MAPS OF MANIFOLDS INDEFINITE METRICS GEOMETRICAL …

IX::ernat. J. Math. i Math. S.(1978) 137-140

137

MAPS OF MANIFOLDS WITH INDEFINITE METRICSPRESERVING CERTAIN GEOMETRICAL ENTITIES

R. S. KULKARNIDepartment of Mathematics

Indiana UniversityBloomington, Indiana 47401

and

(Mathematics Department, Rutgers University)

(Received June i, 1977)

ABSTRACT. It is shown that (i) a diffeomorphism of manifolds with indefinite

metrics preserving degenerate r-plane sections is conformal, (li) a sectional

curvature-preservlng diffeomorphlsm of manifolds with indefinite metrics of dimen-

sion > 4 is generically an isometry.

1. INTRODUCTION.

Let (Mn, g), (n,) be pseudo-Riemannian manifolds. A diffeomorphism f:M /

is said to be curvature-preserving if given peM and a 2-dimensional plane section

o at p such that the sectional curvature K(o) is defined then at f(p) the sectional

curvature K(f,o) is defined and K(o) K(f,u). A point peM is called isotroplc if

there exists a constant c(p) such that K(u) c(p) for any 2-plane section at p

for which K is defined. I studied the notion of a curvature preserving map in the

Riemannian case and showed

THEOREM i. If n > 4 (Mn,g), (n,) Riemannian manifolds and non-lsotropic

138 R.S. KULKARNI

points are dense in M then a curvature-preserving map f:M + M is an isometr.

cf. [i] and for this and other types of "Riemannian" analogues cf. [5], [6]

[2], [3], [4]. The purpose of this note is to point out Theorem 2.

THEOREM 2. Theorem i is valid for pseudo-Riemannlan manifolds.

Unlike certain local results in pseudo-Riemannian geometry Theorem 2 is not

obtained from Theorem i by formal changes of signs. Its proof is actually simpler

but for an entirely different reason which seems to be well worth pointing out.

One of the main steps in Theorem i and its other analogues mentioned above is that

a curvature-preserving map is necessarily conformal on the set of nonisotropic

points. This step is automatic in the case of indefinite metrics due for the next

result. Let us call a subspace A of a tangent space at a point in M degenerate

(resp. nondegerate) if glA is degenerate (resp. nondegenerate). Sectional curva-

ture is defined only for nondegenerate 2-plane sections. So by definition a

curvature-preserving map carries degenerate 2-plane sections into degenerate 2-

plane sections.

THEOREM 3. Let (Mn,g), (,) be indefinite pseudo-Riemannian manifolds,

n 3. Let r i. Let f:M / M be a diffeomorphism which carries degenerate r-

dimensional plane sections of M into those of M. Then f is conformal. (i.e.

there exists a nowhere vanishing smooth function #:M + such that f* #.g.)

Recall that a geodesic on (M,g) whose tangent vector field X satisfies

g(X,X) 0 is called a light like geodesic.

COROLLARY i. Let (Mn, g), (n,g) be indefinite pseudo-Riemannian manifolds.

Then a diffeomorphism f:M / M which preserves light-llke geodesics is conformal.

This is the case r i of Theorem 3. Note that this corollary is an exten-

sion and "Geometrization" of H. Weyl’s famous observation about the conformal

invariance of Maxwell’s equations.

MAPS OF MANIFOLDS WITH INDEFINITE METRICS 139

2. PROOF OF THEOREMS 2 AND 3.

First we prove Theorem 3.

The case r 2 contains the essential ideas so we prove the theorem only in

this case leaving the general case to the reader. Let T (M) denote the tangentP

space to M at p etc. It clearly suffices to show that for each p in M

f, :Tp(M) / Tf(p) (M) is a homothety. Let {ei,ej,e} be an orthonormal set ofP

vectors so that

<el,el> <ej,ej> =-<e,ee>

Let f,ei ei

and g or <,> also denote the canonically induced metric in all

tensor powers and similarly for g. Let x2 + y2 I. Then the 2-dimensional

plane o=span {xei+ yej + e ye + xeo} is degenerate. Hence by hypothesis

i 3

f,o is degenerate i.e.

o g ((x ei+ y ej + ee)A(-y ei + x ej), (x e

i + y ej + e)A(-y ei+ x ej))

g (ei

A j + x e A ?.-y e A e ei

A e + x e A .-y e A el)

{g(ei

A ej, ei

A ej) + x2g(ee A ej, e A ej) + y2(e A el, ee A ei

2xy g(e A el, e, A ej)} + {2x g(ei

A ej, e, A ej) 2y g(ei A ej, ee A ei }"

A similar expression with (x,y) replaced by (-x,-y) is also true. Hence each {,}

is separately zero and since (x,y) are subject to the only relation x2 + y2 i

it follows that

o g(ei

A e A ei) g(ei

A ej, e A ej) g(eiA e, ej A e,)

and

g(ei A ej, e

iA ej) -g(e

iA ee, e

iA ee) -g(ej A e, ej A ee)

140 R.S. KULKARNI

i.e. {e A ej, ei

A ea, ej A ea} is an orthogonal basis of the second exterior

power A2(spa {’, e], ea}). This ans that f induces a hothetic mp of

A2(span {ei, el, e}) onto A2(span {’-t’ ej, ea}). It is then easy to see that f

induces a homothety of span {ei, e], e} onto span {e

i, ej, e}. By varying the

set {el, e,j ea} it is clear that f, is a homothety. This finishes the proof. QEDP

PROOF OF THEOREM 2. By Theorem 3 we have f* .g where is a nowhere

vanishing function on M. Now the proof that f is an isometry i.e. i is exactly

as in [i] or [4] 7. QED

ACKNOWLEDGMENT. This work was partially supported by NSF Grant MPS 71 03442.

REFERENCES

i. Kulkarni, R. S. Curvature and Metric, Ann. of Math. 9__i (1970) 311-331.

2. Kulkarnl, R. S. Congruence of Hypersurfaces, J. Differential Geometry 8_(1973) 95-102.

3. Kulkarnl, R. S. Equivalence of Khler Manifolds, J. Differential Geometry9 (1974) 401-408.

4. Kulkarni, R. S. On the Bianchi Identities, Math. Ann. 19__9 (1972) 175-204.

5. Nasu, T. Curvature and Metric in Riemannian 3-Manifolds, J. Math. Soc.Japan 27 (1975) 194-206.

6. Yau, S. T. Curvature Preserving Diffeomorphisms, Ann. of Math. i0__0 (1974)121-130.

KEF WORDS AND PHRASES. Riemannian and pseudo-Rinannian anifold, diffomor-om o ’o.d

SUBJECT CLASSIFICATIONS (1970). 53C20, 53C25.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

MAPS OF MANIFOLDS INDEFINITE METRICS GEOMETRICAL …

Documents

Transcript of MAPS OF MANIFOLDS INDEFINITE METRICS GEOMETRICAL …