g on V whose curvature K(v) is Lipschitz, but no neighborhood of some fixed point Vo E V admits an isometric C2-immersion into 1R3.

Isometric Immersions into Warped Products. Consider a manifold V wirth a metric go. Let b(t) and r/!(t) for tl < t < t2 be Can-functions, where r/!(t) > 0 for all t E (tl' t2)' Then we define the warped metric r/!(t)go + b(t) dt2 on W = V X (tl' t2), which is positive for b > O.

(A) Examples. (a) Let go be the metric of constant curvature + 1 on sn. Then

(i) the metric t2 go + b dt2, 0 < t < 00, equals the standard flat metric on IRn+1 minus the origin for all constant b > 0;

(ii) the metric (sin t)2go + dt2, 0 < t < ~, is the standard metric of curvature + 1 on the hemisphere S~+l minus the center;

(iii) the metric (sinh t)2 go + dt2, 0 < t < 00, is the standard metric of curvature -1 on the hyperbolic space Hn+1 minus a point, where sinh t = t(e' - e-').

(b) Let go be the standard flat metric on IRn. Then e2tgo + b dt2 is the metric of constant curvature - b for all constants b > O.

(c) Let go be the metric of constant curvature -1 on H" = M"[ -1]. Then the metric (cosh t)2go + dt2, - 00 < t < - 00 is isometric to that on H"+l, where cosh t = !(e t + e- t ).

Metrics on Graphs. Consider a C 1-function h: V ~ (tl' t2) and observe that the graph of h, called r h: V ~ (w, t/J(t)go + o(t) dt2), induces on V the metric g~ = t/J(h(v))go + o(h(v))(dh)2. This generalizes the basic (twisting) formula in 3.1.1 for the flat metric t2go + e2dt2 on 8" x (0,00) = 1R"+1\0.

Let the derivative t/J'(t) be nonvanishing on some subinterval {t1 ,t2] c (tl' t2) and let t/J(to) = lfor some to c [t~, t;J. Let us parametrize [t1 ,t2] by 8 = ! log t/I(t), set 8 i = 8(t;), i = 1, 2, and let the following two conditions be satisfied for some A>l.

(1) 18;! > A, i = 1,2; (2) the C5-norm of the function b(8) = o(t(8))(dt/d8f8 E [81 , 8 2 ], satisfies

Ilb(8)11cs ::;; A-1.

Now, let V be a connected compact surface (with or without boundary) and let {gx} be a smooth family of Coo-metrics on V where the parameter x runs over some Euclidean space IRP. Recall that the conformal equivalence classes of the metrics on V close to go are parametrized by a finite dimensional ball B' [in the Teichmiiller space for some choice of the basis in 11:1 (V)] according to Poincares uniformization theorem. Let us relate to each metric gx its conformal class [gx] E B' for x close to zero and assume that the map x H [gx] is a diffeomorphism on a small neighborhood U c IRP around the origin. (This implies, of course, p = r.) Finally, take a COO-metric 9 in the conformal class [go], which means the existence of a function ho on V for which (V, e2hogo) is isometric to (V, g).

(B) Proposition. lfthe above constant A = A(t/J, 0) is sufficiently large, A:?: e-1 for a small number e = e(V, g, {gx}) > 0, then there exists a COO-function h on V such that the metric g; = t/J(h(v))gx + 0(h(v))(dh)2 is isometric to gfor some x E W close to zero. Furthermore, if 9 and gx are real analytic, then the function h and the implied isometry (V, g) ~ (V, g;) are also real analytic.

Proof Switch to the parameter 8 = ! log t. Then g; = e2hgo + b(h)(dh)2, where Ilb(8)11cs ::;; e for 81 ::;; 8 ::;; 82 • Let g(qJ) = e2'fJg - e2'fJb( -qJ)(dqJf and assume the existence of an isometry f: (V,g(qJ)) ~ (V,gJ. Then the metric g; for h(v) = - qJ(f-l(V)) is isometric to g. Thus our problem is reduced to finding an isometry f for some x E IRP and for some function qJ on V.

Let us write g for the volume form of 9 in case the surface V is oriented and let w(g) denote the curvature form of 9 that is Kgg of 9 for the curvature (function) Kg of g. By a classical formula,

for the Laplace operator L1 = L1g on 2-forms. Since the metric g(qJ) is induced from a warped product, the curvature of g( qJ) is a differential expression of second (not

third!) order in cp by Gauss theorema egregium. Hence,

(**) w(g(cp)) = w(g) + A(cpg) + D.(cp)

where D. is a non-linear differential operator of second order whose coefficients depend on 9 and -g, such that IID.lle3 ~ 0 for e ~ O. (This is the e which bounds II-gllco.)

Fix a function CPo for which e2",og is isometric to go and let cp be C2-close to CPo. Then there exists a unique x = x(cp) such that g(cp) is conformally equivalent to gx. The conformal equivalence f: (V,g(cp)) ~ (V,gx) may be not unique [for X(V) ~ 0] but we agree on some canonical choice off = f",. Denote by I(cp) the induced 2-form f",*(w(gx)) for x = x(cp).

Lemma. If e = e(V, g, {gx}) > 0 is sufficiently small, then the functional equation

w(g) + A(cpg) + D.(cp) = I(cp)

is Coo (Can)-solvable in cp. Moreover, this solution exists and is unique with thefollowing normalization,

(a) Jve2"'g = Jvgx, in case V has no boundary; (b) The 2{ormsf",*(w(gx)) and e2"'g are equal at the points VEOV, in case oV =1= O.

Proof We treat only (a) and leave (b) to the reader. B.: cp ~ w(g) + Acpg + D.(cp) -I(cp) sends functions cp normalized by (a) to 2-forms w for which J w = O. If a function cp is of Holder class Ci ,,., i = 2, 3, ... , 0 < oc < 1, then B.(cp) is Ci - 2'''-smooth. Moreover, the operator B.C'" ~ Ci - 2 ,,. is differentiable. Its linearization (differential) at CPo E Ci ,,. is a small perturbation of the Laplace operator A which sends Ci'''-functions cp with JvCPg = 0 to Ci - 2'''-forms w = Acpg with Jvw = O. Since A is an invertible operator, the operator B. is invertible near CPo by the implicit function theorem. Thus the lemma is proven.

Exercise. Fill in the details in the above argument.

If cp satisfies B.(cp) = 0 then the conformal map f",: (V,g(cp)) ~ (V,gx) is curvature preserving. Therefore the Jacobian'" of this map satisfies A log '" = 0 for A = Ag(",)

and so '" == 1 under conditions (a) and (b). This completes the proof for orientable surfaces V. The non-orientable case is treated by passing to the oriented double covering of V.

(B') Isometric Immersions V ~ W 3 for K(W3) = -&-2, Let V be a compact connectedsurface with a Coo (Can)-metric. Thenfor every positive e < eo = eo(V) > o there exists a 3-dimensional manifold W of constant curvature -e-2 which receives an isometric COO(Can)-immersionf: V ~ W. Moreover, one can take W andf, such that

(i) If V is (homeomorphic to) 82, then W 3 is the hyperbolic space H3 (of curvature _e-2 ) andfis an imbedding.

(ii) If V is p 2 then W is obtained from H3 by dividing H3 minus a point Wo E H3 by the geodesic symmetry in woo This W is homeomorphic to p 2 x ~ and it is not complete (at wo). The map f is an imbedding.

(iii) If V is a closed surface with X (V) ~ 0, then W is homeomorphic to V x IR and complete. The map f is an imbedding.

(iv) If V is orientable with a non-empty boundary, then W = H 3, but one cannot (?) guarantee an imbedding.

(v) If V is non-orientable and av is non-empty, then one may take the above (H3\wo)/7L2 for W One may also take a complete W which is homeomorphic to (Klein bottle) x R In neither case one guaranties an imbedding.

Proof Let V be closed and let gx be the metrics of constant curvature + 1, 0 or -1 on V parametrized by the Teichmiiller space X. Thus every metric g is conformal to some gx and this gx is unique if we specify the homotopy class of the conformal equivalence. Now, we observe [compare (A)J that the warped product W =

(v x 1R,821jJ2gx + 82dt2), whereljJ(t) = sinhtfor K(gx) = + 1, ljJ(t) = etfor K(gJ = 0 and ljJ(t) = cosh t for K(gx) = -1, has constant negative curvature _8- 2. One can easily verify that J( e) is small for 8 small, and (B) applies.

Now let V = (V, g) have a boundary and let Vo = (VO, go) be a complete flat manifold which is non-orientable in case V is non-orient able. Then by the uniformization theory for surfaces with boundaries there exists a family of Can-immersions fx: V --+ Yo, such that the induced flat metrics gx on V satisfy the assumptions of (B). [If V = D2, for example, we need a single conformal immersion (V, g) --+ Vo which exists by the Riemann mapping theorem.J Thus we obtain with (B) an isometric immersion of V into W = (Vo x 1R,82e2tgo + 82 dt2). In a similar way one may use the projective plane p 2 with K(P 2 ) = 1 for Yo.

(B") Exercises. (a) Let V = (V,g) be a non-compact surface and let Vo = (Vo,go) be a closed non-orientable surface for which X(Vo) ~ 0 (i.e. Vo is either p 2 or the Klein bottle). Prove the existence of a conformal immersion V --+ Yo. [Compare GunningNarasimhan (1967).J

(b) Let V c H3 be an immersed elliptic (i.e. locally convex) non-compact connected surface for which the orthogonal projection to a hyperbolic plane, say P: V --+ Ho C H 3, is a proper map such that dist(v,P(v» ~ const < CI) for all VE V. Show that, in fact, V is embedded, the projection P is a homeomorphism and V lies in a halfspace, say in H~ c H3, bounded by Ho.

(b') Let the curvature of the induced metric in V satisfy -1 ~ a ~ K ~ b < 0 and show that 0 ~ Ca ~ dist(v, P(v» ~ Cb for all V E V, where C" denotes the constant for which the hyper surface {wEH3 Idist(w,Ho) = C,,} has curvature K. In particular, if K == const = K, then dist(v, P(v» == const' = C".

(b") Let V = (V, g) be a closed surface of genus ~ 2. Show the warped product manifold W of curvature -8-2 which isometrically contains V [see (iii)] to be unique up to isometry for given V and 8. Then show the isometric immersion V --+ W to be also unique up to an isometry of W Prove a similar uniqueness of isometric immersions of flat tori into hyperbolic manifolds.

(c) Let W+ (x) = (V x IR+, (cosh tfgx + dt2) be the warped product of curvature - 1, where the metrics gx on V of curvature - 1 are parametrized by the Teichmiiller space X '3 X [which is homeomorphic to the Euclidean space of dimension


6(genus V) - 6]. Let fE be the space of pairs (XEX,J: V -+ W+(x)) where f is a strictly elliptic immersion whose projection to V x 0 c W+(x) is homotopic to the identity. Show the differential operator E0: fE -+ G assigning to each (x,f) the metric g E G on V induced by f to be a Fredholm map for appropriate Banach manifold structures in fE and in G.

(c') Show the Fredholm index of E0 (that is the index of the linearization of E0 at any point in g{) to be zero.

(c") Show with (b") every metric go E G on V of constant (negative) curvature to be a regular value of E0 and observe that E0-1 (go) consists of a single point in gr.

(d) Prove [with Pogorelov's (1969) a priori estimates of the extrinsic curvatures and their derivatives of convex surfaces in H 3 ] the operator E0 to be a proper map (in suitable topologies) ofg{ into the subspace G+ c G of met rices g with K(g) > -1.

(d') Prove with (c") and (d) the map E0: fEOO -+ G~ to be of Fredholm degree = 1 and, hence, onto where 00 refers to the subspaces of Coo-maps and Coo-metrics in fE and in G+.

(d") Sharpen (B') by showing eo(V) ~ infvEv ( -Kv(V,g)t2 for closed surfaces V of genus ~ 1 (compare the existence claim in the Alexandrov-Pogorelov theorem of genus = 0)

(dill) Show with (B) in 3.2.3, that no such bound on eo is possible for metrics g on the disk D2.

Question. Let V be a closed surface with finitely many punctures (or a more general open surface) and let W(x) be some family of hyperbolic 3-manifolds. What are the properties of the pertinent operator E0?

Isometric Immersions V -+ M4[K]. There is no single known obstruction for isometric COO-immersions of surfaces into Riemannian 4-manifolds. On the positive side one has the following

(C) Theorem. Let V = (V,g) be a Coo(Can)-surface which is either homeomorphic to the torus T2 or where V is compact orientable with a non-empty boundary. Then V admits an isometric coo(c an ) immersion into an arbitrary small 4-ball with constant curvature K for all K E [R.

Proof First let av i= 0. Then there is a family of Can-immersion fx: V -+ [R2 such that the induced metrics gx satisfy the assumption of (B) (compare the previous proof). Therefore, (V, g) isometrically immerses into w" = (([R2 X [R, (e-1t)2go + dt2) for the flat metric go on [R2 and for all small I' > O. Then we take the isometric Can-immersion of w" into [R4 = M4[0] which is the cone over the obvious isometric immersion of [R2 into the e-sphere S; c [R4. The case K i= 0 is treated in a similar way. Furthermore, the case V = T2 reduces via (B) to the following

Lemma (B. Lawson). Let V be aflat torus. Then there exist isometric Can-immersions V -+ S~ for some sequence ei -+ O.

Proof Consider the Hopf fibration S3 -+ S2 and observe that the pull-back of an arbitrary closed curve in S2 is an (intrinsically) flat torus in S3 = Si. One finds


among these tori and their finite covers the homothety class of each flat metric on T2 (an easy exercise) and the lemma follows.

Exercise. Use an approximation of isometric immersions (V, g) --+ S; by nearly isometric imbeddings (V, g) --+ S;, and show that every compact connected oriented surface V with boundary admits an isometric Coo(Can)-imbedding into [R4.

Question. Let a compact surface V = (V, g) (with or without boundary) admit a non-zero vector field. Does there exist a (nearly) isometric C2-immersion V --+ S.3 for small e :-:::; eO (V, g)?

(C') Isometric Immersions V --+ M4[ -8-2]. An arbitrary compact Coo(can)-surface V = ( V, g) admits an isometric Coo (can )-immersion into the hyperbolic space M4[ -e-2 J of curvature _e- 2 for all positive e :-:::; eo = eo(V) > O.

Proof In order to apply (B) to the warped product M4[ -e-2J = ([R3 X [R,e 2e2tgo + e2 dt 2 ) we need a family of Can-immersions fx: V --+ [R3 whose induced metrics gx uniquely represents all conformal classes of metrics on V near [g]. To construct fx we start with an arbitrary family g~ whose conformal classes run over the Teichmiiller space X. Then we have by the Nash-Kuiper immersion theorem a C1-map F: V x X --+ [R3 which is isometric on each fiber (V = V x x, g~). Finally we C1-approximate F by a Can_map f: V x X --+ [R3. The conformal classes of the induced metrics 9 on V = V x x define a Can_map of X into itself, x H [gxJ EX, which is CO-close to the identity. Since X is a manifold, every small (in the fine CO-topology) perturbation of the identity is a surjective map. We cannot always guarantee this map to be a global diffeomorphism, but it is a local diffeomorphism over a fixed class [gJ E X provided f is generic. Q.E.D.

Exercises. Construct isometric Coo(Can)-imbeddings V --+ M4[ -e- 2 J for surfaces V with a non-empty boundary and for closed ones with even Euler characteristics [if V = p 2 and K(V) > _e- 2 , then no isometric CZ-imbedding V --+ M4[ -e-zJ is possible, see (e') in (E) of 3.2.2].

Prove (all forms of) the h-principle for conformal Coo(can)-immersions V --+ W, dim W > dim V = 2 for arbitrary Coo(Can)-metrics on V and W Study conformal imbeddings V --+ W

Let X be a (possibly singular) real analytic space (e.g. an affine real algebraic variety). Show that every CO-small continuous perturbation of the identity map X --+ X sends X onto X.

Question. Let r be an (integrable or not) 2-dimensional subbundle of T(V), dim V = n > 2. Does the h-principle hold true for those Coo-maps V --+ W, dim W > 2, which are conformal on r for given Riemannian metrics on V and on W?

Isometric Immersions V --+ M5[K]. In the following three cases a compact surface V admits an isometric Coo(Can)-imbedding into the space M 5[KJ of constant curvature for all K E [R. [See (D) below for a more general result.J


(i) V is homeomorphic to S2, T2 or to the Klein bottle. (ii) V is connected with a non-empty boundary.

(iii) V is orientable and there exists an isometric orientation reversing involution I: V --+ V whose fixed point set is non-empty. In this case there is an equivariant isometric imbedding V --+ MS[IC] which is symmetric in a hyperplane in MS[IC].

Proof The sphere S2 with any metric imbeds into M 3 [ _e-2] and each ball in M 3[ _e-2 ] isometrically immerses into MS[IC] [see (D) in 3.2.2]. The resulting immersion can be Coo (Can)-approximated by an isometric imbedding. (This approximation is left to the reader.)

The case of V = T2 was studied in (C). The proof of (C) reduces the case of the Klein bottle and (ii) to isometric Can-immersions of the flat Klein bottles K2 into S4 = M4[e- 2 ]. To obtain these we take the b-neighborhood U6 of the projective line in p.4 = S:1712 whose boundary is a flat K2. If b is small compared with e, then U6 admits an isometric Can-immersion into S: and we find each flat K2 among finite covers of au{) [compare the Exercise (b") in (F) of 3.2.1]. Thus we get isometric immersions into MS[IC]. Passing to isometric imbeddings is an exercise for the reader.

Now, in case (iii), we consider the quotient Vo = VII which is a smooth surface with a boundary. Then for an arbitrary symmetric metric g on V one can easily find a Can-function ifJ on V with the following three properties,

(a) ifJ(l(v)) = - ifJ(v), v E V. (b) The form g - (difJ)2 is positive definite outside the curves fixed by 1. (c) There exist a Coo (Can)-metric go on Vo and a Coo (Can)_map ifJo: V --+ Yo, such that

ifJ~(go) = g - (difJ)2 and ifJo(Iv) = ifJo(v).

Finally, we take an isometric imbedding fo: (VO, go) --+ 1R4 and take (ifJ, ifJo 0 fo): V --+ IRs for the required imbedding for IC = O. The case IC "# 0 is left to the reader.

Corollary. There exists a closed Can-surface V c: IRS with a metric of constant curvature -1. Therefore, there exists an isometric Can-immersion H" --+ IRS"-S for Hn = Mn[ -1] and for all n = 2, 3, .... [Compare (D) in 3.2.2.]

Immersions V --+ M(j[IC]. An arbitrary (possibly non-compact) COO(Can)-surface (V, g) admits an isometric COO(Can)-imbedding into M6[IC] for all ICE IR.

Proof If V is compact the proof is immediate with (B). Indeed, we have a family of Can-immersions fx: V --+ 1R3, XEX, for which the conformal classes of the induced metrics gx on V cover a neighborhood of the class [g] in the Teichmiiller space X. Then we use the isometric immersion I.: 1R3 --+ SS c: 1R6, e > 0, given by e sin e-1Yi and ecose-1Yi for the coordinates Yi' i = 1,2,3 in 1R3. Since g is conformal to gxo for some XoEX, we may write g = ifJ2gxo' We observe that the immersion F. = ifJ(I. 0 fxo): V --+ 1R6 is e-isometricfor ifJ2gxo' as the induced metric is ifJ2gxo + 3e2(difJ)2. Finally, we perturb F. to an isometric immersion F: (V, ifJ2gxo) --+ 1R6 = M6[0] by applying (B). We leave to the reader the case IC "# 0 as well as the elimination of double points of F to obtain imbeddings.

Now let V = (V, g) be non-compact and let.E1 c V be a disjoint union of simple closed double-sided curves in V, such that every component of the complement V\.E1 is relatively compact in V Let .E2 and .E3 in V be small deformations of .E1 such that some tubular neighborhoods Vi c V of .Ei, i = 1, 2, 3 do not intersect.

Sublemma. There exist C"junctions Yi and t/li on V, i = 1, 2, 3, such that g =

Lf=l t/I?(dyy and such that the support oft/li equals V\ VJor i = 1,2,3.

Proof The equation Lf=l t/li2 dyt = g is easily solvable on the union V = V1 U V2 U V3 C V Indeed, V is a didjoint union of cylinders Sl x [0,1J and so there is a conformal Coo-immersion V -+ [R2 which amounts to a Coo-decomposition gl V =

eP(dzi + dzD. On the other hand, the convex integration applies to Yi outside V for t/li > 0 being fixed. Thus one obtains a global decomposition g = Lf=l <Pi2 dzt where the functions <Pi are Coo but Zi, i = 1, 2, 3 are only C1-smooth. Passing to a COO -decomposition is immediate in the case where V is simply connected since for any COO-triple (Y1,Y2,Y3) which is C1-close [in the fine C1-topology to (Zl,Z2,Z3)] the metric .E<p2 dy2 is conformal to g. Hence, (V, g) is isometric to (V, e~.E<p? dyt) for some COO-function rJ. on V Furthermore, if n 1 (V) is finitely generated, then the Teichmiiller space X of the conformal classes of metrics on V is finite dimensional and the approximation argument applies to families gx = .E<p? dz~x for which [gxJ = x E X. Finally, the desired approximation is obtained on an arbitrary surface V by applying the above to an increasing family of compact smooth domains Jj c V,j = 1,2, ... , which exhaust V

Exercise. Fill in the details in the argument. Then prove (all forms of) the hprinciple for COO-solutions t/li and Yi to the equation Lf=l t/I? dYt = g.

Now, let g = Lf=l t/I? dyt and let Si = s;(v), V E V, i = 1, 2, 3, be small positive functions on V which rapidly decay for v -+ 00 and such that each Si is locally constant outside Ui' Then the map F, = (Si' cJ: V -+ 1R6 for Si = eil/ti sin ei 1 Yi and Ci = Sit/l; cos ei1 Yi, i = 1,2, 3, induces the metric g + Lf=l et dt/l? on V which can be made arbitrarily Coo-close (in the fine COO-topology) to g for small Si' We claim the existence of all Coo-perturbations t/Ji and y; of t/I; and Yi respectively for which (V, Lf=l t/Jl dyt - st dt/J2) is isometric to (V, g = Lf=l t/ll dyt) where each function Si is assumed Coo-small and t/I; is required to vanish on the subset where S; is not locally constant. This amounts [compare the proof of (B)] to the existence of a small perturbation (Yl,x,Y2,x,YJ,J of(Yl'Y2,Y3) such that there exists a conformal map f: (V, gx - Lf=l st dt/ll) -+ (V, g), where gx = Lf=l t/I;2 dy~x, which preserves the curvature forms. This can be expressed by the following equation for the conformal factor e2h,

Llhgx + w(gJ + De(h) = fx*(w(g)).

This equation is similar to the one used in the proof of (B) and the solution is obtained on compact subsets in V by the argument in (B). The solution on all of V comes with the standard compact exhaustion trick. Thus a Coo-immersion (V, g)-+[R6 is constructed.

Exercises. Fill in the details in this proof. Extend the argument to immersions V --t M6[KJ for K ¥- O. Make the proof work for Can-immersions of Can-surfaces. Eliminate possible double points and thus obtain an isometric COO(Can)-imbedding (V, g) --t M6[KJ for all KE R

References. Our basic analytic tool, Proposition (B), is a modification of H. Weil's implicit function theorem which claims the existence of isometric immersions (S2,g) --t 1R3 for the metrics g which are Coo-close to the standard one (see Pogorelov's 1969-book for an account of the theory of isometric immersions (S2, g) --t M 3[KJ). Isometric Coo-immersions (D 2,g) --t 1R4 due to Posnyak (1973). Conformal immersions V 2 --t 1R3 were obtained by A. Garsia (1961).

(D) Immersions V 2 --t IRs. We construct here an isometric COO(Can)-imbedding (V, g) --t IRs for all compact surfaces V. We start with the study of the following (k + I)-decompositions,

k k+l

g = L dxl + L <pidxl, i=l i=k+1

where the map X = (x1 , ... ,XH z): V --t IRHI is a Coo-immersion and where <Pi' i = k + 1, ... , k + I, are positive Coo -functions on V. (The case I = 0 amounts to isometric Coo -immersions X: V --t IRk.) Let ifJ stand for the vector functions ifJ = (<p;), i = 1, ... , k + I, with positive components <Pi where <Pi == 1 for i = 1, ... , k. The expression (*) defines the differential operator

k+l

D: (X, ifJ)Hg = L <Pidx? i=l

Definitions. A (k + I)-decomposition defined by a pair (X, ifJ) is called stable if the operator D is infinitesimally invertible at (X, ifJ) in the variables Xi and <Pi' That is the linearized equation L(X, aJ) = g, for the linearization L = Lx,r[> of D admits a solution (X, aJ) = (Xi' iP;), i = 1, ... , k + I where iPi == 0 for i = 1, ... , k. Moreover, there is a differential operator M = Mx,r[> which sends metrics g to the above pairs (X, aJ), such that L 0 M = Id.

A (k + I)-decomposition is called strongly stable if the operator D is infinitesimally invertible in Xi' i = 1, ... , k + I and in <Pi' now for i = k + 2, ... , k + I. This means the existence of a linearized solution (X, aJ) with iPi == 0 for i = 1, ... , k + 1.

(D') Lemma. If a metric g admits a strongly stable (k + I)-composition, then g also admits a stable (k' + I')-decomposition for k' = k + 2 and l' = I - 1.

Proof If a decomposition g = L7~f <Pi dxl is strongly stable, then for every small t: > 0 there exist Coo-small perturbations x; of Xi and <P; of <Pi' such that <P; == <Pi for i = 1, ... , k + 1, and such that

This follows from Nash's implicit function theorem (see 2.3.2). Furthermore, the


decomposition L~~f 'P;(dx;f is also strongly stable since the infinitesimal invertibility is defined (see 2.3.1) in some neighborhood of the pair (X, tP). Now let ~. -1, d C -1, Th s = By 'Pk+l sm B xk+1 an c = By 'Pk+l cos B Xk+1. en

k k+1

L (dx;f + ds2 + dc2 + L 'P;(dX;)2 = g. i=1 i=k+2

Finally, an easy argument (which is left to the reader) reduces the stability of this decomposition to the strong stability of the above L~~I 'P;(dX;)2.

Exercise. Generalize (D') to manifolds vn for all n = 1, 2, 3, ....

(D") Lemma. Let 9 admit a stable decomposition, 9 = Lr~f 'Pi dxt for 'Pi == 1, i = 1, ... , k. If k + 21 ~ 5, then there exists a strongly stable decomposition 9 = L~~I 'P; (dx;f for some coo -small perturbation ('P;, xi) of ('Pi' xJ, where 'P; == 1 for i = 1, ... , k.

Remark. The isometric Coo -immersion problem (V, g) --+ ~5 is reduced with (D') and (D") to the construction of a stable decomposition 9 = dXI + 'PI dx~ + 'P3 dx~.

Proof The linearized equation L(X, cl» = g, where X = (xJ, i = 1, ... , k + I and cl> = (CPi) i = k + 2, ... , k + I (where CPi == 0 for i = 1, ... , k + 1) contains k + 2/- 1 unknown functions and so it is underdetermined for k + 21 ~ 5, as the metric g has three components, say gl''' 1 ~ f,l ~ v ~ 2, in some local coordinates (u1 , u2) on V. Hence, the results in 2.3.8 suggest the strong stability of L~~I 'Pi dxt for generic functions Xi' i = 1, ... , k + I and 'Pi' i = k + 1, ... , k + l. The study of the equation L(X, cl» = g is greatly simplified with the following auxiliary (unknown) functions

(+ ) YI' = (X, ~), f,l = 1,2,

where ~ denotes the vector-functions, ~ = ('Pi0I'Xi), i = 1, ... , k + I, for 01' = %ul" and where ( , ) stands for the (Euclidean) scalar product. Next, we denote by ~v, 1 ~ f,l, v :$; 2, the vector function with the components -2(cpJll'vxi + (OvCPi)(0I'X i», i = 1, ... , k + I, and we denote by ZI'V the vector function whose first k + 1 components are zero and the following I - 1 are (0I'XJ(OvXi), i = k + 2, ... , k + I. Then we introduce the following three equations in the unknowns X, cl> and YI"

(+ +) (X, Yl'v) + (cl>,Zl'v) = gl'v - 0I'Yv - ovYw

A straightforward computation (compare 2.3.1 and 2.3.8) leads to the following conclusion: if for some functions Yl and Y2 the linear algebraic equations (+) and ( + + ) are consistant and, hence, satisfied by some X and iP, then L(X, iP) = g. The five linear equations (+) and (+ +) contain k + 21 - 1 unknowns (these are Xi' i = 1, ... , k + I, and CPi' i = k + 2, ... , k + I) and in the generic case the consistency condition is expressed by m equations in YI' and oVYI' for m ~ 6 - k - 2/. Since k + 21 ~ 5, we are left with at most one differential equation of first order for which the solution is especially easy (see 2.3.8). Recall that the operator Lx,tP on Xi' i = 1, ... , k + I and CPi' i = k + 2, ... , k + I is invertible (by some M = Mx,tP), provided the infinite order jet of the given pair (X, tP) misses a certain subset E in the space

JOO of jets of all such pairs. (We use JOO instead of J' for r < 00 since the order of the operator M plays no role in our discussion. In fact, we always deal with operators of a fixed order and so the space JOO stands for J' for some fixed large integer r < 00

which is not specified here.) If we apply the considerations in 2.3.8 to the Eqs. ( + ) and ( + + ), we immediately conclude to the inequality codim I; 2': 1. But we need the following sharper inequality.

Let J: c r consist of the jets of stable infinitesimal solutions (X, <1» to the equation L~~I <fJi dxr = g. In other words, J: is (the lift to r of) the differential relation which governs stable solutions of this equation. We denote by In J: the intersection I; n J: and we claim the subset I c J: to have codim I 2': 3 for all COO-metrics g on V. (Here again 00 means some integer r < 00.) The proof is straightforward with the Eqs. (+) and (+ +) (compare 2.3.8) and we leave this to the reader.

Now the transversality theorem for infinitesimally invertible operators (see 2.3.2) implies (D").

Exercise. Generalize (D") to n-dimensional manifolds, n 2': 2, for k + 2l 2': n(n + 3)/2.

Our construction of isometric Coo-immersions (V, g) --+ [R5 is concluded with the following

(D"') Proposition. An arbitrary Coo-metric g on V admits a stable decomposition,

(**) g = dxi + <fJ2 dx~ + <fJ3 dx~.

Proof Let [R3 be the (Xl' X 2 , x3)-space and let He [R3 be the (X2' x3)-plane {Xl = O}. For an immersion X = (X 1,X2 ,X3 ): V --+ [R3 we denote by I;1 C V c; [R3 the singularity of the normal projection V --+ H. Throughout the following discussion we require the following two properties of X.

(i) The singularity I; 1 is generic. Namely, I; 1 consists of finitely many smooth closed curves and the projection V --+ H folds along I; 1 \I; 11, where I; 11 is the finite set of non-degenerate cuspidal points. Furthermore, the critical points of the functions x2 1I; 1 and x 31I; 1 are non-degenerate. We denote by I; f c I; 1 the union of these points. Observe that the set I; f is finite and I; 11 C I; f.

(ii) The function Xi has Ilgradg xlll < 1 outside I;1. [If (**) is satisfied with some <fJl

and <fJ2 at a point vEI;l, then Ilgradgx 1 (v)11 = 1].

If the Eq. (**) is satisfied with some continuous functions <fJl and <fJ2' on an open subset in V, then [due to (i)] the functions <fJl and <fJ2 are uniquely determined by Xi' i = 1, 2, 3, and by g. Hence, there exists a unique maximal open subset, say U = U(X) c V, on which g admits a stable decomposition (**) for a given X and some (COO-smooth and positive) functions <fJ2 and <fJ3' We want U = V and we construct an appropriate X in six steps. Step O. We start with an arbitrary map X which satisfies (i) and (ii). If the surface V is orient able, we may assume I;11 = 0 and then the following step may be omitted. Step 1: Making U ::::> I;11. Consider smooth positivefunctions <fJ2 and <fJ3 on [R3. and introduce the metric g = dxr + <fJ2 dx~ + (P3 dx~ on [R3. This generalizes the warped products in (A). In particular, such a metric may have any given constant curvature

K at a given point in 1R3. If the map X: (V, g) ~ (1R3, g) is isometric, then (**) is satisfied with <P2(V) = <P2(X(V)) and <P3(V) = <P3(X(V)). Furthermore, if v E.E 11, then the point v is hyperbolic and the vertical (i.e. normal to H) line through v is tangent to V along an asymptotic direction in T,,(V). To arrive at this situation, we let, with some choice of <P2 and <P3' the metric 9 have large positive curvature K near X(V)E 1R3. Then we construct an infinitesimally isometric immersion X': ((Dil(v), g) ~ (1R3, g), such that X'(v) = X(V)E 1R3. If K > Kg(v), then this X' is hyperbolic and we can make one of the two asymptotic directions vertical at v. Furthermore, with appropriate small perturbations of <P2' <P3 and X' we arrive at a generic situation for which the point v is a non-degenerate cusp and for which the corresponding decomposition (**) is stable near v. The stability allows us to apply Nash's implicit function theorem and thus to arrive at an immersion, which is still called X: V ~ 1R 3, for which the projection V ~ H has non-singular cusps at the points of a given finite subset .E 11 c V and such that U :::> .E 11. Of course, the projection V ~ H may now have cusps besides .E 11. However, maps V ~ H without cusps satisfy (all forms of) the h-principle (see 1.3.1) and so it is not hard (and left to the reader) to arrange cusps at .E 11, such that the required extension outside {Dil.E 11 exists. Step 2: Making U :::> .El. Let v E.El be a non-cuspidal point. First we deform X outside {Dil.E 11 in order to make XI = 0 on a small segment S c .E 1 around v E.E 1.

We can also assume the segment S c .E 1 C Vto be geodesic in (V, g) and then choose a metric lJ = dxi + <P2 + <P3 dx~ in 1R3 near X(V)E 1R3 for which isometric immersions

X': (V, g) ~ (1R 3,g) with X'IS = XIS

would fold along S after projecting to H like X. Such an immersion X' is actually obtained near S by first achieving the stability (for perturbed X and g) and then by applying Nash's implicit function theorem. The extension outside S is obvious and thus we let U3V for all vE.El. Step 3: Making U :::> .E 1• One might do this by generalizing the above argument, but we take a different route to show additional features of the picture. First we regularly homotope the immersion X = (X 1 ,X2 ,X3 ): V ~ 1R3 without changing X2 and X3 at all, also keeping fixed XI I {Dil.El, and such that the homotoped function Xl' called x ~, satisfies

(i) IIgradgx~ II = 1 on.E 1, and.E 1 is a non-degenerate maximum set for Ilgradgx~ 112. That is 1 - Ilgradgx~112 ?: s(dist(v,.E 1 ))2 for some s > 0;

(ii) The field gradgx~ I.E 1 is transversal to .E 1 outside .E 11.

Next we consider a metric g' = (dx~ f + <P2 dx~ + <P3 dx~ on V for some positive COO-function <PI and <P2, such that gil {Dil.El = gl{Dil.El, and observe that gradg'x~ also satisfies (i) and (ii).ltfollows that the quadraticforms g - (dx~f and g' - (dx~f on T(V) are equivalent under some fiber-wise linear diffeomorphism IX: T(V) ~ T(V). Moreover, there is an IX which keeps.E 1 (or rather the zero section.E 1 ~ T(V))

fixed, such that 1X*(g') = g, and such that a*(gradgx~) = gradg'x~. Furthermore, we may require this a to be fixed over {Dil.E l and to be isotopic (in the category of fiber-wise linear diffeomorphisms which are fixed over (Dil.E l) to the identity. Consider the linear forms 1; = fo; dx i , i = 2, 3, let Ii = a*(1;) and observe that (dX~)2 + 1~ + lj = g. Since the forms Ii do not vanish on T(.E 1) c T(V) outside.E l


there exist functions x; and CfJ; > 0, i = 2, 3, on a small neighborhood {!}jzIl c V, such that Ii = ~ dx; andx;II 1 = Xi' i = 1,2. Moreover, these x; and CfJ; are unique on {!}jzIl and they are equal to Xi and CfJi respectively on (!}jzI lev. The functions x; extend to all of V, such that Il(X2'X;) = Il = I 1(X2'X3). Furthermore we achieve the stability of the decomposition [(dx~f + CfJ~(dX2f + CfJ3(dx;)2] 1 {!}jzI 1 = gl {!}jzI 1 by choosing generic x~ and 0(. (Checking this is left to the reader.) Then we return to the notation Xl' CfJi. Step 4: Satisfying (**) outside Il with C1-functions x~, x2, x;. We seek a C1-diffeomorphism P': V -+ V, which is the identity near Il, and a C1-function x~ on V, for which x~ 1 {!} jzI l = xli {!} jzI 1, such that the functions x; where x; = Xi 0 p' for i = 2, 3, satisfy (**) with some continuous functions CfJ[ > 0 on V. This can be expressed outside Il by the following differential relation (of the first order) between x;,

for g = 9 - (dx~f. A straightforward check up (left to the reader) shows the convex integration (see 2.4.7) to apply to (***) and then the existence of P' and x~ is obvious. Step 5: Making x; COO-smooth outside Il. Take a compact region with a smooth boundary in V, say Vo c V, which does not meet Il but comes close to it. Namely, Vo U {!}jzIl = V. Take Coo -smooth functions Xi> i = 1,2, 3 and CfJi' i = 2, 3 on Vo such that xd{!}jz8Vo = x; {!}jz8Vo and CfJd{!}jz8Vo = CfJ;I{!}jz8Vo. Then the Coo-metric gl =

9 - dxi on Vo is CO-close to g2 = CfJ2 dx~ + CfJ3 dx~ [since 9 = (dxD2 + CfJ~(dX2)2 + CfJ3(dx;f on V] and gll{!}jz8Vo = g21{!}jz8V2. If V2 is simply connected, then there exists a conformal mapping A: (VO,gl) -+ (VO,g2) which is CO-close to the identity. In fact, such an A exists for all Vo provided the approximating functions Xi and CfJi are chosen such that the conformal classes [gl] and [g2] are equal. Recall that the conformal classes of metrics on Vo form a finite dimensional (Teichmiiller) space and so the required adjustment of Xi and CfJi is easy with the convex integration applied to families at the previous step [compare the proof in (C')]. The induced metric A*(g2) on Vo equals t/I-lg1 for some Coo-function t/I > 0 and so the functions xt = Xi 0 A and CfJ;* = CfJi 0 A, i = 2, 3, satisfy on Yo,

This solution of(**) on Vo may not agree with what we had before on {!}jz8Vo c {!}jzIl. However, the metrics gl and g2 are equal on {!}jz8Vo and they are Coo-smooth. Hence, the CO-closeness of A to the identity yields the Coo-closeness between A and Id on (!}jz8Vo. This implies the COO-closeness of xt to x; and of t/lCfJ;* to CfJ; on Vo for i = 2, 3. Finally, the stable solutions of(**) on {!}jz8Vo are microflexible (see 2.3.2) and so a small Coo-deformation of our solutions on Vo and on {!}jzIl makes them agree on {!}jz8Vo. To complete the proof we need the decomposition (*0) to be stable on yo. This is achieved by introducing a small generic perturbation (which keeps [gl] =

[g2]) into our approximating functions Xi and CfJi. The check up of this genericity is left to the reader.

Thus the proof of(D"') is concluded and we have a COO-immersion (V, g) -+ ~5.

Furthermore, the stability of our solution to the (isometric immersion) equation 9 = Lf=l dx; makes possible (see 2.3.2) the elimination of possible double points

and the Can-approximation. Hence, we obtain the desired Coo(Can)-imbedding of each compact Coo(Can)-surface into IRs.

Exercise. Construct an isometric Coo (Can)-imbedding (V, g) --+ M S [KJ for allK E IR.

Remarks on Immersions (V, g) --+ IRs for Non-Compact Surfaces V. It is not hard to obtain a stable decomposition g = dxi + CfJ2 dx~ + CfJ3 dx~ for non-compact surfaces V. Unfortunately, this does not lead to an isometric immersion since the form e(djq;)2 is never small in the fine COO-topology (here e > 0 and the support of CfJ is non-compact) to which Nash's implicit function theorem applies. However, one may expect the above proof to generalize to complete non-compact surfaces with a bounded geometry. This means a uniform lower bound on the injectivity radius, Inj Radv(V, g) ~ {) > 0 for all v E V, and an upper bound on the norms of some covariant derivatives of the curvature. For example, II VrKg(v) II ::;; const < 00 for the derivatives v r of the order r = 0, 1, ... , 10.

Another class of metrics g may be handled with a Coo-decomposition

g = dxi + CfJ~dx~ + CfJ~dx~, where the functions CfJj, i = 2, 3, vanish on a small neighborhood Vj C V of a disjoint union of simple closed curves in V, such that the connected components of the support Supp CfJi are relatively compact in V. Such decompositions exist (an exercise for the reader) for all (V, g) and they allow (non-constant!) functions e(v) for Nash's twisting. This implies the density of the immersible into IRs metrics in the space of metrics on V with the fine COO-topology. The presence of zeros of CfJi obstructs the stability, but one may try something else instead of Nash's implicit function theorem. The problem boils down to perturbing the functions Xl' x2 and CfJ2 > 0 in a given decomposition g = dxi + CfJ~ dx~ (near a closed curve in V) to x~, x; and CfJ~, such that (dX~)2 + (CfJ~)2(dx;f = g - e( dCfJ~)2 for a given arbitrary small e > O. This is likely to work if the curvature K(v) does not vanish (near the curve in question) or if the metric g is can. Thus one expects an isometric Coo (Can)-imbedding (V, g) --+ IRs to exist if K(v) does not vanish outside a compact subset in V or if g is real analytic outside a compact subset.

Further Questions. Does every Coo-metric g on a (compact) surface V admit the following Coo-decomposition?

(a) g = dxi + CfJ2(dx~ + dxD, (b) g = dxi + dx~ + CfJ~ dx~. The decomposition (b) amounts (at least locally) to an isometric immersion (V, g) --+

(1R3, g) for g = dxi + dx~ + iP2 dx~. Here Xi are the coordinates in 1R3 and iP = iP(Xl, X2' X3). If iP > 0 (and thus g is positive definite), one defines the ellipticity of the decomposition (b) as the ellipticity (i.e. the local convexity) of the immersion (V, g) --+ (1R3, g). If a metric g on a compact surface V admits an elliptic decomposition (b) then there is an isometric Coo-immersion (V, g) --+ 1R4. (This is an exercise for the reader). For example, let (vo, go) be a complete simply connected surface without

conjugate points (for instance, KgO :::;; 0). Then every closed convex Coo-hypersurface V c (Vo x lR,g = go + dX2) admits an isometric COO-immersion into 1R4. [If KgO :::;; 0, then metric spheres V in (Vo x IR, g) are convex.]

Exercise. Let some metric g on a closed surface V admit an elliptic decomposition (b). Show V to be homeomorphic to 82•

Let g be a positive semidefinite quadratic Coo-form on a closed connected surface V, which degenerate (i.e. has rank <2) on a non-empty closed (connected or not) curve E 1 c V. Under what condition can one Coo-decompose g by

(c) g = CfJ1 dx~ + CfJ2 dx~? or even better, by

(d) g = CfJ(dx~ + dxm

Do the Coo-decomposition (c), (d) abide by the h-principle? The existence of (c) is related to the following question. Consider a Riemannian

Coo-manifold W which is diffeomorphic to the product Wo x [0,1] for a closed manifold Wo. The gradient field of each Coo-function f on W, which has no critical points and is constant on the boundary of W, defines a diffeomorphism, say Df :

(Wo x 0) -+ (Wo x 1). The question is whether every diffeomorphism D: Wo x 0-+ Wo x 1, which is isotopic to some Df , is, in fact, equal to Dr for another function f' on w.

Exercise. Two positive semi-definite Coo-forms on IR", say g1 and g2' are called equivalent (near the origin) if there is a fiberwise linear diffeomorphism IX: T(IR") -+ T(IR") which keeps the origin OEIR" fixed and for which 1X*(g1) = g2 near OEIR". Study this equivalence relation between the forms on IR" which are induced by (generic) Coo-maps IR" -+ IRq from positive definite forms g on IRq.

(E) (k + I)-Decompositions and Isometric Coo-Immersion (V, g) -+ 1Rk+21 for dim V = n ~ 3. The decomposition (*) of (D),

k k+1

g = L: dXf + L: CfJidxf i=1 i=k+1

makes sense for manifolds V of any dimension n. We insist, as earlier, on coo_ smoothness of the functions Xi and CfJi' and on the positivity of CfJi' The decomposition (*)iscallednon-singularatvEVifthequadraticformsdxf,i = k + 1, ... ,k + I span the space of the quadratic forms on the tangent space T,,(V). If I < n(n + 1)/2, then (*) is singular at all points VE V, but for I ~ n(n + 1)/2 the pertinent singularity E = E(Xi) c V, i = k + 1, ... , k + I, has codimE ~ 1- [n(n + 1)/2] + 1 for generic functions Xi'

If a form go admits a non-singular (i.e. nowhere singular) (k + I)-decomposition, then, obviously, all COO-forms g which are CO-close to g also admit such a decomposition. Furthermore, each non-singular decomposition can be perturbed (for n ~ 2) into a stable one by applying the "generic" argumentation of (D). This gives [compare (D)] an isometric COO-immersion (V, g) -+ 1Rk+21 for the above metrics g which are CO-close to go and for all compact manifolds V. This fact has certain merit

only for k + 21 < (n + 2)(n + 3)/2, where the results of 3.1 do not apply, and so the interesting dimensions here are n = 3 and 4, for which 2n(n + 1) < (n + 2)(n + 3).

If V is a closed manifold, then non-singular decompositions do not exist at all for I ~ n(n + 1)/2, because of the critical points of the functions Xi' i = k + 1, ... , k + 1. However, for I > n(n + 1)/2 these decompositions enjoy the h-principle (this is an exercise on the convexe integration) and so they are not hard to come by. Moreover, one has the following

Proposition. An arbitrary e~-metric g on V decomposes into g = dxi + Ll=2 (fJidxt, for 1= n(n + 1)/2, such that this decomposition is stable near the singularity E = E(Xi) C V, i = 2, ... , I.

The proof is a generalization of the steps (0)-(4) in (Dill). Namely, we start with generic Xi' i = 2, ... , 4. Then, by a local argument near E = E(x;) c V we achieve a stable decomposition near 1:. Finally, we extend this with a convex integration outside E. Working everything out is left to the reader.

Corollary. An arbitrary compact Riemannian 3-dimensional eX) (Can)-manifold (V, g) admits an isometric Coo(Can)-imbedding (V, g) -+ 1R13. (The general theorem in 3.1.7 gives an immersion into IR IS.)

Exercise. Show that the Coo-metrics g on the ball B4 which are CO-close to a flat metric go admit isometric Coo-imbeddings (B\g) -+ 1R20. Generalize this to all compact parallelizable manifolds vn with a non-empty boundary.

Question. Does every metric g admit a decomposition g = Ll=l (fJi dxt for I = n(n + 1)/2 which is stable near the singular set E? If this is so, then there are isometric Coo-immersions (V 3, g) -+ 1R12 and (V\ g) -+ 1R20.

Singular Decompositions. Take a compact normally oriented hypersurface H in a Riemannian manifold (V, g), dim V = n. If the induced metric in H, say g', admits a nonsingular decomposition, g' = II=l cpi(dxi)2, then the metric g on a small tubular neighborhood U c V of H also decomposes,

I

g I U = dx~ + I (fJi dxt. i=l Indeed, let P: U -+ H be the normal projection and let xo(u) = ± dist(u, H), where the ± sign distinguishes the two components of U\H. Take Xi = X; 0 P and observe that the functions (fJ; on H extend to some (fJi on U such that Ll=l (fJi dxt = g - dx~. Since every metric g' locally decomposes with 1 = n(n - 1)/2, n - 1 = dim H, we get (**) with this I near each point v E V. In case the decomposition (**) is stable, the genericity consideration suggests for 41> n(n + 1) the existence of an isometric Coo-immersion V -+ 1R21+1. This may be interesting for n ~ 7 where n(n - 1) + 1 < (n + 2)(n + 3)/2. Observe that the stability (and the due genericity) can sometimes be achieved by adding extra terms dxt. Furthermore, the actual size ofthe neighborhood where (**) exists can be estimated from below in terms of the geometry of V. Specific statements are contained in the following


Exercise. Let (V, g) be a complete Riemannian manifold with an absolute bound on the sectional curvature, IK(V, g)1 ::;; C2 for some C > O. Let all geodesic loops at a given point Vo E V have length ~ ~ for some ~ > 0 and let B = Bvo(R) be the ball in V of radius R around Vo. Prove for R ::;; min(~/4, (10C)-1) the existence of a decomposition

I

glB = dx~ + L ((Jjdx;, 1= n(n - 1)/2. j=l

Prove the existence of a stable decomposition

k k+1

glB = L dx; + L ((Jjdx;, k = 2n - 2, I = n(n - 1)/2. i=l j=k+l

Prove the existence of an isometric Coo(Can)-imbedding (B,g) --+ ~k+21+1 for the above k and I. (It is likely that a smaller k will do.)

Final Remarks and Questions. The (k + I)-decompositions seem interesting enough in their own right. One may study them by the techniques of 3.1. This will probably give the global existence theorem for all k and I, such that k + 21 ~ (n + 2)(n + 3)/2. One also expects a (k + I)-decomposition to exist for k + 21 ~ n(n + 1)/2 on a small neighborhood U c V of a given point in case the metric is can. Then (for k = 0) one would try to construct an isometric Can-imbedding of (U, g) into the e-ball B. c ~21 for all I ~ n(n + 1)/4 and for all e > O. An especially interesting decomposition is g = L~=l ((Jj dx~ on non-closed 3-dimensional manifolds (V, g), but little is known about its existence.

3.3 Isometric COO-Immersions of Pseudo-Riemannian Manifolds

Let V be an n-dimensional manifold with a quadratic COO-form g. The dimension of the zero part of g, called [gJo, may be a non-constant function on V, denoted no = no(v) = [gl T,,(V)]o = n - rankvg, and so the positive and negative parts of g may also have variable dimensions, denoted by n+(n) = [gvJ+ and n_(v) = [gvJ- = n - no(v) - n+(v).

We denote by W = (w, h) a COO-smooth pseudo-Riemannian manifold which means [hoJ == O. Hence the dimensions q± = [hJ± are constant and q+ + q- =q.

Our analysis of the isometric immersion equation f*(h) = g for Coo-maps f: V --+ W combines the Riemannian immersions techniques of 3.1 with the theory of sheaves (compare 2.2.2). Some results are even stronger for pseudo-Riemannian manifolds as we are aided in many constructions by the presence of isotropic directions in Won which the metric h is zero. For example, we construct global immersions V --+ W for q ~ [n(n + 3)/2J + 2 for some V and W, (see 3.3.5), where similar Riemannian immersions require (at the present state of knowledge) q ~ (n + 2)(n + 3)/2. However, the h-principle is still unknown for free isometric immersions V --+ W for min(q+, q_) ::;; n(n + 1)/2.


3.3.1 Local Pseudo-Riemannian Immersions

We start with two simple properties of submersions (which are not necessarily fibrations).

(A) Let B be a finite dimensional polyhedron and let oc: A ~ B be a micro fibration (see 1.4.2). If the fiber oc-l(b) is k-connected for all bE B then the space of sections B ~ A is I-connected for I 2:: k - dim B [( - 1 )-connected means non-empty]. Furthermore, if dim B :$; k + 1, then each partial section Bo ~ A extends to B for all sub-polyhedra Bo C B.

Proof. The sheaf tP of sections A ~ B is flexible (see 1.4.2,2.2.1) and the germs tP(b), bE B, are k-connected since oc is a microfibration. The flexibility allows the induction by skeletons which reduces the problem to microfibrations over the m-cube [0, 1]m, m :$; dim B. Then the induction in m (compare 2.2.1) reduces the problem further to m = 1 where the proof is straightforward and left to the reader.

(A') Let B be a smooth manifold stratified by submanifolds Ei c B, i = 0, 1, ... , (compare 1.3.2) such that codimEi 2:: i. Let oc: A ~ B be a smooth submersion such that the fiber oc-l(b) is (k - i)-connected for all b EEi and for all i = 0, 1, .... Then the induced homomorphism on the homotopy groups oc*: niA) ~ nj(B) is an isomorphism for j :$; k and an epimorphism for j = k + 1.

Proof. To lift an element (J E nj(B) to A we realize it by a generic smooth map x: sj ~ B, such that dim X-I (Ei) :$;j - i. Then we lift x to A by applying (A) over the strata X-I (Ei) c sj, i = 0, 1, .... A similar lift applies to the balls Bj, aBj = Sj-l

which implies the injectivity of oc* as well as the surjectivity.

Exercise. Show every submersion with (non-empty!) contractible fibers to be a Serre fibration.

(B) The study of the second jets offree isometric immersion reduces [see (C) below] to the following quasi-linear algebraic equations in (m - n)-tuples of vectors X/l E

IRq+,q-, Jl = n + 1, ... , m. Fix linearly independent vectors Xl' ... , Xn in IRq+,q- and take a subset, say M, of pairs (Jl, v) of integers Jl = n + 1, n + 2, ... , and v = 1,2, ... , such that Jl > v.

Consider the equations

(Jl, v)EM,

where each A/lv is a smooth function in Xl' X 2 , ... , Xv' Denote by R(m) c IRq(m-n)

for m = n + 1, n + 2, ... , the space of those linearly independent m-tuples of vectors Xl' ... , Xn, Xn+1' ••• , Xm in IRq, where Xl' ... , Xn are the fixed vectors and where the vectors Xn+l , ... , Xm satisfy the above equations for Jl :$; m. This R(m) is a smooth submanifold in IRq(m-n) and the natural projection IIm: R(m) ~ R(m - 1) is a submersion whose fibers have dimension s(m) which is the number of the pairs (m, v) in M.


The space R(m) is stratified by the rank of the pseudo-Euclidean form h(X, X) =

(X,X) = LJ~1 xJ - LJ=q++1 xJ on the spans of m-tuples in R(m) as follows. The stratum Rr(m) c R(m) consists of those m-tuples (Xl"'" Xm) where m-rank = r which is equavalent to [hISpan{XI, ... ,Xm}Jo = r. The projection IIm sends Rr(m) to the union of Rr-l(m - 1), Rr(m - 1) and Rr+1(m - 1) in R(m - 1). Then one can easily see the following inequality for the codimension cdAm) = codim Rr(m).

(+) cdr(m);:::o: min(cdr_l(m - 1) + m - s(m),cdr(m - l),cdr+1(m - 1)),

which by induction implies

(+ +) codim Rr(m) ;:::0: r - no

for no = [hISpan{XI, .. · ,Xn}JO' Next, the pullback II;;1 (y) for y E Rr(m) is the difference L \J} of affine subspace

Land L' in [Rq+,q- such that dim L = q - s(m) and dim L' = r. Hence, this pullback is (q - m - r - l)-connected. Thus (A') applies to IIm and an obvious induction on m shows the space R(m) to be at least (q - m - no - i)-connected.

Denote by R(m,A+,A_) c R(m), for some non-negative integers A+ and A_, the subspace of those m-tuples of vectors for which the numbers No =

[hISpan{XI, ... ,Xm}Jo and N± = [hISpan{XI, ... ,Xm}J± abide the inequalities No + N+ ::::;; A+ and N+ + N_ ::::;; A_. Assume [in order to have R(m,A+,A_) nonemptyJ A± ;:::0: n± + no for n± = [hISpan{XI, ... ,Xn}J± and Ao = A+ + A_ -m;:::o:O. Take a point a={XI, ... ,Xm-dERr(m-l) and let N'r.=N'r.(a)= [hISpan{XI,,,,,Xm-dJ±. Since the map IIm sends R(m,A+,A_) into R(m - 1, A+,A_), the intersection R" = II;;I(a)nR(m,A+,A_) is empty unless r::::;; Ao + 1 and N'± + r ::::;; A±. If N'± + r ::::;; A± - 1, then R" = II;;I(a) and so R" is (q - m -r - l)-connected. If r ::::;; Ao, then either N~ + r ::::;; A+ - 1 or N'- + r ::::;; A_ - 1. In the case where N~ ::::;; A+ - 1 and N:" + r = A_, the space RG is homotopy equivalent to the sphere Sk for

k = q+ - N~ - r - 1 = q+ - m - N:" = q+ + A_ - m - r.

Similarly, if N~ + r = A+, then RG is homotopy equivalent to Sk for k = q_ + A+ - m - r. Hence the space R" is (p' - r - m - l)-connected for

p' = min(q,q+ + A_,q_ + A+),

and for all aERr(m - 1) n R(m - 1,A+,A_), where r::::;; Ao. Since codim Rr(m - 1) ;:::0: r - no we conclude (by induction in m) with (A') to the

p-connectivity of the space R(m, A+, A_) for

p = min(Ao - no -l,p'- no - m -1).

(B') Exercise. Let S be the space of linear embeddings s: [Rm -+ [Rq+,q- and set Si = {sESI[s*(h)Jo = i}. Show that codimSi = i(i + 1)/2. Then let

S[A+,A_J = {sESI[s*(h)]o + [s*(h)]± ::::;; A±},

and show this space S[A+,A_J to be p-connected for

. (Ao(Ao + 3) ) p=mm 2 ,q-m-l,q++A_-m-l,q_+A+-m-l,

where Ao = A+ + A_ - m. Finally, prove the [Ao(Ao + 3)/2]-connectedness of the space of those quadratic forms g on [Rm for which [g]o + [g] ± :::; A ±.

(C) Fix local coordinates u1, ... , Un in a neighborhood of a point v E V and consider (germs of) isometric CGO-immersions f: (V, g) -+ (w, h) with a fixed value w = f(v). Then the first covariant derivatives Xi = VJ(V)E Tw(W) = [Rq+,q- satisfy for < , ) = < , )h

(1) <X;, X) = gij(v), i,j = 1, ... , n

and the second derivatives

X k1 = Vkd(v)E Tw(W) = [Rq+,q-,

satisfy the equations

(2) _ klij <XwXij) - D (v) + <Xkj,Xil ),

for the pairs (k, I) and (i,j) for which i :::; j < k :::; I, where Akli = !(8igk1 + 8gki -8kg1;) and where Dklij = 8,Akij - 8jAkil + <XkR((Xj' Xz), X;) for the curvature tensor R(( ... )) in (W,h) (compare 3.1.6). Denote by R the space of those solutions (J = {Xi' X k1 } to (1) + (2) for which the vectors Xi and X k1 are linearly independent. Let Rr c R consist of those (J for which No((J) = [hISpan(J]o = r and let R[A+, A_] c R be the subset of those (J for which No((J) + N +((J) :::; A+. Observe that Span(J c [Rq+,q- = Tw(W) equals the osculating space T,,2(V) c T)W) of the map f and so the independence of Xi' Xij reflects the freedom of f at the point v. (Compare 3.1.6.) Since the space of independent solutions {Xl>"" Xn} to the system (l)(obviously)is(p - I)-connected for p:::; q± - n±(v) - no(v), the discussion in (B) implies with (A) the following two properties of the system (1) + (2).

(C') If p:::; q± - n±(v) as well as p :::; q - [n(n + 3)/2] - no (v), then the space R is (p - I}-connected. Furthermore, if A± ~ n± + no and

p:::; min(Ao - no,q+ + A_ - [n(n + 3)/2] - no,q- + A+ - [n(n + 3)/2] - no),

for Ao = A+ + A_ - [n(n + 3)/2], then the space R(A+, A_) is (p - I)-connected.

(C") The subspace Rr c R has

codimR r ~ r - no.

(D) Let I be the space of (germs at v of) free isometric CGO-immersions f: (DjZ(v) -+ W, such that f(v) = w. The second jet defines a map J: I -+ R by f H Jj(v). This J is a weak homotopy equivalence by the argument in 3.1.6. Moreover, the same argument shows the map J:I(A+,A_)-+R(A+,A_) for I(A+,A_)= rl(R(A+, A_)) to be a w.h. equivalence as well. Hence, the spaces I and I(A+,A_) have the connectivity indicated in (C"). In particular, the space I is nonempty, provided q± ~ n± + no and q ~ [n(n + 3)/2] + no'

Next, we combine (C") with the transversality theorem in 2.3.2 and conclude to the following propositions.

(D') Let fo: (V, g) ..... (w, h) be afree isometric Coo-immersion. Then there is an arbitrary Coo-small isometric deformation of fo to a free isometric Coo-immersion f for which the restriction of the form h on the osculating bundle of f satisfies [hl1f2(v)]o :::; no + n at all points v E V.

(D") Exercises. (a) Prove the local isometric C"-immersibility into W of C"-manifolds (V, g) for the Holder classes C", ex > 2.

(b) Show that codimR, ~ no + n(1/2)+B for every fixed e > 0 and for n ~ n(e). Improve with this the connectivity estimate for R(A+,A_).

Question. What is the first non-trivial homotopy group of the space R(A+, A_)? (The answer depends on gij' The simplest case is gjj = 0, Akli = 0 and Dklij = 0).

(E) Local c;-Extensions. Let fo: (v, g) ..... (w, h) be a free isometric Coo-immersion. A field X in W along V (compare 3.1.2) is called a c;-field for c; = ± 1 or 0 if it is nowhere contained in the osculating bundle T2(V) c T(W)I Voffo and if it satisfies the following equations

(X,X) = c; ,

(3) (X,Vdo) = 0

(X, Vijfo) = O.

The first equation prescribes the length c; to X and the remaining equations express the binormality of X that is the normality to the osculating bundle.

If the form h is non-singular on T2(V), that is No = [hi T2(V)]O == 0, then every non-zero solution X to (3) is nowhere contained in T2(V), but for singular forms hi T2(V) the "nowhere contained" condition is non-vacuous. The inequality No (v) + N+(v) = [hi r.,2(V)]0 + [hi r.,2]+ < q+ clearly is necessary and sufficient for the existence of a I-field on a small neighborhood CDjZ(v) c V, while the inequality N_{v) + No{v) < q_ provides (-I)-fields. If both inequalities are fulfilled, No{v) + N ±(v) < q±, then there is a O-field near v.

An extension of fo to an immersion f: V x IR ..... W, for V = V x 0 c V x IR, is called c;-cylindrical if this f is isometric for the form g EB c; dt2 on V x IR. The "c;-cylindrical" condition implies the Eqs. (3) for X = V,J (compare 3.1.2). Conversely,

If the manifold (w, h) and the map fo are real analytic and if there is a continuous c;-field X along V, then there exists a real analytic c;-cylindrical extension f of fo to a small neighborhood CDjZ V c V x IR.

Proof Continuous c;-fields can be approximated by real analytic ones and so X is assumed can to start with. Next, we recall Janet's equations (see 3.1.2) for the field X' = VttJ,

(4)

(X',V,J) = 0

(X',Vd) =Pj

where Pi = - <Vt!, Vitf) and

Pij = < Vit!. Vjt!) + < Vt!. RWt!' VJ, Vj!)

for the curvature tensor R of(W, h). If Vt!1 V = X, then the Eqs. (4) are (algebraically) solvable in X' on V. Thus the system (4) reduces to the Cauchy-Kovalevskaya form and, hence, it is solvable (in!) near V c V x IR with the initial conditions fl V = fo and Vt!1V = X. These solutions f: (DjZV --+ Ware exactly (and only) c5-cylindrical extensions by the argument in 3.1.2. Q.E.D.

Exercise. Generalize the above to isometric non-cylindrical extensions and thus prove the existence of a local isometric Can-immersion of (DjZ(v) c V into W, provided q± ~ n±(v) + no(v) and q ~ [n(n + 1)/2J + no (v). [Compare Friedman (1961).J

Assume (V, g) = (Vo x IRm,go Ef) 0) for dim Vo = n - m and let q < [n(n + 1)/2J + m. Show that no isometric Coo-immersion (DjZ(v) --+ W exists for generic Can-metrics go on Vo and h on W. [One does not know the conditions which would guarantee isometric Can-immersions (DjZ(v) --+ W for n(n + 1)/2 ~ q ~ [n(n + 1)/2J + no(v) - I.J

A c5-field X is called regular (compare 3.1.2) if the vectors X, ViX, VJo and Vijfo are linearly independent along V, which means the independence of the fields X, ViX from the osculating bundle T2(V) c T(W)I v. A solution X' of (4) along V is called free (relative to a given regular c5-field X) if X' is nowhere contained in Span{T2(V),X, V1X, ... , VnX}.

Lemma. If the immersion fo admits a Cl-smooth regular c5-field X and a continuous free solution X' of (4) then there exists a free c5-cylindrical COO-extension of fo to (DjZ(V) c V x IR.

Proof One can assume the fields X and X' to be Coo-smooth and then one obtains an infinitesimally isometric extension f to V x IR, such that Vt!1 V = X and VttflV = X' (compare 3.1.2). The regularity of X and the freedom of X' imply the freedom off near V c V x IR and then f can be perturbed to an isometric extension near V (see 2.3.6).

Corollary. Let fo admit at some point v E V binormal vectors Y, Y1, ... , Y", Y' in Tw(W) ::J T.,2(V), W = fo(v), which are linearly independent from Tv2 and such that <X, X) = c5. Then there exists a free c5-cylilldrical COO-extension of fo to (DjZ(v) c V x IR.

Proof Let us differentiate (3) and obtain the following equations for the derivatives VkX,

(5)

<VkX,X) = 0

<VkX,VJo) = -<X,Vido)

<VkX,Vijfo) = -<X,Vijdo)·

Observe that adding a binormal field to Vkx does not affect these equations. Hence,

a field X for which X(v) = Y admits a perturbation to a b-field Z, such that VkZ(V) = VkX(V) + IX Y for k = 1, ... , n and for any real number IX. If this IX is sufficiently large, then the vectors VkZ(V) are independent from Z(O) and T/ which implies the regularity of Z on lDft(v) c V. Next, the solutions X' of (4) on V are also unsensitive to adding binormal vectors and so the lemma applies to Z and Z' + IX Y'.

(E') Lemma. If No(v) + N ± (v) < q ± and q > [(n + 1)(n + 4)/2] + No(v), then there is a free O-cylindrical COO-extension of fo to lDft(v) c V x IR.

Indeed the above Y, Yi and Y' obviously exist in this case.

(E") Corollary. Let g be an arbitrary Coo-form on V x IR which extends g on V c V x IR. Then the above inequalities imply the existence of a free isometric Coo_ immersion of lDft(v) c V x IR to W which extends fo.

Proof Let f: V x IR --+ W be a free O-cylindrical immersion and consider the difTeomorphisms Pe: V x IR --+ V x IR given by Pe(v, t) = (v, Bt) for B > o. If B --+ 0 then the induced metric ge = Pe*(g) on V x IR Coo-converges to the O-cylindrical metric go = g EEl 0 = P6'(g) as ?JI V = g. Hence, there is a small Coo-perturbation!. of f which is an isometric immersion of lDftv c (V x lR,ge) to W (see 2.3.6). Of course, one uses for a non-compact manifold Va fast decaying function B = B(V) rather than a constant). Then the composed map f 0 pe- 1 is isometric on (lDft V, g) and thus the isometric immersion problem for g is reduced to that for the O-cylindrical metric of the Lemma.

3.3.2 Global Immersions

We start with an open manifold V which admits a submersion V --+ Vo with open fibers of a positive dimension, such that the form g on V is induced from some COO-form go on Yo. Then free isometric COO-immersions (V,g) W satisfy the hprinciple (see 2.2.2), and we obtain with (A), (D) and (D') of 3.3.1 the following

(A) Theorem. If q± ~ n± + no + n + p and q ~ [n(n + 5)/2] + no + p then the space of free isometric COO-immersions V --+ W is p-connected. In particular, these inequalities with p = -1 imply the existence of a free isometric COO-immersion f: V --+ W in every homotopy class of maps V --+ W. Moreover, one can choose this J, such that No = [hI1:?(V)]o ~ n + no and No + N± ~ A±, provided the numbers A+ and A_ satisfy A+ ~ n+ + no, A+ + A_ ~ [n(n + 5)/2] + no - 1, q+ + A_ ~ [n(n + 5)/2] + no - 1 and q_ + A+ ~ [n(n + 5)/2] + no - 1.

Remark. We have the p-connectedness rather than the (p - 1)-connectedness since the pertinent section of V into the space of jets is needed only over the (n - 1)skeleton of some triangulation of V.

(A') Exercises. (a) Assume the submersion V --+ Vo has contractible fibers and prove the existence of a free isometric COO-immersion f: V --+ W for q± ~ n± + no + dim Vo and q ~ [n(n + 3)/2] + no + dim Yo.


(b) Assume no(v) to be constant in v E V. Then the tangent bundle T(V) splits into the Whitney sum, T(V) = To EEl T+ EEl L, where To = ker 9 and the bundles T+ and T_ (non-uniquely) represent the positive and the negative parts of g. The bundle T(V) is called g-trivial if the bundles To, T+ and T_ are trivial, and this definition obviously extends to an arbitrary bundle with a quadratic form of constant rank. Now, assume T(V) to be g-trivial and consider a continuous map fo: V -+ W, for which the induced bundle fo*(T(W» -+ W is fo*(h)-trivial. (Both conditions are satisfied for contractible manifolds V.) Prove fo to be homotopic to a free isometric Coo-immersion f: V -+ W, provided q+ ~ n+ + no, q_ ~ n+ + no and q ~ [n(n + 3)/2] + no.

(c) Let 9 be an arbitrary quadratic Coo-form on V, let Vo c V be a smooth simplicial subcomplex of positive codimension and let U c V be a regular neighborhood of Yo. Construct a smooth map P: U -+ U, such that P(U) = Yo, and for which the induced form go = P6'(g) on U has the following properties. (i) Free isometric Coo-immersions (U, go) -+ (w, h) satisfy the h-principle for all

manifolds (w, h). (ii) Every free isometric Coo -immersion fo: (U, go) -+ (w, h) admits a perturbation to

a free isometric Coo-immersion f: ({9jlVo,g)-+(W,h) [compare (E") in 3.3.1]. Prove with the above the existence of a small neighborhood {9jl Vo c V which admits a free isometric Coo -immersion ({9jl Yo, g) -+ (W, h), provided q + ~ [g]± + [g]o + nand q ~ [n(n + 5)/2] + [g]o.

(A") Remark. The above theorem applies to the product (V x ~,g EEl 0) for an arbitrary manifold (V, g). Thus one obtains a free isometric COO-immersion (V, g) -+ W for all (V, g), provided q + ~ n+ + n + no + 1 and q ~ [n(n + 7)/2] + no + 3. Moreover, under these assumptions an arbitrary continuous map fo: V -+ Wadmits a fine CO-approximation by free isometric COO-immersions. [This follows from the theory of sheaves in 2.2.2. See (B') below and 3.3.5 for sharper results.]

Exercise. Let the bundles T(V) and fo*(T(V» be respectively g- and f*(h)-trivial. Obtain the above approximation for q± ~ n± + no + 1 and q ~ [n(n + 5)/2] + no + 3.

(B) Let us give a geometric interpretation of the second jets of immersions V -+ W. To do this we introduce the (abstract) osculcting bundle T2(V) -+ V whose sections s: V -+ T2(V) are second order differential operators on functions p on V, say CPl--+s(cp), such that s(const) = O. Ifu1 , .•• , Un are local coordinates in V, then every such operator s is a combination s = I7=1 S;lJi + I7,j=1 SijOij, for some functions Si

and Sij' where 0i = %ui and oij = 02/0UiOUj. Hence, the operators 0i and Oij at VE V form a basis in r.,2(V) which transforms according to the chain rule with a change of coordinates. The bundle T2(V) is a natural bundle (see 2.3.7) but it is not a tensor bundle. However, the tangent bundle (which is spanned by oJ imbeds into T2(V) and the quotient T 2(V)1 T(V) is canonically isomorphic to the symmetric square (T(Vf. Furthermore, every Riemannian (or pseudo-Riemannian) metric on V defines a splitting T2(V) = T(V) EEl (T(V»2 by assigning the covariant derivatives Vij

for a basis in (T(V»2 c T2(V). Each smooth map f: V -+ (W, h) defines the (full) second differen'ial DJ:

T2(V) -+ T(W) which sends T2(V) onto the (geometric) osculating sub-bundle T/ c T(W)IY. Namely, 0if-+ VJand oijf-+ Vijffor the covariant derivatives Vi and Vij in (w, h).

If V is endowed with a quadratic differential form g, then a linear map L: T,,2(V) -+ Tw(W) is called skew-isometric if the vectors Xi = L(Oi) and Xij = L(Oij) satisfy the equations (1) + (2) of (C) in 3.3.1.

The parametric h-principle for free isometric immersions f: (V, g) -+ (w, h) claims, with this terminology, the map ff-+ DJ to be a weak homotopy equivalence of the space of free isometric immersions to the space of fiberwise injective skewisometric homomorphisms T2(V) -+ T(W). This h-principle does hold true for the forms g induced by submersions V -+ (V, go) [see in (A)]. Furthermore,

(B') If q± ~ [n(n + 1)/2] + n± + no and q ~ n(n + 3) + no + 3, then the above h-principle holds true for all manifolds (V, g).

Indeed, under these conditions each free isometric Ceo-immersion (lDfi(v),g) -+ W extends to a free O-cylinder [see (E') in 3.3.1] and the results in 2.2.4 apply.

It follows for the above q and q ± that the space of free isometric ceo -immersions V -+ W is (p - 1 )-connected for all forms g provided

. ( n(n + 5) ) p = mm q ± - n+ - no, q - 2 - no .

[Compare (A).] Unfortunately, this does not yield isometric immersions V -+ W for q <

n(n + 3) + 3 [compare (A")]. These are obtained with the following analysis of skew-symmetric homomorphisms ((J: T2(V) -+ T(W). Let ((J*(h) denote the induced form in T2(V) and let No = No«((J, v) = [((J*(h)1 T,,2(V)]0 and N + = N +«((J, v) = [qJ*(h) I T,,2(V)] ±. Then with the proof of (B') we obtain the following -

Lemma. Let a continuous map fo: V -+ W lift to a fiberwise injective skew-isometric homomorphism ((J: T2(V) -+ T(W), such that No + N + < q+ and No < q[(n + 1)(n + 4)/2]. Then fo admits a CO-approximation- by free isometric Ceo _

immersions f: V -+ W.

Warning. If a free isometric immersion f: V -+ W is extended to a O-cylinder j: V x ~ -+ W to which a diffeomorphism d: V x ~ -+ V x ~ commuting with the projection V x ~ -+ V is applied, then the resulting (free isometric) immersion f' = Jo di V may have the dimensions N + and No different from those of f Thus, there is no control over No(f)~ No(DJ)-and N ±(f) with the techniques of 2.2.4, and there isn't (?) any h-principle in sight to improve the mere approximation claim of the lemma.

The space of skew-isometric injective homomorphisms T.,z(V) -+ Tw(W), w =

!o(v), is identical to the space R in (C) of 3.3.1, while the inequalities No + N± < q± distinguish the subspace R(q+ -1,q_ - 1). Furthermore, the bound No < ro = q - [(n + 1)(n + 4)/2] defines the complement

3.3 Isometric Coo-Immersions of Pseudo-Riemannian Manifolds

Hence, this complement is (n - I)-connected by (C') in 3.3.1, provided

q± ~ n±(v) + no(v) + n

n(n + 5) ( ) q~ 2 +nov

and

(**) codimRr ~ n + 1 for r ~ roo

315

Corollary. If the inequalities (*) and (**) are satisfied, then every continuous map fo: V --+ W can be CO-approximated by free isometric Coo-immersions.

Remark. The bound codimRr ~ no(v) + r in (C") of 3.3.1 reduces (**) to the inequality q ~ [(n + l)(n + 6)/2] + no(v), which already appears in (A"). But the estimate codim Rr ~ no(v) + n(1f2)-e for n --+ 00 (see (D"» leads to more interesting results. See (B") below for another approach.

Exercises. (a) Let V; = {v E VI [gl T,,(v)]o = i} and let codim V; ~ i. [If the form 9 is generic then codim V; ~ i(i + 1)/2, see (B') in 3.3.1.] Approximatefo by free isometric COO-immersions under the assumptions q± ~ n± + 2n and q ~ n(n + 7)/2.

(b) Let V be a parallelizable manifold, let non-negative integers A+ and A_ satisfy A+ + A_ = n(n + 1)/2 and let q± ~ A± + n± + no + n. Show that an arbitrary continuous map fo: V --+ W lifts to a fiberwise injective skew-isometric homomorphism cp: T2(V) --+ T(W) for which N+ ~ A+ + n+ and N_ ~ A_ + n_. Then obtain the CO-approximation, provided q± ~ n± + no + nand q ~ [n(n + 7)/2] + no.

Hint. Use a quadratic form h on T2(V), such that

(i) ker h = T(V) C T2(V) (ii) his 4-symmetric as a quadratic form on (T(V»2 = T2(V)/T(V) (compare 3.1.5

and 3.3.3). (iii) [h]+ = A+ and [h]_ = A_.

(B") Non-free O-Extensions. The topological techniques in 2.2 apply to the (microflexible!) sheaf <P of those isometric COO-immersions f: (V x IR, 9 E9 0) --+ (w, h) at which the metric inducing operator fr---+ f*(h) is infinitesimally invertible. The sheaf <P strictly includes the free isometric immersions V x IR --+ W; moreover, <P is sometimes non-empty for q < (n + l)(n + 4)/2, n = dim V, where no free immersion V x IR --+ W is possible [see (E') in 2.3.8]. Furthermore, a non-free immersion (V, g) --+ (w, h) obtained with <P can be Coo-perturbed for q ~ [n(n + 5)/2] + no to a free isometric one (see the transversality theorem in 2.3.2). Thus one may obtain free isometric immersions V --+ W without (being able to prove) the hprinciple for them. The sheaf <P is expected to be non-empty (and thus useful) for

q > [n(n + 3)/2] + no + 2, but the complexity of the jet structure of cP makes such a result hard to prove.

Exercise. Prove cP to admit a global section for q ± ~ n± + n + no + 1, q ~ [n(n + 7)/2] + no + 2 and then extend (A") to q = [n(n + 7)/2] + no + 2.

3.3.3 Immersions with a Prescribed Curvature and the C1-Approximation

Let us generalize Riemannian constructions of 3.1 to pseudo-Riemannian manifolds.

(A) ((j x (j)-Extensions. Fix (j = -lor 1 and let f: (V, g) -+ (w, h) be a free isometric Coo-immersion. Let binormal mutually orthogonal fields X, X' and Yalong V satisfy

(i) <X,X> = <Y, Y> = (j;

(ii) the field Y is orthogonal to the covariant derivatives ViX, i = 1, ... , n, for all systems of local coordinates U i in V;

(iii) the fields X, ViX, X' and Yare independent from the osculating bundle T2(V) c T(W)I v.

(A') Lemma. There exists a Coo-immersion F: V x ~2 -+ W, such that (a) FI V =

f for V = V x 0 and VtFI V = X, VeFI V = Y, where t and e are the Euclidean coordinates in ~2 = ~ X ~. (b) The map F is infinitesimally isometric of infinite order along V c V X ~2 for the form 9 + (j(dt2 + de2) on V x ~2.

Proof First we extend f to a free infinitesimally isometric map j: (V x ~,g + (j dt2 ) -+ W, such that vtfl V = X and such that the second derivative vltll V is orthogonal to Y. We require, moreover, this derivative to be independent from the span of T2(V) and the vector X, ViX snd Y. The existence of 1 is immediate with the discussion in (E) of 3.3.1 which also insures an extension of 1 to the required F (compare 3.1.7). Q.E.D.

Now, we claim the above tripples of fields X, X' and Y to satisfy the h-principle. Indeed, the only differential relation is imposed on X and it requires the independence of the derivatives ViX from the span of T2(V) and the fields X, X' and Y. The pertinent relation fJlllies over (see 2.4.3) the 1-jets of fields X which are normal to T2(V) and independent from T2(V). Namely, the 1-jets in question are given at each point VE V by the vectors X and Xi' i = 1, ... , n, where X E Tw(W), w = f(v), is a binormal vector independent from T,,z(V), and where the vectors Xi satisfy

<Xi,X> = 0

<Xi' Vd> = - <X, Vid>

<Xi' Vkd> = - <X, Vklif>,

[compare (5) in (E) of 3.3.1]. The relation fJll is given over V by the vectors X, Xi' X' and Y which satisfy (i)-(iii) with Xi in place ViX and where Xi satisfy (*). This

r?t goes to the I-jets by the map (X, Xi' X', Y) -+ (X, X;). A straightforward check up shows &i to be ample over the first jet space (of vectors X) and the CO-dense h-principle follows (see 2.4.3).

Next, we impose an additional condition on our fields over an open subset U c V with local coordinates u1 , •.. , Un. This is done with the linear combinations Zi = Zi(S, c) = S(U;)ViX + c(u;) Vi Y, i = 2, ... , n, by requiring the fields Zi to be independent from the span of T2(V), X, X' and Y for all pairs of continuous functions S(Ul) and c(ud, such that S2 + c2 = 1, (compare 3.1.7).

(A") Lemma. Triples (X, X', Y) which satisfy the above &i over U as well as the additional condition abide by the CO-dense h-principle

Proof The new condition, called &i', is differential in X and in Y and it naturally projects to &i, say by the map p': &i' -+ &i. Thus &i also lies over the I-jets of fields X and it is easily seen to be ample over this space of I-jets. Hence, every section q/: U -+ &i' can be homotoped to a section cp: U -+ &i' whose projection to &i is holonomic which means the holonomy ofthe X-component of (X, X', Y). Now we want the Y-component also to be holonomic. For every fixed X, the condition r?t' becom~s an additional condition on Y, called &i'x, over the I-jets of fields Y. This one also is ample (over the jets of fields Y) and so there is the required homotopy of cp within &i'x to a holonomic [over the space of I-jets of triples (X, X', Y)] section. Q.E.D.

Next, sections U -+ &i' are easily obtained with binormal fields X, Xi> X', Yand Y; over U, i = 1, ... , n, which satisfy the following four conditions:

(1) X, X' and Yare mutually orthogonal and (X,X) = (Y, Y) = <5;

(2) (Xi' Y) = 0, i = 1, ... , n; (3) the fields X, Xi X', i = 1, ... , n, are independent from T2(V); (4) the fields X, X', Yare Zi = sXi + c y;, i = 2, ... , n are independent from T2(V)

for all continuous functions s = s(ud and c = c(ud, such that S2 + c2 = 1.

Such fields are easy to construct [compare (A) and (e') in 3.3.1] for

(n + 2)(n + 3) (f) q 2:: 2 + No + n,

where No = [hi T2(V)]O' provided

(+ ) q+ 2:: N+(f) + No(f) + 2 + n

in the case <5 = + 1, and

(- )

for <5 = -1, where N± = [hIT2(V)]±. Therefore, the Lemmas (A') and (A") yield the following

(A"') Corollary. Let f: (V, g) -+ (w, h) be a free isometric Coo-immersion which satisfies the inequalities (**) and (+) over U c V. Then fl U extends to Coo-map F: U x 1R2 -+ W such that


(a) F is infinitesimally isometric of inffnite order along U = U x 0 c U X 1R2 for the metric g + (j(dt2 + d82), where (j = + 1;

(b) the vectors VjF, V ijF, V,F, VeF and Zk = S(UdV'kF + c(u1)V ekF,for 1 :::;; i,j :::;; n and k = 2, ... , n, are linearly independent. Furthermore, the same is true for (j = - 1 with ( - ) instead of ( + ).

Exercises. (a) Assume No(f) == 0 and prove the above corollary for q ~ (n + 2)(n + 3)/2 and q ± ~ N ± + 2.

(b) Generalize the Riemannian extension theorems in 3.1.6 to the pseudoRiemannian (i.e. [gJo == 0) case.

(B) Adding (jqJ2(dud. Let s = sinBulo c = COSBUl and let !.(Ulo ... ,Un) = F(Ul' . .. , Un' eqJs, eqJc), where F is a smooth map U x 1R2 -+ W which satisfies (a) and (b) of the above Corollary and where qJ: U -+ IR is a COO-function with a compact support. Then a straightforward computation gives us the following formulae (compare 3.1.5, 3.1.7):

Vl!. = Vl F + qJ(cV,F - sVeF) + eBl ,

i = 2, ... , n.

i = 2, ... , n,

Vij!. = VijF + eBij'

where the fields Bj and Bij stay bounded for e -+ O. It follows that the maps!. are uniformely free on U for e -+ 0 (compare 3.1.7).

Furthermore, the induced quadratic form is

!.*(h) = g + (jqJ2 dui + e2 dqJ2 + p(e),

where e-kllp(e)lb -+ 0 as e -+ 0 for all k = 0, 1,2, .... Now, we argue as in 3.1.5 and 3.1.7 and conclude to the following

(B') Lemma. If e > 0 is sufficiently small then there is a COO-perturbation f' of!', such that (l')*(h) = g + (jqJ2 dui and such that f' equals f outside a compact subset in U.

(C) Adding (jqJ2(dud4 to the Curvature off Consider two skew-isometric (see 3.3.2) homomorphisms t/!l and t/!2 of T2(V) to T(V) and let A = A(t/!1,t/!2) = t/!!(h)t/!!(h). This form A (obviously) vanishes on T(V) c T2(V) and so it is, in fact, defined on the symmetric square (T(V»2 = T2(V)/T(V). Furthermore, the quadratic form A on (T(V»2 is 4-symmetric when viewed as a 4-form on T(V) (compare 3.1.5).

Now we use e2qJ instead of eqJ in (+) and we conclude by the argument in 3.1.5. to the following

(C') Lemma. There exists a COO-smooth family of free isometric Coo-immersions f.,,: V -+ W for small 8 > 0 and 0 :::;:; r :::;:; 8, such that 1.,0 equals the above f for all 8> 0 and f." = f outside U for all 8 and r. Furthermore, the maps f." C1-converge to f for 8-+0 and the form L1(DJ,DJJ CO-converges to <5cp2(dud4, while the C2_ distance between f and J, is bounded by const cp2(dU1)4 CO for all 8 and T.

(D) Admissible Deformations of the Curvature. Consider a continuous homotopy of forms Ht on T2(V), tE[O,OO], such that L1 = L1(V,t1,t2) = (Ht2 - Ht,)7;,2(V) for 0:::;:; t1 :::;:; t2 < 00 has ker L1 ::::l T(V) and the associated form on (T(V))2 is 4-symmetric. Denote by 0+ c V x [0,00) the (open) subset of those pairs (v,t), where [Htl r.,2(V)]0 :::;:; q --=- [(n + 2)(n + 3)/2] - n as . well as [Htl r.,2(V)] + + [Htl r.,2(V)]0 :::;:; q ± - 2 - n. Call the homotopy Ht admissible (semi-admissible) if the form L1(v, t, t + t') satisfies for all small t' = t'(V, t) ~ 0,

L1(v, t, t + t') = 0'L1 +,

where the form L1 + (which may depend on v, t and t') is positive definite (semidefinite) on (T(VW = T2(V)/(T(V) and where the number 0' = O'(v, t) abides: 0' ~ 0 for (v,t) outside 0_ and 0':::;:; 0 outside 0+. In particular L1 = 0 for all (V,t)E (V x [0,00))\(0+ U 0_) and all t ' ~ O.

The following fact generalizes the Riemannian curvature theorem [see (A) in 3.1.5].

(D/) Proposition. Let fo: (V, g) -+ (w, h) be a free isometric Coo-immersion and let Ht, o :::;:; t :::;:; 00, be an admissible homotopy of forms on T2(V), such that Ho = (DJo)*(h). Then there exists a Coo-homotopy of free isometric Coo-immersions ft: V -+ W, o :::;:; t < 00, with the following three properties.

(i) The immersions ft, 0 :::;:; t < 00, lie in a given C1-fine neighborhood of fo in the space ofC1-map V -+ W

(ii) The homotopy (DJ,)*(h) lies in a given CO-fine neighborhood of Ht where the homotopies of forms are viewed as CO-section of the pertinent bundle over V x [0,00).

(iii) The immersions ft C2-converge for t -+ 00 to a free C2-immersion foo. This foo is necessarily isometric and DJJh) = Hoo due to (ii).

Proof The forms (du)2 for local coordinates u: U -+ ~ on small open subsets U c V span the bundle of symmetric 4-forms on T(V) (compare 3.1.5). This allows one (by using a partition of unity argument) to CO-approximate the homotopy Ht by a semi-admissible homotopy which linearly interpolates consecutive additions of the forms ± cp2(du)4 for various coordinates u on small subsets U c V and for smooth functions cp: U -+ ~ with compact supports. Since the admissibility condition meets the dimension assumptions in (Alii), one can use (C') to construct homotopies of maps V -+ W which approximately match linear homotopies of forms, like (1 - t)Ho + tcp2(du)4, t E [0, 1]. In fact, since the form L1 (DJ, DJJ, 8 > 0, in (C') only approximates <5cp2(du)4, one should keep compensating the error with appropriate choices of u and cp on the following stages. The positivity part of the admissibility condition


leaves enough room for this. Finally, the controle over distc2(J, i. .• ) in (C') insures the convergence fr --+ foo' while the uniform freedom of i.. yields the freedom of foo.

Exercises. (a) Fill in the detail in this proof. (b) Assume [H,Jo == 0 and show the inequalities q ~ (n + 2)(n + 3)/2 and

[H,J ± ::::;; q± - 2 to suffice for the existence of the homotopy fro

(D") Corollary. Let fo: V --+ W be a free isometric Coo-immersion and let Hoo be a continuous quadratic form on T2(V), such that the kernel of A = Hoo - Ho for Ho = (Djo)*(h) contains T(V) c T2(V) and the form A [now on (T(V»2 = T 2(V)/T(V)J is 4-symmetric. If [HooJo + [HooJ+ ::::;; q+ - 2 - n and if q ~ [(n + 2)(n + 3)/2J + no + 2n + 1, then the immersion 10 admits a fine C1-approximation by free isometric C2-immersions f: V --+ W for which (DJ)*(h) = Hoo.

Proof We may assume No(fo) = [HoJo ::::;; no + n [see (D') in 3.3.1J and also [HooJo ::::;; no + n, since q generic perturbation H'oo of Hoo does satisfy [H'ooJo ::::;; no + n and such a H'oo can be admissibly homotoped to Hoo.

Next we semi-admissibly homotope Ho to a form H1 for which [H1JO ::::;; no + nand [H1JO + [H1J+ ::::;; q+ - 2. This is done by a (locally finite sequence) of the following elementary deformations. Take a form H on T2(V), which may be assumed [for the genericity reason, see (C") in 3.3.1J to have [HoJ ::::;; no + n, and consider a point v_ E V (if there is any) where the inequality [H]o + [HJ+ ::::;; q+ - 2 - n is violated. Then [H]o + [HJ- ::::;; q_ - 3 - n in a small neighborhood U_ c V of v_ and so the form H - [4 for an arbitrary linear form [ on T(V) satisfies

[H - [4Jo + [H - [4J_ ::::;; q_ - 3 - n on U_.

Thus we obtain with a linear interpolation semi-definite negative deformations of H (to H - [4) which will eventually force the inequality [HJo + [H] + ::::;;q -2 - n without destroying [HJo + [H]- ::::;; q_ - 2 - n. In a similar way one achieves the inequality [H]o + [HJ- ::::;; q_ - 2 - n at all VE V, and then one makes the homotopy of Ho to H1 admissible by a generic perturbation

Finally, we observe with (C') in 3.3.1 that the space of pertinent forms H on r./(V) for which [H]o::::;; no + n + 1 and [H]o + [HJ± ::::;; q - 2 - nisn-connected. Hence, there is a homotopy H, of H1 to Hoo which satisfies the above inequality at all points VE V and for all tE [1, 00]. This homotopy clearly is admissible and thus we get an admissible homotopy of Ho to Hoo to which (D') applies.

(E) C 1-Approximation. If q+ ~ n+ + no + 2 + nand q ~ [(n + 2)(n + 3)/2J + no + 2n + 1, then every isometric -C1-immersion fo: V --+ W admits a fine C1_ approximation by free isometric COO-immersions f: V --+ W.

Proof If q_ > q+ we reverse the signs of the forms 9 and h and thus we assume q+ ~ q_. Since q_ > n_ + no and q > n(n + 5)/2, we can C1-approximate f by a free COO-immersion f1: V --+ W, such that the form 9 - f1*(h) is positive definite on V. Then there is a locally finite decomposition 9 - f1*(h) = L/l qJ; du~, JI. = 1,2, ... ,


where u/1 and ({J/1 a COO-functions on V and the support of ({J/1 is a compact subset in a small open set U/1 C V, J1 = 1, ....

Next, by applying (D') and (D") [or rather the first part of the proof of (D")] we find another free Coo-immersion f{ which is arbitrarily C1-close to fl and such that Un*(h) = ft(h), Noun::::::; no + nand NoU{, v) + N+U{, v) ::::::; q+ - n - 2 for all v E V. Hence, according to (Alii) and (B') there exists a free COO -immersion f2: V ~ W whose C1-distance to f{ is bounded by const II ({Ji dui IIco and such that fz*(h) =

N(h) + ({Ji dui· The same operation applies to f2 over U2 then to f3 over U3 and we eventually

arrive at a free COO-immersion f: V ~ W whose C1-distance to fo is bounded by L/1II({J; du;llco and such that f*(h) = fl*(h) + E/1 ({J; du2 = g. Q.E.D.

Questions. How does one improve the above argument in order to obtain the parametric C1-dense h-principle for free isometric Coo-immersions V ~ W for the above q and q ±? How does one reduce the bound on q to q ~ [(n + 2)(n + 3)/2] + no?(Infact, the h-principlemay be true for q± ~ n± + no + 1 andq ~ [n(n + 3)/2] + 1, but this is hardly attainable by the present techniques.)

Exercises. (a) Study (by the convex integration) free Coo-immersions f: V ~ W for which NoU) + N+(f) ::::::; A+ and NoU) + N-U) ::::::; A_.

(b) Let no == 0 and call a free isometric immersion f: V ~ W positive if N+U) = n+ + [n(n + 1)/2]. Assume q+ ~ [(n + 2)(n + 3)/2] - n_ and generalize the hprinciples in 3.1 to positive isometric Coo-immersions V ~ W

3.3.4 Isotropic Maps and Non-unique Isometric Immersions

A C1-map f: V ~ (W', h') is called isotropic if f*(h') == O. An interesting case is that of (W', h') = (WI X W2, hI EB h2), where isotropic maps correspond to pairs of immersions,jl: V ~ WI andf2: V ~ W2, such thatfl*(hd = f2*( -h2)' We specialize further to (W x W, hEEl - h) and obtain (with the results in 3.3.2 and 3.3.3) the following

(A) Theorem. If q ~ [n(n + 9)/4] + 2, then arbitrary CO-maps ({Jl and CfJ2 of V into (w, h) admit CO-approximations by Coo-smooth (Cn-smooth if his cn) immersions fl and f2 respectively, such that fl*(h) = f2*(h). Furthermore, if CfJl and CfJ2 a C1_ immersions for which CfJ!(h) = CfJ!(h), then the maps fl and f2 can be chosen arbitrarily C1-close to CfJl and to CfJ2 respectively.

(B) Remarks. Denote by PI the space of Coo(Cn)-immersion V ~ W, let G be the space of quadratic forms on V and let D: PI ~ G stand for the differential operator D: x ~ x*(h). The above theorem claims the CO-density of the subset E2 C PI x PI of the double points (for which Dx = Dy) of the operator D. This sharply contrasts with the following heuristic argument concerning the COO-density.

Quadratic forms g on V are locally given by s-tuples of functions for s = n(n + 1)/2 while maps x: V ~ Ware given by q-tuples. This may be expressed by

writing "dim" G = s and "dim" fiE = q. Hence, the subset r 2 c fiE x fiE is likely to be nowhere Coo-dense for q < s, and, moreover, to be empty for 2q < s, provided h is a generic metric on W [In fact, the discussion in (F) of 2.3.8 suggests a rigorous approach to these conjectures.]

(C) Exercises. (a) Assume q ~ n(n + 5)/2 and find a free isotropic coo(can)_ immersion in each homotopy class of maps V -+ W, provided V is an open parallelizable manifold and the bundle T(W) is h-trivial.

(b) Prove the CO-density of the subset of triple points of D, called r 3 c

fiE x fiE x fiE, for q ~ is + 4n, where s = n(n + 1)/2. Generalize this to k-multiple points for all k ~ 2.

(c) Call a C-immersionf: V -+ (w, h) C-unique if every Cqmmersionfl: V -+ W for which fl*(h) = f*(h) comes from f by a global isometry of (w, h). Show that an arbitrary C1-immersion V -+ W can be C1-approximated by non-Coo(can)-unique Coo(can)-immersions, provided q ~ [n(n + 9)/4] + 2.

(d) C1-approximate an arbitrary C1-immersion f: V -+ IRq, q ~ n + 1, by nonCoo -unique Coo -immersions.

(d') Assume q ~ n + 2 and then C1-approximate f by those Coo-immersions!" for which the space of isometric Coo-immersions (V, (f')*(h) -+ (IRq, h), h = L{=l dxf, is infinite dimensional.

(d") Find, for every i = 1, 2, ... , and n = 1, 2, ... , a closed Can-hypersurface V c IRn+l which is non-C-unique. (This V is necessarily Can-unique for n ~ 3.) Then find for a given i = 1, 2, 3, ... , a Ci-immersion f: sn-l X Sl -+ IRn+1, n ~ 3, such that the induced metric f*(h) is can but no isometric Can-immersion V -+ IRn+l exists.

(D) Perturbations of Isotropic Immersions. Let V be a compact manifold and let fo: V -+ (w, h) be a free isotropic Coo-immersion. Then by Nash's implicit function theorem every C~-small metric 9 on V for IX > 2 admits a free isometric immersion f: (V, g) -+ W which is C2-close to fo. This also is true for those non-free immersion for which the differential operator D: fH f*(h) is infinitesimally invertible at fo,' provided IX is sufficiently large.

Example. Let 9 be an arbitrary C~-form on V. Then we obtain with a free isotropic immersion fo: V -+ W an isometric C~-immersion f: (V, A? g) -+ W for all small constants A > O. Moreover, if W = IRq+,q-, then we have the isometric map A-1f: (V, g) -+ IRq+·q-.

Exercises. (a) Let (V,g) be a compact C~-manifold for IX> 2. Prove the existence of a free isometric C~-immersion f: (V, g) -+ IRq+,q-, provided q + ~ 2n + 1, q ~ (n2/2) + ~n + 3. -

(b) Assume V to be a compact parallelizable manifold with a non-empty boundary and obtain the above C~-immersion f for q ± ~ nand q ~ (n2/2) + ~n. Show this bound on q to be the best possible for free isometric immersions V -+ IRq+,q-. Prove the existence of a (non-free!) isometric C~-immersion V -+ IRq+,q- for q+ ~ n and q ~ (n2/2) + ~n - 1, provided IX ~ IXo(n). -

3.3 Isometric C'''-Immersions of Pseudo-Riemannian Manifolds 323

(D') Expansion of Isotropic Immersions with Bounded Geometry. Consider an immersed n-dimensional submanifold V' c. [RN and normally project a small neighborhood U' c V' of a point v E V' into the tangent space 1'" ~ [Rn C [RN to V'. Thus the (Euclidean) coordinates in 1'" induce some local coordinates x;, i = 1, ... , n near Vi in V, called standard coordinates. Denote by Gr(V', Vi) the maximum of the norms of the derivatives ai of orders i = 1, ... , r of the immersion V 4 [RN at Vi in the standard coordinates.

Next, fix a proper Coo-embedding I: W ~ [RN and say a family §' of immersions f: V ~ W to have (uniformely) bounded geometry of order r if there is a continuous function C on W x V such that GAl 0 f(v), 10 f(v)) .::;; C(I 0 f(v), v) for all v E V and all f E§'. Observe that this "boundness" of §' does not depend on a choice of the embedding I (though the function C does). The geometry of §' is called bounded if the above holds for each r = 1, ... , (with C = CJ

A family §' is called expanding of order r if for an arbitrary positive continuous function), on V and for an arbitrary embedding J: V ~ [R2n there exists an immersion fE§', such that the map J = J 0 (I 0 f)-1: V' ~ [R2n, for the immersed manifold V' = (I 0 fHV) C. [RN, has Ilaifll(v ' ) .::;; ),(v), v = J(v ' ), where ai, i = 1, ... , r, denote the derivatives of J in the standard coordinates.

The property of being expanding is invariant (like the boundness) under the choice of the embedding I.

Examples. Let V be compact and let fo: V ~ [Rq be a Coo -immersion. Then the family {Afo} for ), E [1, <Xl) has bounded geometry and it is expanding for all r = 1,2, ....

Exercise. Let f: V ~ W be an arbitrary C1-immersion (embedding) and let dim W > dim V. Prove for all r = 1,2, ... , the existence of an expanding family of immersions (embeddings) V ~ W of bounded geometry which lie in a given fine CO-neighborhood of fo.

Finally, call a family §' of immersions f: V -+ (w, h) uniformly free if there is a continuous positive function e on W x V such that all f E ff have Hessv(vJ; Vijf) ~ e(f(v), v), where Vi and Vij' i, j = 1, ... , n, are the pseudoRiemannian [for (w, h)] covariant derivatives in the standard coordinates (in V' ~ V) and where the Hessian is taken relative to the Riemannian metric in W induced by the embedding I.

Lemma. Let fo: V -+ W be a continuous map and assume the existence of a uniformly free expanding family of isotropic Coo-immersions f: V ~ (w, h) of bounded geometry, such that dist(fo(v),f(v)) .::;; p(v) for some distance on Wand for some continuous function p = p(v) > O. If the implied order r satisfies r ~ 5, then an arbitrary Cil-form g on V for 2 < rx < r - 2 admits a free isometric Cil-immersion 1': (V, g) -+ (W,h) such that dist(f',fo) .::;; 2p.

Proof If a map f: V -+ W is sufficiently expanding, then the push-forward form (1 0 f)*(g) on V' ~ V becomes C'-small in the standard coordinates. Hence, the implicit function theorem applies (due to the "locality" of D-1 in 2.3.2).

Example. Let V be the n-torus Tn and let fo: Tn -+ W be a free isotropic Coo-map. Then every CIX-form 9 on Tn, a. > 2, admits an isometric CIX-immersion f': (Tn, g) -+ W.

Proof Consider the (expanding) map E: Tn -+ Tn for E: t 1--+ 2t and apply the lemma to the family :F = {fo 0 Ei}, i = 1,2, ....

Exercises. (a) Generalize this example to the manifolds V which admit expanding endomorphisms [compare (E) in 2.4.4]. Generalize further to branched manifolds with expanding endomorphisms.

(b) Let fo: V -+ W be a free isotropic Coo-immersion and let Vo c V be a submanifold of positive codimension. Prove for all CIX-forms go on Yo, a. > 0, the existence of a free isometric CIX-immersion (Vo, go) -+ W which is homotopic to fo I Yo.

(b') Let 9 be a CIX-form on V, a. > 2. Prove the existence of a free isometric CIX-immersion (V, g) -+ W in each homotopy class of maps V -+ W, provided q ± ~ 2n + 1, q ~ (n2/2) + !n + 3.

(b") Assume the forms 9 in V and h in W to be positive definite and CO-approximate a given strictly short immersion V -+ W by free isometric CIX-immersions, provided q ~ (p + 2)(p + 3)/2 for p = [(n + 2)(n + 3)/2] + [n(n + 5)/2] (compare 3.1.1). One does not know how to obtain an isometric immersion V -+ W for 2 < a. ~ 4 with a more realistic bound on q, say for q ~ (n + 2)(n + 3)/2.

3.3.5 Isometric Coo-Immersions Vn -+ Wi for q ~ [n(n + 3)/2] + 2

The basic construction of immersions (V, g) -+ W (see 3.3.2) which depends on the theory of invariant sheaves requires the existence of (germs of) free O-cylindrical immersions (V x ~,g Ee 0) -+ W. These do not exist for q ~ (n + l)(n + 4)/2; however, we need, in fact, something less than the freedom of the cylinders. Namely, we only need the freedom on some hypersurfaces in V x ~ which are used in building up our immersion V -+ W. Non-free O-cylinders may exist for q± ~ n± + no + 1 and q ~ [n(n + 3)/2] + no + 2. The h-principle for free isometric Coo-immersions V -+ W becomes quite plausible in this range of dimensions. The examples studied below support this conjecture.

(A) Isotropic Immersions V" -+ wq for q ~ [n(n + 5)/2] + 2. We assume V to be a parallelizable manifold and we assume the tangent bundle T(W) to be h-trivial, which is so, for example, for contractible manifolds W.

(A') Theorem. If q + ~ n + 1, q ~ [n(n + 5)/2] + 2 and if the manifold (w, h) is real analytic, than an a~bitrary continuous map fo: V -+ W can be CO-approximated by (possibly non-free) isotropic Can-immersions f: V -+ W.

Corollary. If P ~ [n(n + 5)/4] + 1, then arbitrary continuous maps ((Jl and ((J2 of V into ~p can be CO-approximated by Can-immersions fl and f2 respectively, such that


the induced metrics satisfy !t*(h') = f2*(h') for the form h' = Lf=l dxr in ~p. (Compare 3.3.3.)

Question. Does this corollary hold true for n = 2 and p = 3?

Proof of (A'). The sheaf of free isotropic immersions V -+ W is microflexible and (Diff V)-invariant. Hence the h-principle holds for folded free isotropic immersions (see 2.2.7). Moreover, the h-principle still holds true (for the same reason) if we additionally require O-fields [see (E) in 3.3.1] along our immersions. Thus, using the h-principle and the triviality of the bundles in question, we CO-approximate fo by a COO -map fl: V -+ W which is free isotropic outside a collection of disjoint closed hypersurfaces in V and near such a hypersurface, say V' c V, the map looks as follows. There is a tubular neighborhood U' = V' x [ -1,1] c V of V' = V x 0 which admits a free isotropic Coo-immersion 1': U' -+ W, such that the map fll U' is the composition of I' with the standard folding a: U' -+ U', that is a: (v', t) H

(v', t2 ). Moreover, we may assume the map I' to be can and then we can obtain with a O-field along U' a (non-free) isotropic Can-immersion F': U' x ~ -+ W such that F'I U' = I' for U' = U' x 0 [see (E) in 3.3.1]. Finally, we unfold a to an immersion a': U' -+ U' x ~ that is a': (V',t)H(V',t2,P(t)), where P(t) is a COO-function which is zero outside [ -!,!] and whose derivative does not vanish at t = O. Thus we modify fl to F' 0 a' near each V' and get an isotropic COO-immersion 1': V -+ W Since this map f{ is free away from the neighborhood U' and is free on the hypersurfaces V' x tin U', tE [ -1,1], there is [see (E') in 2.3.8] a Coo-approximation of f{ by a real analytic isotropic map f: V -+ W

Exercises. (a) Assume the above (w, h) to be Coo-smooth and approximate fo by (non-free) isotropic COO-immersions f on which the operator D: fH f*(h) is infinitesimally invertible.

(b) Construct an isometric COO-immersion (Tn, g) -+ W for an arbitrary COO-form g on the torus Tn.

(A") Control of the Freedom. If for every hypersurface H c U' x ~ which transversally meets U' = U' x 0 across some submanifold V' x t, tE [0, 1], the map FIH is free on (()fz Vr' c H, then the map f{ is free for a sufficiently small function p. This freedom can be insured by an appropriate (quasi)-regularity of the O-field in question (compare 3.1.9, 3.1.7) and the (quasi)-regularity is not hard to achieve (an exercise for the reader) for q ~ [n(n + 5)/2] + 4. (Probably, the inequality q ~ [n(n + 5)/2] + 3 suffices.)

Exercise. Prove for q ~ [n(n + 5)/2] + 4 the existence of an expanding uniformely free family of isotropic COO-immersions V -+ W of bounded geometry and then approximate the map fo by free isometric Coo -immersions (V, g) -+ W for an arbitrary Ca-form g on V, provided IX> 2.

(B) Immersions of Pseudo-Riemannian Manifolds. Let V be a (topological) product, V = Vo x S where Vo is compact and where S is a one-dimensional manifold, and

let g be a quadratic Coo-form on V for which no == O. Moreover, assume the form g

to be non-singular on the hypersurfaces V. = Vo x s c V for all s E S and require the bundle T(V.) to be gs-trivial for gs = gl T(V.). We study isometric immersions (V, g) -+ (w, h), where the bundle T(W) is assumed h-trivial.

(B') Theorem. If the manifolds (V, g) and (w, h) are real analytic and if q + ;;:: n+ + 1 and q ;;:: [n(n + 3)/2] + 2, then an arbitrary continuous map fo: V -+ W-can be co_ approximated by isometric Can-immersions f: V -+ W

Proof Let V = (V x IR, g $ 0), and consider the hypersurface V. = (V. x IR, gs $ 0) in V for some s E S and take a small neighborhood Us c V of v., say Us =

V. x (s - e,s + e). We start with a study of COO-immersions P: Us -+ W which are free [compare (A")] along V. (this means the freedom of PI V. as well as the independence of the field VsF from the osculating bundle T 2 (V.) c T(W» and infinitesimally isometric of order three along v.. The freedom insures infinitesimal isometric extension from V. to Us. This, in addition to the microflexibility of free isometric immersions (v., gs $ 0) -+ W, implies the microflexibility of the maps P: Us -+ W Hence, we have the h-principle which provides (with our assumptions on V and on W) a Can-continuous family of can-maps Ps: Us -+ W, S E S, such that

(a) the map Ps is free along V. in the above sense and it is isometric on a (possibly) smaller neighborhood U; c Us of v., say on V. x [s - e', s + e'] for 0 < e' = e'(s) :s;; e, and for all s E S. (b) The map F: V = Vo x S -+ W, for F(vo, s) = ps(vo, s, 0), lies in a given CO-fine neighborhood of fo.

The map Psi Us for Us = Us n V c V is isometric for all s E S but these maps Us -+ V do not agree on intersections of the neighborhoods Us = V. x (s - e', s + e') c V. Let us make them agree on successive neighborhoods, Us, and US2 ' where S2 is sufficiently close to SI' such that SI + e'(sd > S2 - e'(s2). Consider two COO-functions /31 on (SI - e'(sd,sl + e'(sl) and /32 on (S2 - e'(s2),S2 + e'(s2» which range in [0,1] c IR and such that

(i) there is a small subinterval 1" of length e" in the intersection of (SI' S2) with (S2 - e'(s2)' SI + e'(sl» on which /31 = /32 and d/3dds = d/32/ds = 1/2e";

(ii) The functions /31 and /32 are zero away from (a slight enlargement of) the above subinterval 1".

Now we define (isometric) maps f~: USi -+ W, i = 1,2, by Is;: (v, s) ~ Ps.(v, s/3;(s». If S2 -+ SI' then the maps Is;, i = 1,2, on Vo x 1" keep bounded geometry and the uniform freedom, while the Coo -distance between them goes to zero. Hence, there is a small isometric perturbation of the map f:2 on 1" which makes it equal to Is', on 1" and thus we obtain an isometric Coo-map on Us, U US2 = Vo X (SI - e',s2 + e'). This "fitting operation" applies to all pairs of nearby points in S and then it delivers an isometric COO-immersion f': V -+ W, which can be Coo-approximated by desired isometric Can-immersions f

(B") Exercises. (a) Make the above map f free as well as isometric for q;;:: [n(n + 3)/2] + 4 and also generalize (B')(for q ;;:: [n(n + 3)/2] + 4) to COO -manifolds and maps.

3.4 Symplectic Isometric Immersion 327

(b) Drop the assumption of the non-degeneracy of 9 on T(V.) and obtain the isometric immersion f, provided q± ~ n± + 2 and q ~ [n(n + 3)/2] + 4.

(b/) Generalize further by considering a Morse function s: V --+ S (instead of the projection Vo x S --+ S) which may have isolated critical points, and by applying the above construction outside these points. Prove that an arbitrary pseudo-Riemannian manifold (V, g) with a g-trivial bundle T(V) admits an isometric Coo_(Can)-immersion into W for q± ~ n± + 2, q ~ [n(n + 3)/2] + 4.

(c) Prove (B/) for extensions of isometric maps from lDfto(Vo x S) to Vo x S for compact manifolds Vo with a boundary.

(c /) Assume a pseudo-Riemannian manifold (V,g) to admit a codimension one foliation such that 9 is non-degenerate on the leaves and such that the tangent bundle of this foliation, say T' c T(V), is g'-trivial for g' = gl T. Prove in the can-analytic case the existence of an isometric Can-immersion V --+ W for q ± ~ n± + 1, q ~ [n(n + 3)/2] + 2.

(c") Consider an arbitrary Coo-form 9 on V and let T c T(V) be an arbitrary (non-integrable) codimension one Coo -subbundle. Study the sheaf <P' of the Coo -maps F: V x IR --+ W (where W is an arbitrary pseudo-Riemannian manifold), such that

(1) the maps F are infinitesimally isometric of order 3 along V = V x 0 c

(V x lR,h EB 0); (2) the maps F enjoy the above freedom along T', that is for arbitrary independent vector fields Xi on V x IR, i = 1, ... , n + 1, where the fields XilV x 0 lie in T for i = 1, ... , n - 1, the covariant derivatives Vx;F for i = 1, ... , n + 1 and VxkVx,F for k, I = 1, ... , n are linearly independent along V = V x 0 c V x IR.

Prove the restriction <P' 1 Vto be a flexible sheaf and obtain the h-principle for <P' as a corollary. Then derive the following.

Theorem. If q + ~ n+ + no + n + 1 and q ~ [(n + 2)(n + 3)/2] + no + 2, then an arbitrary continuous-map fo: V --+ W can be CO-approximated by free isometric Coo-immersions f: V --+ W

Remark. The techniques developed for pseudo-Riemannian immersions are likely to apply to more general differential relations, [e.g. to isometric COO-immersions of symmetric tensors of degree d ~ 2, compare (B) in 2.4.9 and Gromov (1972)] but no systematic study has been conducted so far.

3.4 Symplectic Isometric Immersion

Here we study isometric immersions f: (V, g) --+ (w, h) for exterior differential forms of a fixed degree deg 9 = deg h. The word "isometric" refers, as earlier, to the form inducing equation f*(h) = g. Though, the convex integration does not apply to this equation (see in 2.4.9), the techniques of 2.2, 2.3 and of 3.3 work with minor modifications. In fact, isometric immersions for forms of degrees one and two (which arise in the contact and symplectic geometry) do not require Nash's implicit function theorem. The h-principle for these immersions only requires the theory of topological sheaves.


3.4.1 Immersions of Exterior Forms

Denote by N L the space of k-linear anti-symmetric forms w on a given linear space L and denote by X· w for x E L the interior product, that is the (k - 1 )-form defined by (x.w)(x 1"",Xk-d = W(X,X1"",Xk-d. The resulting linear map XHX'W is called 1m: L -+ N-1 L. A subspace L' c L is called w-regular if the composition of 1m with the restriction homomorphism N-1 (L) -+ N-1 (L') sends L onto N-1 (L'). A form wEN L is called non-singular or symplectic if the homomorphism 1m: L -+ N (L) = L * is an isomorphism. In general one defines rank w = rank 1m, and one observes that the rank of a 2-form w equals twice the greatest integer r for which the exterior power w r is non-zero. Thus, the existence of a symplectic form on L makes the dimension of L even, say dim L = 2m, and the non-singularity condition amounts to the non-vanishing of the top dimensional (volume) form wm.

The direct sum of an arbitrary linear space Lo with its dual, L = Lo EEl L~, carries a canonical symplectic form, that is Li=1 Xj 1\ Yj for m = dim Lo, where the vectors YjELo constitute a basis of linear forms on Lo and where (X1""'Xm) is the dual basis in Lo.1f L is identified with the complexification CLo = Lo EEl ,J=1Lo, then w(a,b) = (a,,J=1b) for all a and b in L, where the scalar product ( , ) is given by the symmetric form Lf!.l xf + yf on L.

The group of linear transformations of L, dim L = 2m, is transitive on the space of symplectic forms w on L since any such w (obviously) reduces to Wo for some splitting L = Lo EEl L~. The isotropy subgroup, called the symplectic group Yfzt m = Yfzt(L, wo), clearly has dim Yfzt m = 4m2 - m(2m - 1) = 2m2 - 1 and it transitively acts on L\{O}. The maximal compact subgroup in Yfztm can be easily identified with the unitary group Um = U(CLo, Lf!.1 zizj) for Zj = Xj + ,J=1Yj, which is also the maximal compact subgroup in the full linear group GLmC. Thus the groups Yfztm, Um and GLmC have the same homotopy type.

(A) Regular Immersions. Let h be a COO-smooth k-form on a manifold W, that is a COO-section h: W -+ N(W) = N T(W) and let f!) = f!)h: fH f*(h) be the form inducing (non-linear differential) operator on smooth maps f: V -+ (w, h) for a given manifold V. Then the linearized operator L = Llof!) at a given map fo sends vector fields 0 in Walong V (mapped to W by fo) to k-forms on V according to the following formula

L(o) = (d(o' h) + o· dh)1 V,

where d is the exterior differential and o· is the interior product with O. This is an obvious corollary of the well-known expression for the Lie derivative of any form w by a (globally defined) vector field 0,

(**) ow = d(o'w) + o'dw

Definitions. An immersion f: V -+ (w, h) is called h-regular ifthe image ofthe tangent space T,,(V) under the differential of f is hw-regular in the receiving space Tw(W), W = f(v), for all VE V. For example, if h is a nowhere singular 2-form on W, then every immersion is h-regular. We calIf an (h,dh)-regular immersion if the linear map Tw(W) -+ /\~-1 (V) EEl /\~ (V) defined by. H «., h) EEl ( •• dh»1 T,,(V) is surjective for all


V E V, where r E Tw(W) for w = f(v) and T,,(V) is embedded into Tw(W) by the differential of f. Clearly, this regularity implies the h-regularity as well as the dh-regularity of f.

Example. If (w, h) = (W1 X W2 , h1 EB h2 ) and if the projection of f to (W1' hd is h1-regular while the projection to Wz is dh2-regular, then f is (h, dh)-regular.

Lemma. The operator E0: ff--+ f*(h) is infinitesimally invertible on the space of (h, dh)-regular immersions f: V ~ (w, h).

Proof. The regularity implies the solvability in a of the linear system

o·hlV == 0, o·dhlV = g,

for all k-forms g on V, which solves the equation L(o) = g (compare 2.3.1).

Now the results in 2.2 and 2.3.2 yield (compare 3.3) the following

(A') Theorem. If V = Vo x IR and if the form g on V is induced from a Coo-form on Vo by the projection V = Vo x IR ~ Vo, then isometric (h, dh)-regular Coo-immersions f: (V, g) ~ (w, h) satisfy the parametric h-principle. Furthermore, if V is an open manifold, then (h, dh)-regular isotropic (i.e. f*(h) == 0) Coo-immersions f: V ~ (w, h) satisfy this h-principle.

Exercise. Let W = IRq for q = Mk + N(k + 1) and h = Ii'!l dxi, 1 A ... A dXi,k + 'if=l Xj , 1 dXj ,2 A ... A dXj .k+1' Analyse the above h-principle for M ~ [2(n !)/ k!(n - k)!] + 2n and N ~ [2(n!)/(k + 1)!(n - k - 1)] + 2n, where n = dim V and show for k ~ 1 that any continuous map f: Vo ~ IRq admits a fine CO -approximation by isometric Coo-immersions (Vo, go) ~ (IRq, h) for an arbitrary Coo-form go on V. Prove this property of the form h on IRq to be stable under small C1-perturbations.

The Cauchy Problem. Consider an arbitrary k-form g on V = Vo x IR and let a Coo-map f: V ~ (w, h) satisfy the following equations with the field 0 in Walong V (mapped to W by f) which is the image of the field Ot = %t in V = Vo x IR under the differential of f,

(1) a . hi J!; = Ot' g I J!;

o'dhlVr = Ot·dglVr

for J!; = Vo x t c V x IR, t E IR. The system (1) implies with (*) and (**) that oJ*(h) = 0tg and hence each solution of (1) with the initial condition f*(h)1 Vo = gl Vo is an isometric map (V, g) ~ (w, h). If the map fo = fl Vo: Vo ~ W is (h, dh)-regular as well as isometric for go = gl Vo, then the Eqs. (1) can be resolved formally in 0 which reduce them to the Cauchy-Kovalevskaya form. This yields the following

Proposition. Let (V, g) and (w, h) be real analytic and let fo: (Vo, go) ~ (w, h) be a (h, dh)-regular can-immersion. Then fo extends to a (possibly non-regular) isometric can-map f: (U, gl U) ~ (w, h) of some small neighborhood U c Vof Vo c V.


In order to control the regularity of f which is needed for an application of the micro extension theorem (see 2.2.4), we introduce the following

Definition. A subspace L c Tw(W) is called (h, dh)-biregular if there exists a vector " E Tw(W) outside L for which " . hwlL = 0 and such that the span L' = Span(L, ,') c Tw(W) is (h,dh)-regular (i.e. the pertinent map Tw(W) ~ N- l L' Et> N L' is onto). Then we obviously define the biregularity of immersions f: V ~ W in terms of the subspaces T.,(V) c; Tw(W), w = f(v) and we obtain with 2.2.4 and 2.3.2 (compare 3.3) the following

(A") Theorem. Let go be an arbitrary COO-smooth k-form on Vo. Then isometric (h, dh)-biregular Coo-immersions (Vo, go) ~ (w, h) satisfy the CO-dense h-principle.

Exercise. Give examples where (A") yields isometric immersions but where (A') does not apply.

(B) Immersions of Closed Forms. Let the form h on Wbe closed, that is dh = O. Then no map into W is (dh)-regular and so the above considerations do not apply.

Exercise. Show for dim V~ I and k ~ 2 that the sheaf of h-regular isometric Coo_ immersions (V, g) ~ (w, h) contains no open non-empty microflexible sub sheaf whenever dh = O.

Now, if the forms g and h are exact, g = dg l and h = dh l for some Coo-smooth (k - I)-forms gl on V and hl on W then the equation f*(h) = g can be rewritten as f*(h l ) + dcp = gl with an auxiliary (unknown) (k - 2)-form cp on V.

Lemma. The (non-linear differential) operator 'scl: (f, cp) ~ f*(h l ) + dcp is infinitesimally invertible at those pairs (f, cp) where f is an h-regular immersion V ~ (w, h).

Proof The linearized operator Ll = Lf'scl applies to the pairs (0, iP), where 0 is a field in Walong V (mapped to W by f) and iP is a (k - 2)-form on V, by the formula

Ll(o,iP) = o·h + d(o'hd + diP.

If f is h-regular, then the system

(2) o'hlV = 9

o'hl+iP=O

is solvable for all (k - I)-forms 9 on V and every solution (0, iP) of Eqs. (2) satisfies L(o, iP) = g.

Generalization. Let [g]EHk(V;IR) and [h]EHk(W,IR) denote the respective cohomology classes of the closed (but now not necessarily exact) forms g and h. Fix a continuous map fo: V ~ W for which fo* [h] = [g]. Denote by To c X = V x W the graph of fo and let g and Ii be the pull-backs of the forms g and h to X under

the obvious projections. Take a small neighborhood Y c X of ro which contracts to ro and observe that the form Ii - g is exact on Y. Write Ii - gl Y = dh for some COO-smooth (k - I)-form h on Y and let .@cT. cp) = f*(h) + dcp for sections 1: V .... Y and (k - 2)-forms cp on V. Observe that the maps f: V .... W underlying f satisfy f*(h) = 9 + dh = 9 + d.@(J, cp). It easily follows that the space of sections 1: V .... Y for which f*(h) = 9 + dg1 for a given (k - I)-form g1 on V has the same homotopy type as the space of solutions to the equation .@(J, cp) = g1 for the pull-back g1 of g1 to Y under the projection Y .... V. In particular, the equation f*(h) = 9 reduces to the equation .@(J,cp) = 0, in so far as the unknown map f is CO-close to fo

Lemma. The operator .@ is infinitesimally invertible at those pairs (1, cp) for which the underlying maps f: v .... Ware h-regular.

Indeed, the proof of the previous lemma applies.

Exact DifJeotopies. A vector field 8 on an open subset U c V is called g-isometric if 8g = 0, which is equivalent (g is closed!) to d(8· g) = 0. We call 8 exact if the (k - I)-form 8· 9 on U is exact, that is 8· 9 = da for some (k - 2)-form a on U.

Consider a Coo-diffeotopy of g-isometric diffeomorphisms 15,: U .... V, tE [0, 1], 150 = Id, and call 15, exact if the vector field 15; = (d/dt)t5, on t5,(U) c V is exact for all t E [0, 1]. Then there is a smooth family of forms a, on t5,(U,) c V, such that da, = 15; • g. Call 15, strictly exact if one can choose these forms a, equal to zero on the maximal open subset Uo c U, where the diffeotopy 15, is constant in t, that is t5,(u) = t5o(u) for u E Uo. Lift 15, to U x We X by ~(u, w) = (t5r(u), w) and observe ~ to be (Ii - g)-exact (strictly exact) for each g-exact (strictly exact) diffeotopy 15,.

Take an open subset Y' c U x W such that ~(Y') c Y for all tE [0, 1] and let h; be the pull-back of the form h on Y under the diffeomorphism ~: Y' .... Y. Since <5;h = d(C5;· Ii) - <5;. g, there exists, for a strictly exact diffeotopy 15" a smooth family of (k - 2)-forms cp; on Y', t E [0,1], which vanishes near the subset where 15, is constant in t and such that Ii; = h + dcp;. Moreover, if 15, is constant for t ~ to on all of U, then one can (and does) choose qJt = 0 for t ~ to.

Let iPo be the sheaf of sections V .... Y. Then there is a (partially defined) action of diffeotopies in V on iPo (in 2.2.3). In particular, if ~(Y') c Y, t E [0,1], and if a section 10 E iPo(U) sends U to Y' then the underlying diffeotopy 15,: U .... V does act on 10. Now, if the diffeotopy 15, is strictly exact, then this action extends to the sheaf iP of the COO-solutions of .@(J,cp) = ° by 15,*(1, cp) = (15,*1, cp - f(cp;)). In fact,

this extension agrees with .@, as .@(t5r*J, cp - f*(cp;)) = f*(h;) + d(cp - f*(cp;)) =

f*(h) + dcp = .@(J,cp).

(B') The h-Principle for Regular Isometric Immersions. Let Vo be a COO-smooth submanifold in V which is sharply movable (see 2.2.3) by strictly exact diffeotopies in V.

Theorem: The h-principle for regular isometric COO-immersions f: «(OftVo,gl(OftVo) .... (w, h) (where (Oft Vo c V is an arbitrarily small neighborhood of yo) is dense in the space of those continuous maps fo: Vo .... W for whichfo*[h] = [g]!VoEH'(Vo; IR).


Proof Let (Preg be the sub sheaf in the above cP for which the maps f: V ...... W underlying the sections 1: V ...... Yare h-regular. The infinitesimal invertibility of ;!j implies the microflexibility of (Preg and hence (see 2.2.3) the flexibility of (Preg on Vo. This gives us the h-principle for (Pregl Vo. The (obvious) homomorphism of (Preg

to the sheaf cPreg of h-regular isometric immersions f: V ...... W for which J sends V to Y clearly is a weak homotopy equivalence. The resulting h-principle for cPreg for the small neighborhoods Y c X = V x W of the graphs of the maps fo: V ...... W easily yields the required dense h-principle. Q.E.D.

Examples (Sharply Movable Submanifolds). (a) Take an arbitrary closed form g on Vo and let V = (Vo x IR, go Ef> 0). Then, clearly, Vo = Vo x 0 c V is sharply movable by strictly exact diffeotopies in V.

(b) The total space of the bundle Ak- I Vo ...... Vo (obviously) admits a unique (k - l)-form, Ak- I on Ak-I(VO)' such that every (k - l)-form J1 on Vo is induced from Ak- I by the form J1 itself viewed as a section J1: Vo ...... Ak- 1 Vo, that is J1 = J1*(Ak- I ). Then every (k - 2)-form v on Vo defines an exact diffeotopy bt on (Ak- 1 Vo, dAk- 1) by the formula bt : (v, x) f--+(v, (l - t)x + t(dv)(v)), for all VE Vo and xE(Ak - 1 Vo)v c Ak- 1 Vo, and where the (k - l)-form dv is viewed as a section dv: Vo ...... Ak-1(VO). It is easily seen that the zero section Vo c. Ak - 1 Vo is sharply movable by these diffeotopies.

(c) Let OJ be a symplectic (see 2.4.2) form on V and let g = OJI for some I ~ 1. Then (see 2.4.2) every submanifold Vo c V of positive co dimension is sharply movable by strictly exact diffeotopies. (d) Let g be a non-vanishing n-form on V for n = dim V. Then every submanifold Vo c V of positive co dimension (obviously) is sharply movable.

Exercise. Let (w, h) = (IRMk, ho + ehd, e ~ 0, where ho = It!1 dXi.1 /\ ••• /\ dXik and where hI is an arbitrary exact COO-smooth k-form. Let k ~ 2, M ~ [2(n + l)!/ k!(n + 1 - k)!] + 2n + 2 and let e be small, 0 ::; B ::; Bo(h1) > o. Prove for an arbitrary COO -smooth exact k-form if on V, dim V = n, the existence of a regular isometric COO-immersion (V, g) ...... (w, h).

(B") Biregular Immersions. An immersion f: V ...... W is called h-biregular if there exists a vector r' E Tw, for w = f(v) and for all v E V, which lies outside the image of the tangent space T,,(V) in Tw(W), for which r'· hwl T,,(V) = 0 and such that the span of T,,(V) and r' is an hw-regular subspace in Tw(W).

Theorem. Let g be a closed Coo-smooth k-form on V. Then the h-principle for hbiregular isometric COO-immersions f: (V, g) ...... (w, h) is CO-dense near those continuous maps fo: V ...... W for which fo*[h] = [g].

Proof Combine (B') with the proof of (A").

Exercises. (a) State and prove the parametric h-principle for biregular isometric immersions. Study extensions of these immersions and prove the pertinent hprinciple.


(b) Let h be a k-form, k ~ 2, on IRq with constant coefficients (which is equivalent to the invariance of h under parallel translations of IRq). Let fo: V --+ IRq be an h-biregular Coo-immersion and let go be the induced form fo*(h) on V. Show an arbitrary continuous map V --+ IRq to admit a fine CO-approximation by isometric COO-immersions (V, go) --+ (IRq, h).

(b') Study (bi)regular isotropic immersions yn --+ IRq [compare (D') in 3.3.4].

3.4.2 Symplectic Immersions and Embeddings

Let h be a symplectic 2-form on W that is dh = 0 and h is nowhere singular on W The h-principle [see (B') in 3.4.1] for isometric immersions (V, g) --+ Cw, h) is especially useful in this case due to an abundance of symplectic (i.e. h-isometric) diffeotopies of W Indeed, since h is non-singular, the homomorphism 1h: T(W) --+ A 1 W =

T*(W) for 1h(8) = 8· h is an isomorphism. Hence, every closed I-form ({J on W defines a symplectic (i.e. h-isometric) field by 8 = r1(1), as 8h = d(8· h) = dh(8) = dl = 0, and the one-parameter subgroup generated by 8 is a symplectic diffeotopy. In particular, every smooth function, called a Hamiltonian, H: W --+ IR defines an exact symplectic field 8 = 1;;l(dH), which gives us a one-to-one correspondence between exact fields on Wand smooth functions modulo (additive) constants. This agrees with the following heuristic "dimension" count [compare (B) in 3.3.4]. The "dimension" of Diff W equals q = dim W, while the space Q of closed 2-forms [which are I-forms/d(functions)] has "dim" Q = q - 1. Hence the isotropy subgroup Symplh c Diff W should have "dimension" one.

Exercise. Study ill-isometric fields for closed (q - I)-forms ill on W, for q = dim W

Now, let (V, g) be another symplectic manifold (i.e. 9 is a symplectic form) and let Vo c V be a submanifold of positive codimension.

Lemma. The strictly exact diffeotopies of (V, g) sharply move yo.

Proof To move a closed hypersurface S lying in a small neighborhood Vo c Vo we start with a vector 00 E T"o(V) transversal to Vo at some point Vo E Vo. This 00

(obviously) extends to an exact field 8 = 19-1(dh) on V which is transversal to Vo, since the neighborhood Vo c Vo is chosen small. In order to make the corresponding exact isotopy (jt for dMdt = a sharply (see 2.2.3) move S, we take the union SE = Ut (jt(S) c V over t E [0, B] and then multiply the Hamiltonian H by a Coo _ function a on V which vanishes outside an (arbitrarily) small neighborhood @ftSE c

V and which equals one in a smaller neighborhood of SE. This makes the diffeotopy corresponding to the field 1;;1 (d(aH)) as sharp as we want. Q.E.D.

Since every immersion into the (symplectic!) manifold (w, h) is h-regular, the lemma shows the h-principle (B') of 3.14. to apply to isometric immersions f: @ft Vo --+

W for all symplectic manifolds (V, g). Now, the implied sections of the pertinent jet bundle are fiber-wise injective isometric homomorphisms Fo: (To, gl To) --+ (T(W), h),

for To = T(V)I yo. Hence, such an Fo is homotopic to the differential Dfl To of the same isometric immersion (!JpVo --+ W if and only if the continuous map fo: Vo --+ W underlying Fo sends the cohomology class [h] E H2(w, h) to [g] 1V0 E H2(VO; ~). Furthermore, the h-principle (B") of 3.4.1 is refined in the symplectic case by the following

(A) Theorem. Let 9 be an arbitrary (possibly singular) closed COO-smooth 2-form on V and let Fo: (T(V), g) --+ (T(W), h) be a fiberwise injective isometric homomorphism for which fo* [h] = [g]. If dim V < dim W, then the map fo admits a fine CO-approximation by isometric Coo-immersions f: (V, g) --+ (w, h) whose differentials Df : T(V) --+

T(W) are homotopic to Fo in the space of fiberwise injective isometric homomorphisms.

Proof Assume without loss of generality the homomorphism Fo to be Coo-smooth, consider the quotient bundle X = fo*(T(W))jT(V) --+ V and embed V into X by the zero section V c; X. Then the homomorphism Fo extends (non-uniquely) to a fiberwise isomorphic homomorphism Fo of the bundle T(X) I V::::) T(V) into T(W). Since Fo is isometric, the induced form FJ'(h) on T(X)I V satisfies FJ'(h) I T(V) = g. Therefore, there exists a closed Coo-form iI on X such that illV = g and which, moreover, equals FJ'(h) on T(X) I v. Indeed, start with the pull-back p*(g) on X for the projection p: X --+ V and let iI = p*(g) + dl for some I-form I on X whose differential on T(X) I V equals FJ'(h) - p*(g). This iI is clearly symplectic (i.e. nonsingular) near V c X and hence, the above h-principle applies to isometric immersions (!Jp V --+ W Furthermore, the microextension theorem of 2.2.4 which works here as well as in (B") of 3.4.1, "descends" the h-principle from (!Jp V to V. Q.E.D.

Corollaries. (a) Let the form 9 on V be exact and let rank(gl T.,)) ~ r for some o ::0;; r ::0;; n = dim V and for all v E V. Then there exists an isometric immersion

provided 2m ~ 3n - r.

Proof The space of injective isometric homomorphisms T.,(v) --+ ~2m is (easily seen to be) k-connected for k = 2m - 2n + r - 1 and (A) in 3.3.1 applies

(b) An n-dimensional manifold V admits an isotropic (i.e. f*(h) == 0) immersion f into (~2n, h = Ii'=1 dXi A dYi) if and only if the bundle T(V) Ee T(V) --+ V is trivial. [Compare Lees (1976).]

Proof Identify ~211 with 1C" = ~n Ee J=J: ~n and observe that the isotropy condition to the orthogonality of the bundle J=t T(V) c T(cn) I V to the tangent bundle T(V). This makes the "only if" part obvious, and the "if" claim follows from (A).

Remark. Isotropic immersions V --+ (w, h) for dim W = 2 dim V are called Lagrange immersions. The self-intersection points of these play an important role in the symplectic geometry (see 3.4.4).

(c) Let h be the standard symplectic form on cpm (which is uniquely characterized by the invariance under the action of the unitary group U(m + 1) on cpm and by the normalization < [h], [CP1 J> = 1» and let a closed form g on V be integral, which means the integrality of fz h for all2-cycles Z in V. If rankvg ~ 3n - 2m for all v E V, then there exists an isometric immersion f: (V, g) -+ (cpm, h).

Proof. The integrality (obviously) is equivalent to the existence of a continuous map fo: V -+ cpm for which fo*[h] = [g] and then the proof of the above (a) applies

Exercises. (1) [Compare Tischler (1977)]. Show every symplectic manifold (V, g) with the integral form g to admit an isometric embedding into (cpm, h) for m = 2n + 1.

Hint. Use a generic symplectic perturbation of the immersion f in the above (c).

(2) Prove the parametric and the extension versions of the h-principle claimed by (A).

(2') Consider a (non-closed) COO-smooth 2-form on V which is closed on the leaves of a given COO-foliation. Generalize (A) to Coo-maps V -+ (w, h) which are isometric immersions on the leaves.

(3) Let V be an open manifold and let 2m = dim V = dim W. Homotope the map fo: V -+ W, which satisfies the assumptions of (A), to an isometric COO-immersion f: (V, g) -+ (w, h) which is no longer required to be CO-close to 10. In particular, construct an isometric immersion of V into a small ball in ([R2m, Li"=1 dx; A dy;), provided V is topologically contractible.

Hint. Study isometric immersions of a symplectic manifold with a boundary into itself.

(3') Prove the h-principle for those microflexible sheaves over a symplectic manifold (V,g) which are invariant under g-isometric diffeomorphisms (compare 2.2.2). Apply this to submersions f: (V, g) -+ X for which the submanifold f-1(X) c

V is symplectic [Le. g If- 1 (x) is non-singular for all x E X], where X is an arbitrary manifold of dimension q :$ dim V.

(3") Since the linear symplectic group is homotopy equivalent to GLmC, each symplectic structure g on V defines a (unique up to a homotopy) classifying map Cg : V -+ GrmCN for 2m = n = dim V and for all N > 2n. Let (V,g) be an open symplectic manifold and let th~ map Cg be (m + I)-contractible. Construct [compare (C) in 2.2.7] an (integrable!) complex structure J: T(V)+=> such that J is a g-isometric automorphism and such that the quadratic form g(x,Jy) is positive definite.

(B) Isometric Embeddings. Let (V, g) be an n-dimensional symplectic manifold and let fo: V -+ W be a COO-embedding such that fo[h] = [gJ. Let ({Jt: T(V) -+ T(W), t E [0, 1], be a homotopy of fiberwise injective homomorphisms whose underlying maps V -+ W equals fo for all tE [0, 1] and such that ({Jo equals the differential DIo'

while the homomorphism ({J1 is isometric for the forms g on T(V) and h on T(W).

Theorem. (1) If q = dim W ~ n + 2 and if the manifold V is open, then there exists a homotopy of COO-embeddings It: V -+ W, such that the embedding f1 is isometric and

the differential Df 1 can be joined with <P1 by a homotopy of isometric homomorphisms (T(V), g) --+ (T(W), h). (2) Let q 2 n + 4. Then the above homotopy ft exists for all (possibly closed) manifolds V. Moreover, one can choose the embeddings ft, t E [0, 1 J as CO -close to fo as one wishes.

Corollary. If V is a contractible manifold and if n ::; q - 4, then every (nonisometric!) C1-embedding V --+ W admits a fine CO-approximation by isometric Coo-embeddings.

Proof The construction of ft proceeds in two steps. First fo is isotoped to a symplectic embedding 1 for which the induced form l*(h) is nowhere singular (symplectic) on V. Then 1 is isotoped to the desired f1 by a homotopy of symplectic embeddings.

Since the symplecticity of 1 (i.e. non-singularity of l*(h)) is an open differential relation, the existence of 1 in the case (1) follows from the results in (e') of 2.4.6. Moreover, those results insure the existence of such an 1 for which the induced form go = l*(h) admits a homotopy of non-singular (possibly nonclosed) forms gt, t E

[0, 1J with gl = g. The existence of gt implies [since V is open, see (B"') 2.2.3J the existence of an exact homotopy of symplectic form gt between g = go and g = gl' where "exact" means the existence of a homotopy of I-forms, say (Xt on V, such that gt = go + d(Xt for all t E [0,1]. Now, an arbitrary homotopy (Xt can be reduced by a small perturbation to a piecewise linear homotopy which interpolates a locally finite sequence of forms, say (Xi = 0, 1, ... , such that (Xi+1 = (Xi + Xi dYi' where Xi and Yi are COO-functions on V with support in a small open subset Vi C V. Indeed, the forms dy over small open subsets V c V span the cotangent bundle A1(V) and the partition of unity applies. Thus the second step in the proof of (1) is reduced to the following.

(B') Lemma. Let gt = go + tdx 1\ dy, tE[O, IJ be symplectic forms on V, where the supports of the functions x and y lie in a contractible open subset V c V, and let 10: (V, go) --+ (w, h) be an isometric embedding of positive codimension. Then there exists a homotopy of isometric embeddings (V, gt) --+ (w, h) which equal 10 outside U and which lie in a given CO -neighborhood of fo for all t E [0, 1].

Proof. Let da 1\ db be the standard area form on 1R2 with the coordinates a and b and let ri be a COO-immersion of 1R2 into the disk De = {a2 + b2 ::; a} c 1R2, such that ri(da 1\ db) = da 1\ db. (The existence of re is obvious for alIa> 0.) Compose the map (x,y): V --+ 1R2 with re and thus obtain a Coo-map, say Ze: V --+ De for which zi(da 1\ db) = dx 1\ dy. Denote by I;: V --+ V X De the graph of the map tz., tE [0, IJ, and observe that I;*(g = go + da 1\ db) = go + t2 dx 1\ dy. Thus, the lemma follows (compar~ 3.1.2) from the existence of an isometric extension J:: (V x De' g) --+

W of the map fo. Since the pertinent operator ij is infinitesimally invertible [see (B) in 3.4.1J, the local extension problem satisfies the h-principle. This reduces the existence of 1. (for small a> 0) to that of an isometric homomorphism F: T(V x De) --+ T(W), such that FI T(V) equals the differential of 10. for V = V x 0 c

V X De. Since V is contractible, F obviously exists and the lemma follows.

Proof of (2). The above argument provides the required homotopy in a small neighborhood of the (n - 1 )-skeleton of a given triangulation of V (for this q ~ n - 2 suffices). This reduces (2) for n ~ 4 to the following

Relative h-Principle. Let V be homeomorphic to the n-ball, let the map fo be isometric near the boundary oV ~ sn-1 and let the homotopy CPt: T(V) ~ T(W) be constant near ov. If n ~ 4 and q ~ n + 4, then there exists a homotopy fr as required in (2) which, moreover, is constant near a v.

Proof Let v" = (V x De> 9 + da /\ db) ::> V = V x 0. Then the map fo on {OftoV ::> V extends, for small e > 0, to an isometric embedding Jo: {OftoV x De ~ W Next, by applying (C') of 2.4.6 as previously we extend this Jo to a symplectic embedding 1 of a small neighborhood (Oft V ~ v" into W Since n > 2 the relative cohomology H2(V, oV) vanishes which implies [see (ell!) in 2.2.3] the existence of an exact symplectic homotopy of forms gt on {Oft V, such that the (properly chosen) 1-forms at vanish on {Oft V c v". Then the earlier argument provides an isometric embedding {Oft V ~ W, whose restriction to V c {Oft V is the desired fl'

Exercise. Fill in the detail in this argument.

The Case n = 2. If q ~ 6, then generic isometric immersions obviously are embeddings and the above (A) applies.

Exercise. Prove the parametric h-principle for isometric embeddings with a special consideration for the case n = 2.

Remark. The embedding h-principles (1) and (2), probably hold true for singular closed forms 9 on V if rank 9 ~ 2n - q + 2 in the case (1) and rank 9 ~ 2n - q + 4 for (2). However, codimension 2 symplectic embeddings of closed manifolds are related to some problems in the symplectic geometry which go beyond the hprinciple.

Exercises. (a) Let 9 be a closed form of constant rank on V. Prove the h-principle (1) for rankg ~ 2n - q + 2 and prove (2) for rankg ~ 2n - q + 4.

(b) Let V be diffeomorphic to IRn and let U be an open subset in (1R2P, h =

If=l dXi /\ dy;) with a smooth (possibly empty) boundary. Prove the existence of a proper isometric embedding (V, g) ~ (U, h) for an arbitrary symplectic form 9 on V, provided 2p ~ n + 2.

(C) Stability of Symplectic Forms. The Nash implicit function theorem can be replaced in the symplectic geometry by the following more elementary.

Stability Theorem (Darbaux-Moser-Weinstein, see Weinstein 1977). Let gt, t E [0,1], be a COO-homotopy of symplectic forms on V such that gt = go + dat where the 1-forms at vanish outside a compact subset U c V. Then there exists a Coo -diffeotopy <>t: V ~ V, for tE[O, 1], <>0 = Id, which is constant in t outside U and such that <>1(gt) = go for


t e [0, 1 J Furthermore, if the forms CXt vanish on a given submanifold Vo c V, while dcxt vanish on the bundle T(V)I VO, then one may choose bt constant on Vo as well.

Proof Perturb the forms CXt, if necessary, in order to make them zero on T(V)I VO. Then the vector field cx~ = - 1;/(dcxt/dt) (for which cx~· gt = dcxt/dt) vanishes on Vo as well as outside U and cx~gt = dgt/dt. Since the field cx~ has compact support, it integrates to a unique diffeotopy bt such that dbt/dt = cx~ and bo = Id. Then (d/dt)(b*(gt}} = bt*(cx~gt + (dgt/dt}} = O. Q.E.D.

Exercises. (a) Prove the results (A) and (B) using the stability theorem instead of Nash's theorem.

(b) Construct local coordinates Xl' ••• , Xm, Yl' ••• , Ym in a small neighborhood of a given point in a 2m-dimensional symplectic manifold (V, g), such that g = L:"=l dXi /\ dYi'

(c) Let gt, t e [0, 1], be a homotopy of symplectic forms on a closed manifold V, such that [go] = [gl] e H2(V; ~). Assume H2(V;~) ~ ~ and construct an isometry (V, go) -+ (V, g d·

(d) Let go and gl be symplectic Can-forms on V. Show that every Coo-isometry (V,go) -+ (V,gd admits a fine Coo-approximation by Can-isometries.

(e) (Moser 1965; Green-Shiohama 1979). Let Wo and Wl be non-vanishing Coo(Can)-smooth n-form on a connected oriented n-dimensional (possibly noncompact with or without boundary) manifold V, such that Iv Wo = Iv W1 > O. Furthermore, if I v Wo = 00, assume

fu Wo = 00 -- fu Wl = 00,

for all open subsets U c V with compact boundaries au = (CIU)\ U c V. Construct a Coo(Can)-diffeomorphism f: V -+ V, such that f*(wd = Wo'

State and prove a similar result for smooth measures (which are locally given by non-vanishing n-forms) on non-orientable manifolds V.

3.4.3 Contact Manifolds and Their Immersions

Let L be a Coo-subbundle of the tangent bundle T(W), let L' = T(W)/L and let k T(W) -+ L' be the quotient homomorphism. There (obviously) exists a unique anti-symmetric bilinear map of L to L', called d' A: A2 L -+ L' such that every linear form cx on L' satisfies d(cx 0 A) = cx 0 (d' A). For example, if codim L = 1, k = 1 and if the bundle L' is identified with the trivial line bundle, then A reduces to a I-form on Wand d'A = dAIL. [Compare Gray (1959).]

Define for every vector leLw c Tw, we W, the homomorphism 1(1) = I d';.(l): Lw -+ L'w by 1(1)(1') = (1. d' A)(l') = d' A.(l, I') for all I' e Lw' A subspace Ko c Lw is called (d'A)-regular if the homomorphism Lw -+ Hom(Ko,L') defined by 1-+ 1(l)IKo is surjective.

Next, consider a Coo-subbundle K c T(V) and call a smooth map f: (V, K)-+ (w, L) contact if the differential Df sends K to L. Such an f is called regular if Df is injective on Kv and the Drimage of Kv in Lw, w = f(v) is (d'A)-regular for all ve V.

(A) Lemma. Regular contact CO-maps f: V --+ W form a microj7exible sheaf.

Proof. Fix a local basis of I-forms ai: L' --+ IR, i = 1, ... , r = codimL and define a differential operator !!fl: (CD-maps V --+ W) --+ (r-tuples of I-forms on K) by !!fl: f* f--+ {f(ai ° A)IK}. The linearization of!!fl acts on the tangent fields 0 in T(W)I V by [see (A) in 3.4.1].

Of--+{do·(aioA) + o·d(aioA)}IK,

which reduces for fields 0 in LfV c T(W)fV to Of--+ {aioI(o)}IK. Therefore [compare (A) in 3.4.1 J, the operator !!fl is infinitesimally invertible at regular contact maps f. Since these are solutions of !!flf = 0, the lemma follows.

Exercise. Define biregular contact maps [compare (A') in 3.4.1J and prove the h-principle for them.

(B) Let, L = Ker A c T(W) for a I-form A on W Then the subbundle L (as well as the form A) is called contact if the 2-form d' A = dAI L is nowhere singular. Then KerdA is a one-dimensional subbundle in T(W) transversal to L, called Lt c T(W).

A vector field 0 on W is called contact if it preserves the subbundle L. This is equivalent to

OAIL = (0· dA + d(o· A))IL = O.

Let 00 and o~ be the respective Land Lt-components of o. Then oAIL = 1(00 ) + d(A(O ~))IL. Hence, every function H on W defines a unique contact field 0 (depending on the choice of A) for which A(O~) = H, and 00 = - rl(dH). It follows, (compare 3.4.2) that diffeotopies of contact diffeomorphisms (w, L):::J sharply move all submanifolds in W Thus, we conclude [compare (A) in 3.4.2J to the following

(B') Theorem. Let L c T(W) be a contact subbundle of codimension one. Then for every subbundle K c T(V) contact COO-immersions (V, K) --+ (w, L) satisfy the Codense parametric h-principle, provided dim V < dim W

Corollaries and Exercises. (a) If K = T(V), then contact immersions V --+ Ware everywhere tangent to L. For dim W = 2n + 1, n = dim V, these are called Legendre immersions. According to the CO-dense h-principle, one obtains (Duchamp 1983) the following

Approximation Theorem. Let Fo: T(V) --+ K be a fiberwise injective homomorphism, such that F6'(d' A) == O. Then the underlying continuous map fo: V --+ W admits a fine CO-approximation by Legendre immersions f: V --+ W

(b) Let W = (1R 2 p+1, L = Ker A) for A = dz + Il=l Xi dYi and let K be a coorientable [i.e. K' = T(V)jK is orientableJ contact subbundle in T(V). If P :::;; m + n for dim V = n = 2m + 1, then an arbitrary continuous map fo: V --+ W admits a fine CO-approximation by contact immersions. Indeed, the inequality p ~ m + n insures the existence of a homomorphism Fo: (T(V), K) --+ (T(W), L) which pulls back the

340 3. Isometric Coo -Immersions

form d'), on L to (a non-zero multiple of) the respective form d'l( on K (where Kerl( = K).

(b') Prove (b) by the techniques of (E) in 2.1.3. (c) Extend (C) of 3.4.2 to contact manifolds.

3.4.4 Basic Problems in the Symplectic Geometry

We have seen that the h-principle applies to symplectic isometric embeddings for codim ~ 4. Furthermore, the construction of symplectic forms on open manifolds V is possible with the h-principle [see (B"') in 2.2.3]. But the further study of symplectic manifolds brings forth new phenomena.

(A) The Existence Problem. Let a closed manifold V of dimension n = 2m satisfy

(i) there is a cohomology class a E H2 (V, ~) such that am # 0; (ii) there exists a (possibly non-closed) nowhere singular 2-form go on V. [This is

equivalent to a reduction of the structure group of the tangent bundle T(V) to the complex linear group GLmC, that is an almost complex structure on v.J

Does there exists a homotopy of non-singular forms gt, t E [0, IJ, for which gl

is symplectic? (The h-principle predicts such a homotopy but one may expect further obstructions to the existence of gt.) Does V support any symplectic form at all?

(A') A similar problem arises for the contact structure. Namely, let V admit a co dimension one subbundle K c T(V) with a nowhere singular 2-form go: K --+

T(v)IK (if K is co-orientable one speaks of ordinary ~-valued 2-forms). Does K admit a homotopy to a contact subbundle? [The answer is "yes" for dim V = 3, see Lutz (1971), Martinet (1971).J

(B) Standard Examples of Symplectic and Contact Manifolds. Let h be the standard symplectic form on cPq [which is U(q + 1)-invariantJ and <h, [Cpl J > = 1. Then every complex submanifold V c cPq is symplectic, since the induced form hi V is nowhere singular on V. Thus every complex algebraic manifold admits a symplectic structure.

(B') Let a Lie group L admit a left invariant symplectic form g and let r c L be a discrete subgroup. Then, by passing to the quotient, we get a symplectic form on V = LI r. This is especially interesting if V is compact.

Exercise. Let L be a nilpotent Lie group and let some a E H2(V;~) have am # 0 for 2m = dim L. Prove the existence of an invariant symplectic form on L and thus obtain a symplectic form on V = VI rfor all discrete subgroup r c L. [See CorderoFernandez-Gray (1985) for further results of this type.J

(B") Let V be a strictly pseudo-convex hypersurface in a complex manifold W Then the subbundle T(V) n J=1 T(V) c T(V) c T(W) is (easily seen to be) contact. For


example, the unit sphere s2m+l C Cm+ l carries a U(m + I)-invariant contact structure which, moreover, is invariant under the group U(m + 1,1) (of automorphisms of the form ZlZl - Ii';;:? ZiZi in Cm+ 2 ) which acts on s2m+l by linear fractional transformations. This structure agrees (in an obvious way) with the symplectic form on cpm for the Hopf map s2m+1 --+ cpm and the pull-back of every symplectic submanifold Vo c cpm is a contact submanifold in s2m+1. Furthermore, if X is a complex analytic submanifold in Cm+ l which transversally meets s2m+l (or any pseudo-convex hypersurface for this purpose) then the intersection X n s2m+1 is a contact submanifold in s2m+1. In particular, every (germ of a) subvariety X c Cm

with an isolated singularity at the origin meets every small sphere s;m+1 C Cm +1

over a contact submanifold. [In fact, Eliashberg (1983) proved e~ contact structure on a 3-dimensional manifold V to be the above T(V) n -J -1 T(V) for some complex structure on V x IR :::; v.]

Exercises. (a) Recall [see (C) of3.2.2] the canonical contact structure on the oriented Grassmann bundle Q = Grq_1P for a given q-dimensional Can-manifold P and obtain this structure by a pseudo-convex embedding of P into a complexification CP:::;P.

(b) Find local coordinates z, Xi' Yi' i = 1, ... , m, on s2m+l minus a point, such that Ker(dz + I;"=l Xi dy;) equals the sub bundle T(S2m+l) n j=t T(S2m+1).

(c) Construct a contact structure on s2m X S\ m = 1, 2, ... , which equals r:---...J

Ker(dz + I~=l Xi dy;) on the universal covering s2m X Sl = 1R2m +1 \ {O}.

(C) Symplectic Fibrations. Let (V, g) be a symplectic manifold, let p: X --+ V be a smooth fibration and let hv be a symplectic form on the fiber Xv = p-l(V) c X which is smooth in v for all v E V.

(C') (Thurston 1973). If there is a cohomology class f3 E H2(X, IR), such that f3IXv =

[hv], then X admits a symplectic structure.

Proof There obviously exists a smooth closed 2-form h on X, such that hlXv = hv for all v E V. Then, assuming X is compact, (the open case is covered by the h-principle) the form p*(g) + eh is non-singular on X for all small e > O. Q.E.D.

Example. (a) Let Y --+ V be a k-dimensional complex vector bundle and let PY --+ V be the associated projective bundle with the fiber Cpk - l • Then one has, with a Hermitian metric in Y, the canonical symplectic forms on the fibers Xv = Cpk - l •

Furthermore, the Chern class Cl of the canonical complex line bundle X --+ PY satisfies cliP¥" = [hvJ and, hence, (C') applies.

(b) (Thurston 1973). Let X be a bundle of closed oriented surfaces over a symplectic manifold V, such that the homology class [XvJ E H 2 (X) has an infinite order. Then the above hv and f3 obviously exist and yield a symplectic structure on X.

Exercise (Thurston). Find a closed symplectic manifold X of dimension 4 which admits no Kahler structure.

(D) Blow up. Take the canonical line bundle p: Xo -+ cpm and consider the obvious holomorphic map O"t: Xo -+ Cm+1. This 0"0 is biholomorphic outside the origin OECm+1 and 0"0'1(0) equals cpm (embedded into Xo by the zero section). Then the pull-back O"t(go) on X of the standard form go = Li!.il dXi /\ dYi on Cm+1 can be perturbed to a symplectic form on Xo as follows. Take the standard symplectic form g1 on cpm and observe that the form P*(g1) on Xo is cohomologous to zero outside cpm c Xo. Hence, there exists a closed form g1 on Xo which vanishes away from cpm and which equals p*(gd near cpm c Xo. Then, obviously, the form O"t(go) + eg1 is nowhere singular on Xo for all small e > O.

Next, take a point Vo in an arbitrary symplectic manifold (V, g) of dimension 2m + 2. Then there is a small neighborhood Uo c V of Vo which is isometric to the 2e-ball B2, in (Cm+1, Li=i1 dXi /\ dYi) around the origin. Delete the neighborhood U c Uo of Vo which corresponds to the e-ball in Cm+1 and then attach eo1(B2 ) to V by the map 0"0 over Uo \ U = B2 \B •. The resulting blow up manifold V (whi~h is diffeomorphic to the connected su~ V # cpm+1) comes with a natural map uo: V -+

V which is a diffeomorphism of uO'1(V\{VO}) onto V\{vo} and for which uO'1(vo) ~ cpm. The above local construction provides a symplectic form on V which equals u*(g) outside any given neighborhood of u-1(vo) c V.

Let us apply the "blow up" 0"0: Xo -+ Cm+1 to the fibers of a complex vector bundle Y -+ Vo. That is, we take the associated projective bundle PY -+ Vo with the canonical line bundle X -+ PY and then take the obvious map 0": X -+ Y. This 0" is diffeomorphic outside Vo 4 Y (embedded by the zero section) while 0"-1(VO) = PY. If Vo is a symplectic manifold, then we equip PY with a symplectic form according to the above Example (a) and we extend this form to X by a closed form g vanishing away from PY c X. Then we lift the symplectic form go on Vo to X and observe (an exercise to the reader) that the form O"*(go) + eg is nowhere singular on X for small e > O.

Finally, we take a symplectic submanifold Vo c (V, g) (i.e. go = gl Vo is symplectic) and let Y -+ Vo be the g-normal bundle of Vo in V. We equip Y with a complex bundle structure for which the multiplication by .j=-i is isometric for 9 1 Y and we blow up this Y to the above X -+ Y. Then, as earlier, we delete a small neighborhood U c Vof Vo and attach some neighborhood of the submanifold PY eX. Thus we blow up V along Vo and we construct (the detail is up to the reader) a symplectic form on the blown up manifold.

Exercises. (a) (Mc Duff 1984). Construct, by blowing up Cps along some 4-dimensional symplectic submanifold Vo c Cps, a closed simply connected symplectic manifold which admits no Kiihler structure.

(b) An immersed symplectic submanifold Vo -+ V, by definition, has a symplectic crossing if the self-intersection is transversal and if the k-intersection, say v,. -+ V symplectically immerses into V. Assume v,. to be empty for k ~ ko + 1 and blow up V along the (embedded!) symplectic submanifold v,.o c V. Show that Vo lifts to an immersed symplectic submanifold Vo in the blown up manifold P, such that the self-intersection ~ of Vo is symplectic for k < ko and v,. is empty for k ~ ko. Then blow up V along ~-1 and keep blowing up until the self intersection is completely eliminated.


(b') Let VI be a symplectic submanifold in (vo, go), such that codim VI =

dim V - dim Vo ~ 4 and let the group 71..2 = 71../271.. freely and isometrically act on VI. State and prove the embedding type h-principle for isometric immersions (VO, go) --+ (V, g) which symplectically self intersect along VI according to the given 71..2-action and have no self intersection apart from VI.

Question. Can one define singular symplectic (sub) varieties? Can one resolve their singularities?

(c) Let (V, K) be a contact manifold and let Vo c V be a contact submanifold, i.e. the subbundle Ko = K n T(Vo) c T(V) is contact. Then the normal bundle Yof Vo in V equals the (d'K)-orthogonal complement of Ko in KI Yo. Equip the bundle Y with a complex structure compatible with d'KI Y [i.e. the multiplication by J=1 is (d'K)-conformal] and define the blow up of V along Vo as earlier. Assume the existence of a free contact Sl-action on (VO, Ko) which lifts to a fiberwise complex linear action on Y and construct a contact structure on the blow up manifold V. In particular, prove the existence of a contact structure on the blow up V, in case (VO, Ko) is isomorphic to S2k+1 with the standard [U(k + 1)-invariant] contact structure.

(E) Ramified Coverings. Let Vo be a codimension two symplectic submanifold in a symplectic manifold (V, g) and let p: V --+ V be a ramified covering which ramifies along Yo. (V is obtained from a finite covering of the complement V\ Vo by completing with respect to a metric induced from some Riemannian metric in V.) The induced form p*(g) on V can be easily perturbed to a symplectic form g on V which equals p*(g) away from p-I(VO) c V. Unfortunately, this construction gives few new examples, as symplectic embeddings Vo --+ V violate the h-principle (codim Vo = 2!) and so the submanifolds Vo c V are hard to come by.

Exercise. Study symplectic ramified coverings with a singular ramification locus Vo c V.

(E') Consider two disjoint codimension 2 symplectic embeddings of Vo into V and let Y1 and Y2 be their respective normal bundles. Take small tubular neighborhoods TI and T2 of these embeddings, let 15: aTI --+ aT2 be a diffeomorphism induced by an orientation reversing isomorphism Y1 --+ Y2 (the forms gl Y1 and gl Y2 endow the bundles with natural orientations), and let V be obtained from V by deleting TI U T2 and gluing the boundary of V\(T1 U T2) by b. Then the form g outside TI U T2 easily extends to a symplectic form on the manifold V.

Exercises. (a) Generalize the above to immersed submanifolds Vo c V with g-normal crossings.

(b) Generalize (E) and (E') to codimension 2 contact submanifolds in (V, K). (c) (Meckert 1982). Construct a contact structure on the connected sum of two

contact (2m + 1 )-dimensional manifold.

Hint. Use the contact action of U(m + 1,1) on s2m+l.

(F) The results indicated in (I) below provide many pairs of open subsets Vi and V2

in (1R2m, ho = Li!=i dXi A dYi), m ~ 2, which admit a volume preserving diffeomorphism V1 ~ V2 but not a symplectic (i.e. ho-isometric) one. For example, the open bicylinders

U { 2 2 2 2 2 b2} 111>4 i = Xl + Yl < ai ,x2 + Y2 < i C ~ , i = 1, 2, and 0 < ai < bi'

are symplectically diffeomorphic if and only if a1 = a2 and b1 = b2. A similar fact is known for contact 3-manifolds.

(F') (Bennequin 1983). There exists a contact subbundle K on 1R3 which admits no contact embedding into (1R3, Ker(dz + x dy)).

(G) Double Points of Lagrange Submanifolds. The h-principle provides Lagrange immersions f: V ~ (W,h), dim W = 2 dim V = 2n, which have, after a generic Lagrange perturbation, a discrete subset of transversal double points. The lower bound on the possible number of these points is an interesting invariant of the symplectic form h (and of the topology of V) which is not reducible to the h-principle.

If the form h is exact, h = dlX, then the formf*(IX) is closed on V [since f*(h) = 0] and the cohomology class [f*(IX)J EH1(V, lR)depends onfand h but not on IX. Then one distinguishes exact Lagrange immersions for which [f*(IX)J = O.

Example. The obvious Lagrange embedding ofthe torus Tn into (1R 2m L7=1 dXi A dYi) is not exact. In fact, no closed manifold admits an exact Lagrange embedding into (1R2n, L7=1 dXi A dYi) (see Gromov 1985).

Exercises. (a) Show that the circle Sl admits no Lagrange embedding into 1R2. (b) Let V be a parallelizable manifold. Prove for 7 -:F n = dim V ~ 2 (compare

totally real embeddings in 2.4.5) the existence of a symplectic form h on 1R 2n which receives an exact Lagrange embedding f: V ~ (1R2n, h).

(b') Show for the above V the existence of a (non-exact!) Lagrange embedding V x Sl ~ (1R2n+2, L7~l dXi A dyJ

Hint. Start with an isotropic embedding V ~ (1R211+2, L7~l dXi A dYi)'

(bl!) Let V admit a Morse function with k critical points. Then construct an exact Lagrange immersion V x Sl ~ (1R211+2, L7~l dXi A dYi) with precisely k transversal double points.

(c) Construct for an arbitrary closed manifold V a closed symplectic manifold (W,h), dim W = 2 dim V which receives a Lagrange embeddingf: V ~(W,h).

Hint. Represent V by the IR-points of a complex algebraic manifold W defined over IR.

(G') Fixed Points of Exact Diffeomorphisms. An isometric diffeomorphism of a symplectic manifold,f: (V,g)~ is called exact if there exists an exact diffeotopy (see 3.4.1) fr: (V,g)~, such that fo = Id and fl = f For example, if a is an exact field, a = I;;l(H) for a COO-function H on V, then the one parameter subgroup of diffeo-


morphisms fr: V ~ is clearly exact. Then every critical point of H is fixed under the diffeomorphisms fr and so the number of the fixed points of f1 can be estimated from below by Morse theory, assuming V is a closed manifold. Observe that the isometry condition dli8) = 0 is equivalent to (the graph of) the I-form Ig(8): V ..... N V to be a Lagrange embedding for the canonical symplectic form ho on N v. In fact, Laudenbach and Sicorav (1985) extended the Morse theory to Lagrange submanifolds in N V obtained from the zero section by an exact diffeomorphism.

Furthermore, fixed points of an arbitrary g-isometric diffeomorphism f: V ..... V are exactly the intersection of the following two Lagrange submanifolds in (V x V, g EEl - g): the first is the diagonal and the second is the graph of f

Exercise (Arnold 1974; Weinstein 1976). Let an exact diffeomorphism f of a closed symplectic manifold V be sufficiently C 1-close to the identity. Then there exists a COO-function H = HI: V ..... [R (classically known as a generating function for f) whose critical points are exactly the fixed points of f

The above C 1-closeness condition was relaxed by Weinstein (1983) to the CO-closeness of the pertinent exact diffeotopy to the identity. The crucial new ingradient is due to Conley and Zehnder (1983) who solved the following

Arnold's Conjecture. Let (T2m, g) be a symplecting torus obtained from ([R2m, Ii'=l dXi /\ dy;) by dividing by a lattice lL2m C [R2m and let f be an exact g-isometric COO -diffeomorphism of T2m. Then there exists a Coo -function H = HI on V whose all critical points are fixed under f. Moreover, if the fixed points of f are non-degenerate, then the critical points of H also are nondegenerate. (In the latter case the number of the fixed points of f is bounded from below by 22m and in the former by 2m + 1.)

The idea of the proof (due to Conley and Zehnder) is indicated in the following

Exercise. (a) Consider an N-dimensional vector bundle over a closed manifold, say z ..... V and let A: z ..... IR be a COO-function whose restriction to each fiber Zv ~ [RN, VE V, is a non-singular quadratic form (i.e. I~=1 zf - Il"=k+l zf in some linear basis in zv). Let h be an arbitrary bounded COO-function on X. Show that the number of critical points of the function A + h on Z is bounded from below by the Morse theory for functions on V. In particular, A + h has at least two critical points (and at least 2m + 1 if V ~ T2m).

(a') Generalize (a) by allowing fibrations X with infinite dimensional Hilbert space fibers Zv ~ [ROO.

(1)) Let fr, t E [0,1] be an exact diffeotopy of a symplectic manifold (V, g), such that dfr/dt = 1;;I(dH), where H = H(v, t) is a COO-function in VE V and tE [R, which is periodic with period = 1 in t, and where the differential dH applies to the v-variable. Establish a one-to-one correspondence between the fixed points of fl and those Coo-maps z: S1 ..... V, for SI = [R/lL, which satisfy

dz(t) = 1;;1 (dH(z(t), t). dt


(b') (Hamilton principle). Assume V = T 2m = 1R2m;Z2m and define for contractible maps z, which lift to maps z: Sl -+ (1R 2m, D"=l dXi /\ dYi), the action A(y) =

Ssd*(IX) for the 1-form IX = Li!=l xidYi on 1R2m. Show (*) to be the Euler-Lagrange equation for the functional A(z) + h(z) for h(z) = Ss! H(z(t), t) dt.

(c) Prove, by suitably adopting (a), that the above functional has as many critical points z, as Morse theory predicts for functions on T2m. Prove Arnold's conjecture by approximating the space Zv ~ lRoo of maps z: Sl -+ 1R2m with Ss! z(t) dt = v E 1R2m

by the (finite dimensional!) spaces of truncated Fourier expansions of z. Prove the above-mentioned result by Weinstein for CO-diffeotopies [see Chaperon (1983, 1984) for further results in this direction].

Remark. Arnold's conjecture for all closed surfaces was proven by Eliashberg (1978) by a more direct (and more complicated) method. [Compare Floer (1984), Sikorav (1984), Fortune-Weinstein (1984), Laudenbach-Sikorav (1985).]

(Gil) Contact Embeddings. One expects the codimension two contact embeddings of closed manifolds to display certain "rigidity" (similar to that of Lagrange embeddings) which is not accountable for by the h-principle. The following result by Benniquin (1983) [compare Eliashberg (1981)] confirms this belief.

Theorem. Let Sl c (1R3, K = Ker(dz + x dy)) be a closed unknotted Legendre curve. Then Sl has a strictly negative linking number with its small parallel translate in the direction of the z-coordinate.

Exercise. Derive (F') from this theorem.

(H) The Group Diffy of Sympletic Diffeomorphisms of (V, g). Eliashberg (1981) announced (among many deep facts) that Diff;' is a CO-closed subgroup in the group Diffgm of diffeomorphisms preserving the volume form gm where 2m = dim V. One can recapture Eliashberg's result by combining the techniques indicated in (I) with the following.

Maximality Theorem. Let (V, g) be a closed connected symplectic manifold. Let a (possibly non-closed) subgroup '§ c Diff;' contain all g-exact diffeomorphisms and let some element tjJ E '§ be neither g-isometric nor anti-isometric, that is tjJ*(g) =1= ± g. (If m is odd, then no tjJ E Diff;' is anti-isometric.) If H1(V; IR) = 0, then the subgroup '§ contains the connected component of the identity element Id E Diff;'. Furthermore, if H1 (V; IR) # 0, then '§ contains the subgroup of gm-exact diffeomorphisms which are defined in the course of the proof

Proof A vector field 0 on V preserves gm iff ogm = d(o· gm) = mo· 9 /\ gm-1 = 0. Such a 0 is called gm-exact if the cohomology class [0· gm] E H 2m- 1(v, IR) is zero. Thus every gm-exact field 0 equals I;;J.(dcp) for some (2m - 2)-form cp on V.

Next consider N-tuples of symplectic forms {gv} on V such that g'{' = gr;! = ... = g;; = OJ. Call an N-tuple {gv} large [compare (D) in 2.3.8] if the forms g:-l span the bundle A 2m- 2 v.

(H') Lemma. Let {gv} be a large N-tuple. Then every w-exact field 0 on V is a sum, o = L~=l Ov where Ov is a gv-exact gv-isometric field for all v = 1, ... , N.

Proof Let o· w = d<p = 1",(0). Then, by the largeness, there are COO-functions Hv on V such that <P = m L~=l Hvg';'-l. Put Ov = /;,1 (dHv) and compute,

o = 1;;1 (d<p) = ml;;l Ct1 dHv A gm-1) = ml;;l Ct1 ov' gv A g';'-l)

= ml;;l ( m-1 vt1 Ov' g,;,) = 1;;1 (v~ 1",(Ov)) = Vt10v. Q.E.D.

(HI!) Remark. One obtains a canonical decomposition 0 = Lv Ov with a canonical choice of <P for o· w. This is possible, for example, with the Hodge-De Rham theory.

Lemma. There exist diffeomorphisms Iv E '§, v = 1, ... , N, for some N, such that the N-tuple of the induced forms gv = fv*(g) is large.

Proof Consider diffeomorphisms in '§ which keep a given point Vo E V fixed and let Go consist of the linear transformations of the tangent space Tvo(V) ~ ~2m which are the differentials of all these diffeomorphisms at Vo' Since g-exact diffeomorphisms transitively act on V, one reduces the Lemma with a standard partition of unity argument, to showing that the Go-orbit of the form gO'-2 = gm-21 T.,o(V) linearly spans the space Azm-zT.,o(V)' Since '§ does not preserve ±g, the Go-orbit of go contains a 2-form g' which is not a scalar multiple of go. Then (an exercise in linear algebra) the span of the orbit of g' under linear symplectic transformations of (Tvo(V), go) equals AZT.,o(V). Hence, the span of the orbit Go(go) also equals A2T.,o(V) which implies (another exercise) the equality Span GO(g:f-2) = Azm-zT.,~(V). Q.E.D.

Let :1£ be the Cartesian product of N copies of Diff;' and let f0: :1£ --+ Diff;;" be defined by f0: (Xl> x z, .. ·' xN) --+ <PI . Xl . <P2 . X z ..... <PN' X N for some fixed <Pv E '§ and for the group product in DifT;;". If we identify the tangent space 7[(Diff;;") for f = f0(Id) = f = <PI . <pz ..... <PN with 1id(Diff;;") by the right translation 15 1---4 Jf-1 in f0iff;;", then the linearization (differential) L of f0 at IdE.or will send N-tuples of g-isometric fields to gm-isometric fields by the formula

N

L: {ov} 1---4 L Di.10., v=l

for f1 = <P1,fZ = <Pl' <PZ, .. ·,fN = f = <Pl' <p2· .. ·· <PN' We arrange the diffeomorphisms fv to make the N-tuple Uv*(g)} large and then

we solve the equation 0 = L{ ov} for all gm-exact fields 0 on V by using (H') and by observing that the differential DJ , bijectively maps gv-isometric fields, gv = f/(g), onto g-isometric ones. Hence, the linearization L is surjective at Id E.or and the above argument also shows this L to be surjective near Id E.or. Furthermore, (HI!) provides a right inverse to L, to which the implicit function theorem in 2.3.2 applies. Strictly speaking, the results in 2.3.2 do not apply to our f0 which is not a differential operator. However, this operator (obviously) satisfies the estimates needed for


the construction of (non-local) inversion 2.C-1• Thus we come to the following conclusion.

Let Jt E2.Ciff;", 0::;; t::;; 1, be a gm-exact dif!eotopy, which means the exactness of the field dJtl dt for all t E [0, 1], such that 150 = Id. Then Jt for t E [0, e], e > 0, lies in the image of 2.C. If Hl(V, IR) ~ H 2m- 1(v, IR) = 0, then these diffeotopies (obviously) generate the connected component of Id E 2.Ciff;". In general, they generate a subgroup of codimension ::;; dim Hi (V, IR) in 2.Ciffgm which by definition consists of all gm-exact diffeomorphisms. Q.E.D.

Exercises. (a) Fill in the details in this proof. Then extend the maximality theorem to the subgroups 2.Ciff: c 2.Ciffoo, for an arbitrary volume form (J) on V, and to 2.Ciff~ c 2.Ciffoo for contact manifolds (V, K).

(b) Let a subgroup f§ c 2.Ciff;", which is not assumed to contain 2.Ciff;" , be normalized by 2.Ciff;", that is ff§f- 1 = f§ for all f E 2.Ciffg , and let f§ contain a diffeomorphism '" which is neither g-isometric nor anti-isometric. Show all gm-exact diffeomorphism to lie in f§.

Hint. Show f§ to be normalized by all exact diffeomorphisms and apply Thurston's theorem (1974) on normal subgroups in 2.Ciffgm.

Remark. If one tries a direct proof of (b) one arrives, after the linearization, to a linear system of difference equations to which the formalism of 2.3.8 does not apply. It would be interesting to extend the technique of 2.3.8 to difference equations (e.g. by an approximation of differences by differentials), in order to prove (b) without (in fact, together with) Thurston's theorem. See Mc Duff (1984) and Banyaga (1978) for a further study of symplectic diffeomorphisms.

(I) Pseudo-Holomorphic Curves in Symplectic Manifolds. A 2-dimensional subset S c V is called a J-curve for a given almost complex structure J: T(V) --+ T(V) [See (C) in 2.2.7] if there exists a (connected or not) smooth surface 8 with an almost complex structure J: T(S} --+ T(S} and a Coo-map f: 8 --+ V, such that

(i) f sends 8 onto S and the set of double points, {(x,y)E8 x 8If(x) = f(y)}, is discrete,

(ii) fis pseudo-holomorphic, which means J 0 Df(T) = Df 0 J(T) for all tangent vectors TE T(8).

Such an (8, J), if it exists, clearly is unique up to an isomorphism; the parametrizing map f is unique up to an automorphism of (8, J).

If the structure J is Coo -smooth, then the space L of closed J -curves in V is (easily seen to be) locally finite dimensional. This is also true for compact J-curves S with the boundary condition 8S c V', where V' is totally real submanifold in V, that is T(V') n JT(V') = 0. Furthermore, if V is a closed symplectic manifold whose 2-form 9 is J -positive, which means g( T, h) > ° for all non-zero vectors T E T(V), then the area of closed J-curves S c V (relative to a fixed Riemannian metric in V) obviously abides

AreaS::;; constll[S]II,

where [S] EH2 (V) is the fundamental homology class of S and II II denotes a fixed norm on the homology of V (A similar inequality holds true for compact J-curves S whose boundary as lies in a fixed Lagrange submanifold V' c V) The inequality (*) insures a weak compactness (in the current topology) of the space I' of closed J-curves S c V with II [S] II ~ const' < 00, which leads [compare (B") in 3.2.4] to certain existence theorems of closed J-curves in V (and of compact J-curves whose boundaries lie in a prescribed Lagrange submanifold V' c V). Here is a simple example [see Gromov (1985) and Mc Duff(1985) for further applications of J-curves to symplectic geometry].

(I') Let the symplectic (J-positive!) form 9 split,

(V, g) = (S2 X V1,go + gd,

where (V1,g1) is a closed symplectic manifold and where go is an area form on the sphere S2. Then, for each point VE V, there exists a connected J-curve S = Sv c V which contains v and such that [S] = [S2 x v1] E H2(V).

Exercise. Let a round ball B(r) of radius r in the standard symplectic space (IRn, wo), n = dim V, admit a symplectic embedding (B(r), wo) --+ (V, g). Show that nr2 ~ IS2 go.

(I") Remark. The above J-curve Sv may be singular (it may even have Sv disconnected) and, in general, it is not unique. However, if dim V = 4, if the surface V1 is connected and if Iv 1 g1 = IS2 go, then Sv is unique. Moreover, this Sv c V is a smoothly embedded sphere which is also smooth as a function in the variable J (as long as the inequality g(r,1r) > 0 holds).

Exercise. Show the group of symplectic diffeomorphisms of (S2 x S2, go + 9 1) to be weakly homotopy equivalent to the subgroup of orientation preserving isometries of S2 x S2 (which is a Zj2Z-extension of SO(3) x SO(3)), provided IS2g1 = IS2g0. (If IS2 gl #- IS2 g2 the two groups are not w.h. equivalent.)

References

Adachi, M. (1979) Construction of complex structures on open manifolds, Proc. Jap. Acad. 55, Ser. A, pp. 222-224.

Ahlfors, L.V. (1935) Zur Theorie der Dberlagerungsfliichen, Acta Math. 65, pp. 157-194. Alexander, J. (1920) Note on Riemann spaces, Bull. Am. Math. Soc. 26, pp. 370-372. Alexandrov, A. (1944) Intrinsic metrics of convex surfaces in spaces of constant curvature, Dokl. Akad.

Nauk S.S.S.R., 45: 1, pp. 3-6. Allendoerfer, C.B. (1937) The embedding of Riemann spaces in the large, Duke Math. J. 3, pp. 317-333. Aminov, lA. (1975) On estimates of the volume and diameter of a submanifold in Euclidean space,

(Russian), Ukr. Geom. Sbornik 18, pp. 3-15, 1982 Aminov, J.A. (1982) Embedding problems: Geometric and topological aspects, Itogi Nauki Tekh. Ser.

Probl. Geom. (in Russian). Arnold, V. (1957) On functions of three variables, Dokl. Akad. Nauk U.S.S.R., 114, #4, pp. 679-681. Arnold, V. (1974) Mathematical methods in classical mechanics, Moscow. Arnold, V. (1980) Lagrange and Legendre cobordisms, J. Funct. Analysis and Applications, 14, pp. 1-14. Asimov, D. (1975) Round handles and non-singular Morse-Smale flows, Ann. Math. (1) 102, pp. 41-54. Asimov, D. (1976) Homotopy to divergence free vector fields, Topology 15, pp. 349-352. Atiyah, M. (1976) Elliptic operators, discrete groups and Von Neumann algebras, Asterisque 32-33,

Soc. Math. de France, pp. 43-72.

Atiyah, M., Patodi, V. and Singer I. (1975) Spectral asymmetry and Riemannian geometry, Math. Proc. Camb. Phil. Soc. 77, pp. 43-69.

Audin, M. (1984) Quelques calculs en cobordisme Lagrangien, Prepublication, Orsay. Banyaga, A. (1978) On fixed points of symplectic maps, Preprint. Baldin, Y. and Mercuri, F. (1980) Isometric immersions in codimension two with non-negative curvature,

Math. Zeit. 173, pp. 111-117. Barnette, D.A. (1973) Proof of the' lower bound conjecture for convex polytopes, Pac. J. Math. 46:2,

pp .349-354. Barth, W. (1975) Larsen's theorem on the homotopy groups of projective manifolds of small embedding

codimension, Proc. Symp. in Pure Math., XXIX, A.M.S., pp. 307-315. Bennequin, D. (1983) Entrelacements et equations de Pfaff, Soc. Math. de France, Asterisque 107-108,

pp.87-161. Berger, E. (1981) The Gauss map and isometric embedding, Theses, Harvard. Berger, E., Bryant, E. and Griffiths, P. (1983) The Gauss equation and rigidity of isometric embeddings,

Duke Math. Journ. 50:3, pp. 803-892. Bierstone, E. (1973) An equivariant version of Gromov's theorem, BAMS 79, pp. 924-929. Blank. S. (1967) Thesis, Brandeis. Blanusa, D. (1955) Dber die Einbettung hyperbolischer Raiime in euclidische Riiume, Mon. Math. 59,

# 3, pp. 217-229. Boardman, J. (1967) Singularities of differentiable maps, Publ. Math. IHES n033, pp. 21-57. Borisenko (1977) Complete I-dimensional surfaces of non-positive extrinsic curvature in a Riemannian

space, Math. 56. (N.S.) 104 (146):4, pp. 559-576. Borisov, Ju. (1965) C""-isometric immersions of Riemannian spaces, Dokl. Akad. Nauk, S.S.S.R., 163,

#, pp. 11-13. Bott, R. (1969) Lectures on K(X), W.A. Benjamin.

References 351

Bott, R. (1970) On topological obstructions to integrability, Proc. Symp. Pure Math. 16, A.M.S., pp. 127~131.

Briant, L., Griffiths, P. and Yang, D. (1983) Characteristics and existence of isometric embeddings, Duke Math. Journ., 50:4, pp. 893~995.

Brown, E. (1962) Cohomology theories, Ann. Math. 75, pp. 467~484. Burago, Yu. (1968) Inequalities of isoperimetric type in the theory of surfaces of bounded exterior

curvature. Proc. Leningrad, Steklov Inst., v. 10. Burago, Yu. and Zalgaller, V. (1960) Polyhedral imbedding of a net-vest., Leningrad University, 15, # 7,

pp. 66~80. Burago, Yu. and Zalgaller, V. (1980) Geometric Inequalities, Leningrad (The English translation by

Springer-Verlag is in preparation). Burlet, O. (1976) Propos au sujet des applications dilTerentiables, Singularites d'applications dilTeren

tiables, Springer Lect. Notes Math., 535, pp. 187~204. Chaperon, M. (1983) Quelques questions de geometrie symplectique, Sem. Bourbaki, Asterisque 105~ 106,

pp. 231~249. Chaperon, M. (1984) Questions de geometry symplectique, Sem. Sud-Rhodanien IV, Balarue, to appear

in Travaux en cours, Hermann, Paris. Cartan, H. (1953) Varietes analytiques complexes et cohomologie, 2° Coli. de geometrie algebrique.

(Liege) Centre Beige de Rech. Math. pp. 41~55. Cartan, H. (1957) Varietes analytiques reelles et varietes analytiques complexes, Bull. Soc. Math. Fr. 85,

pp. 77~99. Cartan, H. (1958) Espaces fibres analytiques, Symp. Int. de Top. Alg., Univ. Nac, de Mexico, pp. 97~121. Cheeger,1. and Gromoll, D. (1971) The splitting theorem for manifolds of non-negative Ricci curvature,

1. DilT. Geom. 6, pp. 1l9~128. Chen, K.T. (1971) DilTerential forms and homotopy groups, 1. DilT. Geom., 6, #2., pp. 231~246. Chen, K.T. (1973) Iterated integrals of dilTerential forms and loop space homology, Ann. Math. (2) 97,

pp. 217~246. Cerf, 1. (1984) Suppression des singularites de codimension plus grande que 2 dans les families de

functions dilTerentiables reelles, Sem. Bourbaki, Juin, Paris. Chern, S.S. and Lashosf, R. (1957) On the total curvature of immersed manifolds, Am. 1. Math. 79,

pp. 306~318. Chern, S.S. and Kuiper N.H. (1952) Some theorems on the isometric imbeddings of compact Riemannian

manifolds in Euclidean space, Ann. Math. 56, #, pp. 422~430. Clarke, C. (1970) On the global isometric embeddings of pseudo-Riemannian manifolds, Proc. Royal

Soc. Lond. A-314, pp. 417~428. Cohen, R.L. (1984) The homotopy theory of immersions, Proc. ICM in Warszawa, p. 627~639, North

Holland. Cohn-Vossen, S. (1933) Sur la courbure totale des surfaces ouvertes, C.R. Ac. Sc. (Paris) 197,

pp. 1l65~1l67. Conley, C. and Zehnder, E. (1983) The BirkholT-Lewis fixed point theorem and a conjecture of V.I.

Arnold, Invent. Math. 73, pp. 33~49. Connelly, R. (1977) A counterexample to the rigidity conjecture for polyhedra, Publications Mathe

matiques IHES n° 47, pp. 333~338. Cordero L., Fernandez M. and Gray A. (1985) Symplectic manifolds with no Kiihler structure, Preprint. Cowen, M. and Griffiths, P. (1976) Holomorphic curves and metrics of negative curvature, J. An. Math.

29, pp. 93~153. D'Ambra (1985) Construction of connection inducing maps between principal bundles I, Preprint

M.S.R.I., Berkeley. Dajczer, M. and Gromoll, D. (1984) On spherical submanifolds with nullity, Preprint, SUNY at Stony

Brook. Deligne, P. and Sullivan, D. (1975) Fibres vectoriels complexes a groupe structural discret, C.R. Ac. Sc.

(Paris) 281, Ser. A-108!. Duchamp, T. (1984) The classification of Legendre immersions, Preprint. Duplessis, A. (1975) Maps without certain singularities, Comm. Math. Helv. 50, pp. 363~382. Duplessis, A. (1976) Contact invariant regularity conditions, Singularites d'Applications DilTerentiables,

Springer Lee!. Notes Math. 535, pp. 205~236.

352 References

Efimov, N. (1949) Qualitative problems in the local deformation theory of surfaces, Proc. Steklov Inst. xxx.

Efimov, N. (1964) Generation of singularities on surfaces of negative curvature, Math. Sb. (N.S.) 64, pp. 286-320.

Eliashberg, J. (1970) On maps of the folding type, Izv. Akad. Nauk S.S.s.R. 34, pp. 1111-1127. Eliashberg, J. (1972) Surgery of singularities of smooth maps, Izv. Akad. Nauk S.S.S.R. 36, pp. 1321-1347. Eliashberg, J. (1978) Estimates of the number of fixed points of area preserving maps, preprint, Syktyvkar

(in Russian). Eliashberg, J. (1981) Rigidity of symplectic and contact structures, Preprint. Eliashberg, J. (1984) Cobordisme des solutions de relations differentielles, Seminaire Sud-Rhodanien de

geometrie symplectique I, Hermann, Paris. Farrell, F.T. and Jones, L.E. (1981) Expanding immersions on branched manifolds, Am. Journ. of Math.

103, pp. 41-101. O'Farrell, A.G. (1986) Coo-maps may increase COO-dimension, Preprint IHES, France. Feit, S. (1969) k-mersions of manifolds, Acta Math. 122, # 3-4, pp. 173-195. Feldman, E. (1968) Deformations of closed space curves, J. Diff. Geom. 2, pp. 67-75. Feldman, E. (1971) Nondegenerate curves on a Riemannian manifold, J. Diff. Geom. 5, pp. 187-210. Ferus, D. (1975) Isometric immersions of constant curvature manifolds, Math. Ann. 217: 2, pp. 155-156. Ferus, D., Karcher, H. and Munzner, H. (1981) Cliffordalgebren und neue isoparametrische Hyperflachen,

Math. Z. 177, pp. 479-502. Fiala, F. (1941) Le probleme des isomerimetries sur les surfaces ouvertes a courbure positive, Comm.

Math. Helv. 13, pp. 293-346. Floer, A. (1984) Proof of the Arnold conjecture for surfaces and generalizations for certain Kahler

manifolds, Preprint, Bochum. Forster, O. (1970) Plongements des varietes de Stein, Comm. Math. Helv. 45, pp. 170-184. Forster, O. (1971) Topologische Methoden in der Theorie Steinscher Raume, ICM-1970, Nice,

pp.613-618. Fortune, B. and Weinstein, A. (1984) A symplectic fixed point theorem for complex projective spaces. To

appear. Friedman, A. (1961) Local isometric imbeddings of Riemannian manifolds, with indefinite metrics, J. Math. Mech. 10, #4, pp. 625-649. Friedman, A. (1965) Isometric imbeddings into Euclidean spaces, Rev. Mod. Phys. 37, #1, pp. 201-203. Fuchs, D. (1977) Non-trivialite des classes caracteristiques de g-structures, C.R. Ac. Sc. (Paris) 284,

pp. A-ll05-1109. Fuchs, D. (1981) Foliations, Itogi Nauki Tekh., Ser. Probl. Geom., 18, pp. 191-213. Garsia, A. (1961) An embedding of closed Riemann surfaces in Euclidean space, Comm. Math. Helv. 35,

#2, pp. 93-110. Gluck, H. (1975) Almost all simply connected closed surfaces are rigid, Geom. Topology (Proc. Conf.

Park-City, Utah), Springer Lect. Notes Math. 438, pp. 225-239. Godbillon, C. and Vey, J. (1971) Un invariant des feuilletages de codimension 1, C.R. Ac. Sci. (Paris) Ser.

A-B 273, pp. A 92-A 95. Godement, R. (1958) Topologie algebrique et theorie des faisceaux, Hermann, Paris. Golubitsky, M. and Guillemin, V. (1973) Stable mappings and their singularities, graduate Text in Math.

Springer-Verlag. Grauert, H. (1957) Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen, Math. Ann.

133, pp. 450-472. Grauert, H. (1958) On Levi's problem and the embedding of real analytic manifolds, Ann. Math. (2) 68,

pp.460-472. Gray, A. (1969) Isometric immersions in symmetric spaces, J. Diff. Geom. 3, pp. 237-244. Gray, J.W. (1959) Some global properties of contact structures, Ann. Math. 69, pp. 512-540. Greene, R. (1970) Isometric embeddings of Riemannian and pseudo-Riemannian manifolds, Mem. Am.

Math. Soc. # 97. Greene, R. and Jacobowitz, H. (1971) Analytic isometric embeddings, Ann. Math 93, pp. 189-203. Greene, R. and Wu, H. (1975) Whitney's imbedding theorem by solutions of elliptic equations and

geometric consequences, Proc. Symp. Pure Math. XXVII, #2, A.M.S., pp. 287-297.

References 353

Greene, R. and Shiohama, K. (1979) DifTemorphisms and volume preserving embeddings of non-compact manifolds, Trans. Am. Math. Soc. 255, pp. 403-414.

Griffiths, P. (1974) Entire holomorphic mappings in one and several complex variable, Ann. of Math. Studies, 85, Princeton.

Gromoll, D., Klingenberg, W. and Meyer, W. (1968) Riemannsche Geometrie im Grossen, Springer Lect. Notes Math. 55.

Gromoll, D. and Meyer, W. (1969) On complete open manifold of positive curvature, Ann. Math. 90, pp.75-9O.

Gromov, M. (1969) Stable maps of foliations into manifolds, Izv. Akad. Nauk, S.S.S.R. 33, #4, pp.707-734.

Gromov. M. (1971) A topological technique for the construction of solutions of differential equations and inequalities, ICM. 1970, Nice, vol. 2, pp. 221-225.

Gromov, M. (1972) Smoothing and inverting differential operators, Mat. Sbornik, pp. 383-441. Gromov, M. (1973) Convex integration of differential relations, Izv. Akad. Nauk S.S.S.R. 37, #2,

pp.329-343. Gromov, M. (1981) Curvature, diameter and Betti numbers, Comm. Math. Helv. 56, pp. 179-195. Gromov, M. (1983) Filling Riemannian manifolds; I. DifT. Geom. 18, pp. 1-147. Gromov, M. (1985) Pseudo-holomorphic curves in symplectic manifolds, 82, pp. 307-347 Inv.

Math. Gromov, M. and Eliashberg, 1.(1971), Construction of nonsingular isoperimetric films, Trudy Steklov

Inst. 116, pp. 18-33. Gromov, M. and Eliashberg, 1.(1971), Removal of singularities of smooth maps, Izv. Akad. Nauk, S.S.S.R.

35, #5, pp. 600-627. Gromov, M. and Eliashberg, 1.(1971) Nonsingular maps of Stein manifolds, Func. Anal. and Applications

5, # 2, pp. 82-83. Gromov, M. and Eliashberg, 1.(1973) Construction of a smooth map with prescribed 1acobian, Func.

Anal. and Applications 7, #1, pp. 27-32. Gromov. M. and Lawson, B. (1980) The classification of simply connected manifolds of positive scalar

curvature, Ann. of Math. 111, pp. 423-434. Gromov, M. and Lawson, B. (1983) Positive scalar curvature and the Dirac operator on complete

Riemannian manifolds, Publications Mathl:matiques IHES n° 58, pp. 295-408. Gromov, M. and Rochlin, V. (1970) Embeddings and immersions in Riemannian geometry, Uspechi,

XXV, # 5, pp. 3-62. Gunning, R. and Narasimhan, R. (1967) Immersion of open Riemann surfaces, Math. Ann. 174,

pp.l03-108. Gunning, R. and Rossi, H. (1965) Analytic functions of several complex variables, Prentice-Hall Inc.

New 1ersey. Haefliger, A. (1962) Plongements differentiables dans Ie domaine stable, Comm. Math. Helv. 37,

pp.155-176. Haefliger, A. (1962) Varietes, feuilletes, Ann. Sc. Norm. Sup. Pisa (3),16, pp. 367-397. Haefliger, A. (1970) Homotopy and integrability, Springer Lect. Notes Math., 197, pp. 133-164. Haefliger, A. and Hirsch, M. (1962) Immersions in the stable range, Ann. of Math. (2) 75, pp. 231-241. Halpern, B. and Weaver, C. (1977) Inverting a cylinder through isometric immersions and isometric

imbeddings, Trans. Ann. Math. Soc. 230, pp. 41-70. Hamenstiidt U. (1985) Non-degenerate closed curves in Sft. To appear in Composito Mathern. Hamilton, R. (1982) The inverse function theorem of Nash and Moser, BAMS 7: 1, pp. 65-222. Hartman, P. (1971) On the isometric immersions in Euclidean space of manifolds with non-negative

sectional curvatures II, Trans. Am. Math. Soc. 147:2, pp. 529-540. Hayman, W. (1964) Meromorphic functions, Oxford Math. Monographs, Oxford. Hironaka, H. (1973) Subanalytic sets, Numb. Theory, Alg. Geom. and Comm. Alg., in honour of

Y. Akizuki, Tokyo, pp. 453-493. Hirsch, M. (1959) Immersions of manifolds, Trans. Am. Math. Soc. 93, pp. 242-276. Hirsch, M. (1961) On imbedding differential manifolds into Euclidean space, Ann. Math. 73, pp. 566-571. Hitchin, N. (1974) Harmonic spinors, Adv. Math. 14, pp. 1-55. Hormander, L. (1963) Linear partial differential operators, Springer, Berlin Heidelberg New York.

354 References

Hormander, L. (1976) The boundary problems of physical geodesy, Arch. Rat. Mech. Anal. 62, # 1, pp.I-52.

Hopf, H. (1925) Uber die Curvatura Integra geschlossener Hyperfliichen, Math. Ann. 95, pp. 340-367. Huber, A. (1957) On subharmonic functions and differential geometry in the large, Comm. Math. Helv.

32, #1, pp. 13-72. Ionin, V. (1969) Isoperimetric and certain other inequalities for manifolds of bounded curvature, Sib.

Math. J. 10, pp. 329-342. Jacobowitz, H. (1972) Implicit function theorems and isometric embeddings, Ann. Math. 95, pp. 191-225. Jacobowitz, H. (1973) Isometric embedding of a compact Riemannian manifold into Euclidean space,

Proc. Ann. Math. Soc. 40: 1, pp. 245-246. Jacobowitz, H. (1974) Extending isometric imbeddings, J. Diff. Geom. 9, pp. 291-307. Jacobowitz, H. (1976) Equivariant embeddings of flat tori, Preprint. Jacobowitz, H. (1982) Local isometric embeddings, Sem. on Diff. Geom., pp. 381-393, Ann. of Math.

Studies 102, Princeton. Janet, M. (1926) Sur la possibilite de plonger un espace riemannien donne dans un espace euclidien, Ann.

Soc. Pol. Math 5, pp. 38-43. Jost, J. and Karcher, H. (1982) Geometrische Methoden zur Gewinnung von A-priori-schranken fUr

harmonische Abbildungen, Manuscripta Math. 40, pp. 27-77. Jorge, L.P. and Xavier, F.V. (1981) An inequality between the exterior diameter and the mean curvature

of bounded immersions, Math. Z. 178, pp. 77-82. Kalai, G. (1986) Rigidity and the lower bound theorem I, preprint Hebrew University, Jerusalem. Kiillen, A. (1978) Isometric embedding of a smooth compact manifold with a metric of low regularity,

Arch. Mat. 16, #1, pp. 29-50. Kazdan, J. and Warner, F. (1975) Prescribing curvatures, Proc. Symp. Pure Math. XXVII, #2, A.M.S.,

pp.309-321. Kervaire, M. and Milnor, J. (1960) Bernoulli numbers, homotopy groups and a theorem of Rohlin, Proc.

ICM 1958, pp. 454-458, Cambridge Univ. Press New York. Kirby, R. and Siebenmann, L. (1977) Foundational essays on topological manifolds, smoothing and

triangulations, Ann. Math. St. Princeton. Kobayashi, S. and Nomizu, K. (1963, 1969) Foundations of differential geometry, v. I and II, Wiley-Inc.

New York. Kodaira, K. (1964) On the structure of compact complex analytic surfaces, I, Am. J. of Math. 86,

pp.751-798. Kolmogorov, A. (1956) On representation of continuous functions of several variables by superpositions

of continuous functions of fewer variables, Dokl. Akad. Nauk, S.S.S.R. 108, # 2, pp. 179-182. Kuiper, N.H. (1955) On Cl-isometric imbeddings, I. Proc. Koninkl. Nederl. Ak. Wet., A-58, pp. 545-556. Kuiper, N.H. (1958) Immersions with minimal total absolute curvature, Coli. Geom. Diff. Glob., Centre

BeIge de Rech. Math., pp. 75-88. Labouri, P. (1984) Thesis, Paris. Landweber, P. (1974) Complex structures on open manifolds, Topology 13, pp. 69-75. Lannes, J. (1982) La conjecture des immersions, Sem. Bourbaki, Asterisque 92-93, Soc. Math. de France,

pp.331-347. Laudenbach, F. and Sikorav J-e. (185) Persistance d'intersection avec la section nulle au cours d'une

isotopie hamiltonienne dans un fibre cotangent, Preprint, Orsay. Lawson, B. (1974) Foliations, Bull. Ann. Math. Soc. 80, # 3, pp. 417. Lees, J. (1976) On the classification of Lagrange immersions, Duke Math. J. 43, pp. 217-224. Levin, H. (1965) Elimination of cusps, Topology 3, pp. 263-296. Levin, H. (1971) Blowing up singularities, Springer Lect. Notes Math. 192, pp. 90-104. Levin, H. (1971) Singularities of differentiable mappings, Springer Lect. Notes Math. 192, pp. 1-90. Little, J. (1970) Nondegenerate homotopies of curves on the unit 2-sphere, 1. Diff. Geom. 4, pp. 339-348. Little, J. (1971) Third order nondegenerate homotopies of space curves, J. Diff. Geom. 5, pp. 503-515. Lojasievicz, S. (1965) Ensembles semi-analytiques, Lecture Notes IHES, Bures-sur-Yvette. Lutz, R. (1971) Sur quelques proprietes des formes differentielles en dimension 3, These Strasbourg. Martinet, J. (1971) Formes de contact sur les varietes de codimension 3, Springer Lect. Notes Math. 209,

pp.142-163.

References 355

Mather, J. (1971) On Haefliger classifying space, Bull. Ann. Math. Soc. 77, pp. 1111-1115. Mather, J. (1973) Solutions of generic linear equations, Dynamical System, ed. Peixoto. Acad. Press. Mattila, P. and Rickman, S. (1979) Averages of the counting function of a quasiregular mapping, Acta

Math. 143, pp. 273-305. Mc Duff, D. (1978) On the group of volume preserving diffeomorphisms of IR". Mc Duff, D. (1979) Foliations and monoids of embeddings, Geom. Top. (ed. Contrell) Ac. Press. Mc Duff, D. (1981) On groups of volume preserving diffeomorphisms and foliations with transverse

volume form, Proc. London Math. Soc. 43, pp. 295-320. Mc Duff, D. (1984*) Application of Convex integration to symplectic and contact geometry, Preprint,

SUNY at Stony Brook. Mc Duff, D. (1984**) Examples of simply connected symplectic manifolds which are not Kahler, Preprint,

SUNY at Stony Brook. Mc Duff, D. (1985) Examples of symplectic structures, Preprint SUNY. Meckert, C. (1982) Forme de contact sur la somme connexe de deux varietes de contact de dimension

impaire, Ann. Inst. Four. 32:3, pp. 251-260. Milnor, J. (1963) Morse theory, Ann. Math. St. 51, Princeton. Milnor, T. (1972) Efimov's theorem about complete immersed surfaces of negative curvature, Adv. Math.

8, pp. 474-543. Mishachev, N. (1979) Constructions ofllags of foliations, Uspechy 34, #1 (205), pp. 237-238. Mishachev, N. and Eliashberg, J. (1977) Surgery of singularities offoliations, Func. Anal. and Applications

11, # 3, pp. 43-53. Montogomery, D. and Zippin, L. (1955) Topological transformation groups, Interscience Publ., Wiley

and Sons N.Y. Moore, J.D. (1972) Isometric immersions of space forms in space forms, Pac. J. Math. 40, pp. 157-166. Moore, J.D. (1978) Codimension two submanifolds of positive curvature Proc. A.M.S. 70, p. 72-74. Moore, J.D. and Schlally, R. (1980) On equivariant isometric embeddings, Math. Z. 173, pp. 119-133. Morin, B. (1965) Formes canoniques des singularites d'une application differentiable, C.R. Ac. Sci. (Paris)

260, pp. 5662-5665. Morse, A.P. (1939) The behavior of a function on its critical set, Ann. Math. 40, pp. 62-70. Moser, J. (1961) A new technique for the construction of solutions of nonlinear differential equations,

Proc. Nat. Ac. Sci. U.S.A., 47, # 11, pp. 1824-1831. Moser, J. (1965) On the volume elements on a manifold, Trans. Am. Math. Soc. 120, pp. 286-294. Narasimhan, R. (1967) On the homology groups of Stein spaces, Inv. Math. 2, pp. 377-385. Narasimhan, M. and Ramanan, S. (1961) Existence of universal connections, Am. J. Math. 83,

pp.563-572. Nash, J. (1954) C1-isometric imbeddings, Ann. Math. 60, #3, pp. 383-396. Nash, J. (1956) The imbedding problem for Riemannian manifolds, Ann. Math. 63, #1, pp. 20-63. Nash, J. (1966) Analyticity of the solutions of implicit function problems with analytic data, Ann. Math.

84, #2, pp. 345-355. O'Neil, B. (1962) Isometric immersions ofllat Riemannian manifolds in Euclidean space, Michig. Math.

J. 9, pp. 199-205. Novikov, S.P. (1965) The homotopy and topological invariance of certain rational Pontryagin classes,

Dokl. Akad. Nauk, S.S.S.R., 162, pp. 1248-1251. Otsuki, T. (1953) On the existence of solutions of a system of quadratic equations and its geometric

application, Proc. Jap. Ac. 29, pp. 99-100. Pasternak, J. (1975) Classifying spaces for Riemannian foliations, Proc. Symp. Pure Math., A.M.S.,

v. 27, part 1, pp. 303-310. Phillips, A. (1967) Submersions of open manifolds, Topology 6, # 2, pp. 170-206. Phillips, A. (1969) Foliations on open manifolds, Comm. Math. Helv. 44, pp. 367-370. Phillips, A. (1974) Maps of constant rank, Bull. Ann. Math. Soc. SO, # 3, pp. 513-517. Poenaru, V. (1966) Regular homotopy in codimension 1, Ann. Math. (2) 83, pp. 257-265. Poenaru, V. (1968) Extensions des immersios en codimension 1 (d'apres S. Blanck) Sem. Bourbaki

342-01-342-33. Poenaru, V. (1970) Homotopy theory and differential singularities, Springer Lect. Notes Math. 197,

pp. 106-133.

356 References

Pogorelov, A. (1969) Exterior geometry of convex surfaces, Nauka, (Russian). Pogorelov, A. (1971) An example of a 2-dimensional metric admitting no local realization in E3 , Dokl.

Akad. Nauk S.S.S.R. 198, 1, pp. 42-43. Poznjak, E. (1973) Isometrical immersions of 2-dimensional metrics into Euclidean spaces, Uspechy 28,

#4, pp. 47-76. Poznjak, E. and Sokolov, D. (1977), Isometrical immersions from a Riemannian space into Euclidean

spaces, (Itogi Nauki), Tekh. Ser. Probl. Geom. pp. 173-211. Ramspott, KJ. (1962) Dber die Homotopieklassen holomorpher Abbildungen in homo gene komplexe

Manningfaltgkeiten, Bayer. Ak. Wiss. Math. Natur. Kl. S.-B. Abt II, pp. 57-62. Reinhart, B. (1983) Foliation, Erg. Math. Springer-Verlag. Ritt, J. (1950) Differential algebra, Ann. Math. Soc. Publications, V. XXXIII. Rolfsen, D. (1976) Knots and links, Math. Lect. Series, Publish or Perish, Inc. Rosendorn, E. (1960) Realization of the metric ds2 = du2 + P(u)dv2 in the five-dimensional Euclidean

space, Dokl. Akad. Nauk Arm. S.S.S.R. Rosendorn, E. (1961) Construction of a bounded complete surface of non-positive curvature, Uspechy

16, #2, pp. 149-156. Rosendorn, E. (1966) Weakly non-regular surfaces of negative curvature, (Russian), U.M.N. 5 (131),

(Russian Math. Surveys), pp. 59-116. Sard, A. (1942) The measure of the critical values of differentiable maps, Bull. Ann. Math. Soc. 48,

pp. 8883-890. Schaft, U. (1984) Einbettungen Steinscher Mannigfaltigkeiten, Manuscripta Math. 47, pp. 175-186. Schilt, H. (1937) Dber die isolierten Nullstellen der Fliichenkriimmung Compo Math. 5, # 2, pp. 105-119. Schoen, R. (1984) Minimal surfaces and positive scalar curvature. ICM 1983, Warszawa, p. 575-579,

North-Holland. Schoen, R, and Yau, S.T. (1979) Existence of incompressible minimal surfaces and the topology of three

dimensional manifolds of positive scalar curvature, Ann. Math. 110, pp. 127-142. Schoen, R. and Yau, S.T. (1979) On the structure of manifolds with positive scalar curvature, Manuscripta

Math. 28, pp. 159-183. Schur, F. (1886) Dber die Deformation der Riiume konstanter Riemannschen Kriimmungsmass, Math.

Ann. 27, pp. 170. Schwartz, J. (1960) On Nash's implicit function theorem, Comm. Pure and Appl. Math. 13, pp. 509-530. Seeley, R.T. (1964) Extensions of COO-functions defined in a half space, Proc. Ann. Math. Soc. 15,

pp. 625-626. Segal, G. (1978) The classifying space for foliations, Topology 17, #4, pp. 367-383. Serre, J.-P. (1953) Groupes d'homotopie et classes de groupes abeliens, Ann. Math. 58, pp. 258-294. Sergeraert, F. (1972) Un theoreme de fonctions implicites sur certains espaces de Frechet et quelques

applications, Ann. Ec. Norm. Sup., 4eme ser., v. 5, pp. 599-660. Siebenmann, L. (1973) Approximating cellular maps by homeomorphisms, Topology 11, pp. 271-294. Singer, I.M. (1960) Infinitesimally homogeneous spaces, Comm. Pure Appl. Math. 13, pp. 685-697. Smale, S. (1958) A classification of immersions of the two-sphere, Trans. Ann. Math. Soc. 90, pp. 281-290. Smale, S. (1959) The classification of immersions of spheres in Euclidean spaces, Ann. Math., pp. 327-344. Shub, M. (1969) Endomorphisms of compact differential manifolds, Am. J. Math. XCI. #1, pp. 175-199. Shulman, H. (1972) Characteristic classes and foliations, thesis, u.e. Berkeley. Sikorav, J.e. (1984) Points fixes d'un symplectomorphisme homologue a l'identite, C.R. Ac. Sc., to appear. Spanier, E.H. and Whitehead, J.H.e. (1957) Theory of carriers and S-theory, Alg. Geom. and Top. (A

symp. in Honour ofS. Lefschets), Princeton Univ. Press, Princeton N.J. Spivak, M. (1979) A comprehensive introduction to differential geometry, Publish or Perish Inc.,

Berkeley. Spring, D. (1983) Convex integration of non-linear systems of partial differential equations, Ann. Inst.

Fourier, 33: 3, pp. 121-177. Spring, D. (1984) COO-solutions to non-linear undetermined systems of partial differential equations,

Preprint York University, Canada. Stiel, E. (1965) Isometric immersions of manifolds of non-negative constant sectional curvature, Pac. J.

Math. 15, pp. 1415-1419. Sullivan, D. (1973) Differential forms and topology of manifolds, Proc. Conf. on Manifolds, Tokyo.

References 357

Szuecs, A. (1982) The Gromov-Eliashberg proof of Haefliger's theorem, Studia Sci. Math. Hung. 17, pp.303-3/8.

Szuecs, A. (1983) On detached immersions, Ibid, to appear. Szuecs, A. (1984) Cobordism groups of immersions with restricted self-intersections, Osaka Journ. of

Math. 21, pp. 71-80. Szuecs, A. (1976) Cartan-De Rham homotopy theory, Asterisque 32-33, pp. 227-253. Thorn, R. (1955, 1956) Les singularites des applications differentiables, Ann. Inst. Fourier 6, pp. 43-87. Thorn, R. (1959) Remarques sur les problemes comportant des inequations differentielles globales, Bull.

Soc. Math. Fr. 87, pp. 455-461. Thurston, W. (1973) Some simple examples of symplectic manifolds, Preprint. Thurston, W. (1974) Foliations and groups of diffeomorphisms, Bull. Ann. Math. Soc. 80, #2,

pp.304-307. Thurston, W. (1974) The theory of foliations of codimension greater than one, Comm. Math. Helv. 49,

pp.214-231. Thurston, W. (1976) Existence of codimension-one foliations, Ann. Math. 104, pp. 249-268. Thurston, W. (1978) Geometry and topology of 3-manifolds, Lecture Notes, Princeton. Tischler, D. (1977) Closed 2-forms and an embedding theorem for symplectic manifolds, J. Diff. Geom.

12, pp. 229-235. Toda, H. (1962) Composition methods in homotopy groups of spheres, Princeton Univ. Press. Tomkins, C, (1939) Isometric imbedding offlat manifolds in Euclidean space, Duke Math. J. 5, pp. 38-61. Verner, A. (1970) Non-boundedness of a hyperbolic horn in Euclidean space, (Russian) Sib. Math. J ourn.

9:1, pp. 20-29. Wall, C.T.C. (1971) Stratified sets, Springer Lect. Notes Math. 192, pp. 133-141. Weinstein, A. (1970) Positive curved n-manifolds in [Rn+Z, Joum. Diff. Geom. 4, pp. 1-4. Weinstein, A. (1973) Lagrangian manifolds and Hamiltonian systems, Ann. Math. 98: 3, pp. 377-410. Weinstein, A. (1977) Lectures on symplectic manifolds, North Carolina, Regional Conf. Ser. in Math.,

N 29, A.M.S., Providence. Weinstein, A. (1984) CO-perturbation theorems for symplectic fixed points and Lagrangian intersections,

Geometrie Symplectique et Contact, Edited by P. Dazord and N. Desolneux-Moulis, Hermann, Paris, pp. 140-145.

Wells, R. (1966) Cobordisms of immersions, Topology 5, #3, pp. 281-294. Whitney, H. (1934) Analytic extensions of differentiable functions defined in closed sets, Trans. Ann.

Math. Soc. 36, pp. 63-89. Whitney, H. (1936) Differentiable manifolds, Ann. Math. 37, pp. 645-680. Whitney, H. (1937) On regular closed curves in the plane, Compo Math. 4, pp. 276-284. Whitney, H. (1944) The singularities of a smooth n-manifold in (2n - I)-space, Ann. Math. 45,

pp.247-293. Whitney, H. (1955) On singularities of mappings of Euclidean spaces I. Mappings of the plane into the

plane, Ann. Math. 62, pp. 374-410. Whitney, H. (1957) Elementary structure of real algebraic varieties, Ann. Math. 66, pp. 545-556. Wintgen, D. (1978) Dber von haherer Ordnung regulare Immersionen, Math. Nachr. 85, pp. 177-184. Yoshifumi, A. (1982) Elimination of certain Thorn-Boardman singularities of order two, Joum. Math.

Soc. Jap. 34:2. pp. 241-267. Yau, S.T. (1976) Parallelizable manifolds without complex structure, Topology 15, pp. 51-53. Zalgaller, V. (1958) Isometric imbedding of polyhedra, Dokl. Akad. Nauk. S.S.S.R. 123, pp. 599-601. Zeghib, A. (1984) Feuilletages geodesiques de varietes localement symetriques et applications, These de

3e cycle, Dijon.

Author Index

Adachi, M 103 Ahlfors, L.V. 288 Alexander, J. 30 Alexandrov, A. 289 Allendoerfer, C.B. 239 Aminov, J.A. 259, 282 Arnold, V. 11, 109, 345 Asimov, D. 183 Atiyah, M. 16,58 Audin, M. 109

Baldin, Y. 282 Banyaga, A. 348 Barnette, D.A. 211 Barth, W. 6 Bennequin, D. 344, 346 Berger, E. 239 Bierstone, E. 79 Blank, s. 28 Blanusa, D. 277 Boardman, J. 35 Borisenko, A. 272, 275 Borisov, Ju. 219 Bott, R. 100, 193 Briant, R. 239, 255 Brown, E. 105 Burago, Yu. 64, 188,213,

259, 284 Buriet, o. 50

Cartan, H. 4,6,47 Chaperon, M. 346 Cheeger, J. 264 Chen, K.T. 100 Chern, S.S. 270, 272 Clarke, C. 145 Cohen, R.L. 8 Cohn-Vossen, S. 189 Conley, C. 345 Connelly, R. 210 Cordero, L. 340 Cowen, M. 278

D'Ambra, G. 95 Dajczer, M. 268

Deligne, P. 7 Duchamp, T. 339 Duplessis, A. 30

Efunov, N. 256, 276 Eidlin, V. 220 Eliashberg, J. 20, 27, 30, 50,

55,346

Farrell, F.T. 187 Feit, S. 26, 43 Feldman, E. 9 Fernandez, M. 340 Ferus, D. 165, 263 Fiala, F. 283 Floer, A. 346 Forster, O. 71 Fortune, B. 346 Friedman, A. 311 Fuchs, D. 102

Garsia, A. 298 Gluck, H. 210 Godbillon, C. 101 Godement, R. 58 Golubitsky, M. 35 Grauert, H. 6, 73 Gray, A. 272, 340 Gray, J.W. 338 Greene, R. 39, 74, 117, 145,

147,338 Griffiths, P. 239, 278, 288 Gromoll, D. 110,120,268 Gromov, M. 20, 39, 50, 52,

103, 112, 195, 268, 349 Gunning, R. 4, 70

Haefliger, A. 51, 106 Halpern, B. 268 Hamenstiidt, U. 9 Hamilton, R. 145 Hartman, P. 264 Hayman, W. 288 Hironaka, H.

Hirsch, M. 7, 8, 45, 51 Hitchin, N. 110 Hopf, H. 14 Hormander, L. 38, 145 Huber, A. 188

Ionin, V. 283

Jacobowitz, H. 143,145,255, 267, 272

Janet, M. 12 Jorge, L.P. 284 Jost, J. 229

Kiillen, A. 219 Karcher, H. 165,229 Kazdan, J. 111 Kervaire, M. 16 Kirby, R. 7 Klingenberg, W. 110 Kobayashi, S. 93 Kodaira, K. 107 Kolmogorov, A. 11 Kuiper, N.H. 10, 11, 62, 270,

272

Labouri, P. 268 Landweber, P. 103 Lannes, J. 8 Lashof, R. 62 Laudenbach,F. 345,346 Lawson, B. 52, 102, 110,294 Lees, J. 334 Levin, H. 31, 64 Little, J. 9 Lojasievicz, S. 31 Lutz, R. 340

Martinet, J. 340 Mather, J. 64 Mattila, P. 288 McDuff, D. 242,348,349 Meckert, C. 343

360

Meyer, W. 110 Mercuri, F. 282 Milnor, J. 4, 5, 16 Milnor, T. 276 Mishachev, N. 106 Montgomery, D. 166 Moore, J.D. 272, 282 Morin, B. 63 Morse, A.P. 31 Moser,1. 145, 183, 338 Munzner, H. 165

Narasimhan, M. 95 Narasimhan, R. 5, 70 Nash, 1. 10, 12, 13, 22, 33,

114, 129, 130,223 O'Neil, B. 263 Nomizu, K. 93 Novikov, S. 15

Otsuki, T. 270, 271

Pasternak, J. 113 Patodi, V. 16 Phillips, A. 26, 43, 53, 99 Poenaru, V. 28, 30, 57 Pogorelov, A. 289 Poznjak, E. 223, 259, 276

Ramspoll, KJ. 6 Reinhart, B. 102

Rickman, S. 288 Rill, 1. 156 Rochlin, V. 145 Rolfsen, D. 30 Rosendorn, E. 276 Rossi, H. 4

Sard, A. 32 Schaft, U. 71 Schilt, H. 255 Schlafly, R. 223 Schoen, R. 110 Schur, F. 277 Schwartz, 1. 145 Seeley, R.T. 129 Segal, G. 104 Sergeraert, F. 145 Serre, J.-P. 15, 17 Shub, M. 187 Shulman, H. 101 Siebenmann, L. 7, 19 Singer, LM. 16, 165 Smale, S. 7, 18,42,43 Sikorav, J.C 346 Sokolov, D. 259 Spanier, E.H. 36 Spivak, M. 239 Spring, D. 195 Stiel, E. 264 Sullivan, D. 7, 101 Szuecs, A. 51, 193

Author Index

Thorn, R. 32, 23, 109 Thurston, W. 106, 341, 348 Tischler, D. 335 Toda, H. 15 Tomkins, C 262

Verner, A. 188, 283 Vey,1. 101

Wall, CT.C 31 Warner, F. 111 Weaver, C 268 Weinstein, A. 282, 337, 345 Wells, R. 15, 109 Whitehead, 1.H.C 36 Whitney, H. 8,14,31,33,98,

130 Wintgen, D. 9 Wu, H. 74

Xavier, F.V. 284

Yau, S.T. 107, 110 Y omdin, Y. 124 Y oshifumi, A. 55

Zalgaller, V. 213, 259 Zeghib, A. 268 Zehnder, E. 345 Zippin, L. 166

Subject Index

Algebraic h-Principle 71 Almost complex structure 102 Ample relation 180

Biregular immersions 328, 339

Canonical connection 94 Canonical partition 34 Canonical stratification 31 Cauchy problem 142 Cauchy-Kovalevskaya theorem 38 Cauchy-Riemann relation 4, 18,22, 66 Classifying sheaf 105 Classifying space 103 Closed manifold 7 Cobordism group of immersions 15 Completely integrable systems 163 Complex manifolds 107 Complex structure 102 Compressibility 80 Conformal maps 284 Conformal measure 285 Connection 93 Connection homomorphism 93 Contact form 60 Continuous sheaf 75 Contractibility theorem 77 Convex decomposition 170 Convex hull extension 175 Convex integration 168 Convex relations 24 Critical dimension 7 Curvature relations 109, 110 C1-approximation 230 C.L-approximation 170, 172 Ci-approximation 22 C.L-dense h-principle Can-immersion theorem 39 C"-immersions 230 Coo (can)-solutions 3

Decomposition of metrics 222, 223, 304 Defect of infinitesimal inversion 115 Degenerate forms 100 Degenerate maps 99 Dense k-principle 18, 20

Derivatives along V 21 Differential condition 2 Differential relation 2 Directed cycles 109 Directed embeddings 189, 193 Directed immersions 46, 108, 186, 188 Distribution of values 287 Divergence free vector fields 182

Efimov's lemma 256 Elliptic spaces 71 Embeddings 50 Enlarging maps 23 Equidimensional folding theorem 59 Equisingular maps 33 Equivariant sheaves 78 Euclidean connections 93 Evolution equations 195 Exact diffeotopies 331 Extension of relations 174 Extension of sheaves 84~86

Extra dimension 7,41

Fine topology 18, 117 Fibration 40 Fibre preserving diffeomorphisms 78 Flat direction 260, 273 Flat I/J-bundle 104 Flexibility theorem . 78 Flexible sheaf 75 Folded cylinders 253 Folded maps 27, 28, 54, 55, 112 Folding 27 Folding theorem 27 Foliation 101, 106 Free isometric immersions 12, 21, 22, 42 Free maps 8, 9, 21, 76 Frobenius' theorem 38 Functionally closed relations 22

Generic maps 31, 35 Geodesic lamination 265 Global analytic inversion 143

Harmonic immersions 74 Holomorphic curves 189

362

Holomorphic embeddings 71 Holomorphic h-principle 66 Holomorphic h-principle for immersions 73 Holomorphic immersions 71 Holomorphic Lagrange immersions 71 Holomorphic section 1 Holonomic splitting 2 Holonomy dimension 266 Homomorphism theorem 77 Homotopy principle 3 h-principle 3 h-principle for extensions 17, 39, 59 h-principle for folded maps 27, 57 h-principle for free maps 9 h-principle for isometric immersions 246 h-principle for maps of constant rank 99 h-principle for open manifolds 79 h-principle for regular isometric immersions

331 h-principle for sheaves 76 h-principle for submersions 53 h-principle for the Cauchy problem 47 h-principle near fo 18 h-principle near subset 37 h-principle of Grauert 6 h-principle of Small-Hirsch 6, 49 h-principle over embeddings 191 h-principle over pairs 39 Hopfs formula 63 h-stability theorem 175 h-stable extensions 174 hyperbolic immersions 269

Imbedding theorem 223 Immersion 6, 14, 16, 18,48, 52, 79, 181 Immersion of Stein manifolds 66, 70 Immersion relation 6 Immersions of closed forms 330 Implicit function theorem 114, 118 Inequality of Burago 284 Infinitesimal inversion 115 Infinitesimally enlarging maps 23 Infinitesimally homogeneous manifolds 164 Infinitesimally invertible operator 115, 117 Infinitesimally isometric maps 143 Initial value problem 142 Integrable subbundles 100 Invariant relations 29, 30 Inversion of zero order 152 Isometric cylinders 250, 521 Isometric embeddings 222, 223, 335 Isometric immersion 43, 79, 162, 221, 223, 259,

276, 279, 289 Isometric immersion theorem 44 Isometric C1-immersions 10, 16, 17, 19,201 Isoperimetric inequality 283, 284 Isotropic map 301, 322, 324

Janet's equations 226 Jets 1

k-mersions 26, 27, 48, 181

Lagrange embeddings 61 Lagrange immersions 60, 61 Lefschetz theorem 4 Legendre maps 25, 60, 109 Legendre transform 273 Linearization 115 Local COO-immersion theorem 39 Local compression 91 Local h-principle 37 Local isometric immersions 38 Local w.h. equivalence 76, 119 Localization lemma 79

Subject Index

Localization of Diff-invariant relations 37 Locally flat immersions 100 Lower bound theorem 211

Micro-compressibility 80, 81 Micro-extension of sheaves 84, 85, 88 Micro-extension theorem 85 Micro-fibration 40, 182 Micro-flexible sections 41 Micro-flexible solutions 41 Modified Nash process 139 Multi-parametric h-principle 16, 18

Nash approximation theorem 222 Nash process 121, 122, 131 Nash's formula 222 Natural bundles 145, 154 Non-singular sections 48

Oka's principle 4, 66, 70, 73 Open extension theorem 86 Open manifold 7 Open relation 3, 41 Opening m fz 35, 36 Operator of polynomial growth 141 Osculating space 8 Over-determined operator 157, 166

Parabolic immersions 260 Parallelizable manifolds 7 Parametric h-principle 13, 16, 17,50 Parametric sheaf 75 Poenaru's pleating lemma 51 Principal convex extension 178 Principal extension 177 Principal stability theorem 179 Principal subspace 172

Quasi-conformality relation 23 Quasi-topological sheaf 76 Quasi-topology 35

Subject Index

Regular field 95, 311 Regular homomorphism 95 Regular homotopy 13 Regular immersion 329, 339 Regular section 95 Relation 1 Relation over x(r) 174 Representations of Diff 145 Rigidity 211

Sard theorem 32 Secondary classes 101 Semi algebraic set 31 Semi regular homomorphisms 97 Semitransversality 45, 66 Sharp diffeotopies 82, 83 Sharply moving diffeotopies 83, 332, 333 Sheaf 74 Sheaf of solutions 74 Short maps 19, 191, 264 Signature of a hypersurface 15, 16 Signature theorem 58, 114 Singular h-principle 112 Smale paradox 14 Smoothing estimates 125 Smoothing operators 121, 123, 129 Solution of a relation 2 Solution near a point 37 Stability of symplectic forms 337 Stable relation 3 Stably parallelizable manifold 7 Stein manifolds 4, 66, 73 Stratification of Whitney-Thorn 26 Stratified set 30 Submersion 26, 52 Surrounding map 169

Symplectic fibration 341 Symplectic form 60 Symplectic isometric immersions 327, 333

Theorem of Grauert 4, 6 Theorem of Hirsch 7 Theorem of Nash and Kuiper 10, 203 Theorem of Smale 16 Theorem of Whitney 14 Thorn equisingularity theorem 34 Total curvature 62 Totally degenerate maps 27, 28 Totally real embeddings 192 Transversality theorem 32, 120 Transversal maps 26, 53 Triangular relations 185, 199 Triangular systems 192

Umbilic immersions 269 Under-determined operator 148, 153, 156 Under-determined system 41

Weak flexibility lemma 111 Weak homotopy equivalence 16 Weak h-principle 105 Wi'P-approximation 19, 50 Whitney cusp 27 Whitney's theorem 33, 130

Zero-dimensional h-principle 66 Zero-dimensional relation 44

4>-bundle 104 4>-cycle 109 l'-non-singular section 48

363

Ergebnisse der Mathematik und firer Grenzgebiete, 3. Folge A Series of Modem Surveys in Mathematics

Editorial Board: E.Bombieri, S.Feferman, N.H.Kuiper, P.Lax, R.Remmert, (Managing Editor), W.Schmid, J.-P.Serre, J. TIts

1. A. Frohlich: Galois Module Structure of Algebraic Integers. ISBN 3-540-11920-5

2. W.Fulton: Intersection Theory. ISBN 3-540-12176-5

3. 1. C. Jantzen: Einhiillende A1gebren halbeinfacher LieA1gebren. ISBN 3-540-12178-1

4. W.Barth, c.Peters, A. Van de Ven: Compact Complex Surfaces. ISBN 3-540-12172-2

5. K Strebel: Quadratic Differentials. ISBN 3-540-13035-7

6. M Beeson: Foundations of Constructive Mathematics. Metamathematica1 Studies. ISBN 3-540-12173-0

7. A Pinkus: n-Widths in Approximation Theory. ISBN 3-540-13638-X

8. RMaiie: Ergodic Theory. ISBN 3-540-15278-4

9. M Gromov: Partial Differential Relations. ISBN 3-540-12177-3

10. A. L. Besse: Einstein Manifolds

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo

Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge A Series of Modem Surveys in Mathematics

Editorial Board: E.Bombieri, S.Feferman, N.H.Kuiper, P.Lax, R.Remmert, (Managing Editor), W.Scbmid, J.-P.Serre, J. TIts

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo

Forthcoming titles:

11. M. D. Fried, M.larden: Field Arithmetic ISBN 3-540-16640-8

12. lBochnak, M.Coste: Geometric Algebrique Reelle ISBN 3-540-16951-2

K Diedrich, 1 E. Fomaess, R P. Pflug: Convexity in Complex Analysis ISBN 3-540-12174-9

E. Freitag, R Kiehl: Etale Kohomologietheorie und Weil-Vennutung ISBN 3-540-12175-7

G. A. Margulis: Discrete Subgroups of Lie Groups ISBN 3-540-12179-X

Partial Differential Relations ||

Documents

Transcript of Partial Differential Relations ||