Journal of Mathematical Sciences, Vol. 99, No. 6, 2000

FOLIATIONS, SUBMANIFOLDS, AND MIXED CURVATURE V. Rovensk i i UDC 514.762

I N T R O D U C T I O N

This survey is based on the author's results on the Riemannian geometry of foliations with a nonnegative mixed curvature and on the geometry of submanifolds with generators (rulings) in a Riemannian space of

nonnegative curvature. The main idea is that such a foliated (sub)manifold can be decomposed into a product if the dimension of its leaves (generators) is large. The methods of study are local in the sense of directions transversal to leaves and mostly synthetic in the case of submanifolds with generators.

The paper is divided into four chapters. Chapter 1 (introductory) contains facts concerning foliated smooth manifolds. Chapter 2 deals not only with a foliated Riemannian manifold, but also with the slightly more general case

of a pair of complementary orthogonal distributions. The latter is sometimes called a Riemannian "almost- product structure." Starting from the "co-nullity operator" of D. Ferus, P. Dombrowski, and others, in Chap. 2, we introduce a pair of structural tensors that are similar to O'Neill's or Gray's pairs but are more convenient for our purposes; these tensors also satisfy a Riccati-type PDE with mixed curvature. Chapter 2 concludes with a survey of (1) constructions of totally geodesic and umbilic foliations, (2) recent studies of Riemannian

foliations, and (3) the relationship between the result by V. Toponogov and that of the author, which asserts that if a complete simply connected Riemannian manifold with a sectional curvature ~_ 4 and injectivity radius > 1r/2 has the extremal diameter lr/2, then it is isometric to CROSS and large-spheres foliations, in particular, skew-Hopf fibrations.

Chapter 3 is devoted to a key problem of this work, namely, we discuss the role played by the Riemannian curvature in studies of foliated (sub)manifolds. Chapter 3 begins with the Ferus theorem (1970) on the optimal (largest) dimension of a totally geodesic foliation with constant positive Kmix (mixed sectional curvature) on a given manifold. It contains a striking relationship between the Riemannian geometry of foliations and the topological concept of the number of vector fields on the n-sphere. For a nonconstant K~x > 0, the dimension of a compact totally geodesic foliation can be easily estimated from above by half of the dimension of the Inanifold itself: the idea (by T. Frankel) is that two compact totally geodesic submanifolds (for instance, large spheres in a round sphere) in a space of positive curvature necessarily intersect each other if the sum of their dimensions is not less than the dimension of the whole space. In Sec. 3.7, we extend this result to the case of positive partial mixed curvature.

The problem by V. Toponogov is to obtain a Ferus-type estimate for the dimension of a foliation over a (compact) manifold with /(mix > 0. A local counterexmnple containing a fibration over closed geodesics that presupposes the necessity of additional assumptions concerning a foliation is given in Sec. 3.4. In Sec. 3.5, we introduce a variation procedure based on the concept of the volume (in particular, area) of an L-parallel vector field and the turbulence of a foliation along a leaf, which allows us to obtain rigidity and splitting theorems for foliations with Km~ > 0 and under some additional conditions; in particular, we generalize Fetus' result.

In Sec. 3.6, the Riccati-equation technique is extended to foliations. Combining some ideas, we obtain integral formulas and inequalities with mixed scalar curvature along a compact manifold or a complete leaf.

In Sec. 3.8, we introduce some classes of foliations that generalize Riemannian foliations and apply to them the concept of an index of relative nullity.

Systematic studies of the local and global structure of submanifolds in (pseudo)Riemannian spaces in- clude a study of the relationships between their intrinsic and extrinsic geometries, tests for totally geodesic and cylindrical submanifolds, estimates of codimension, etc. The attention given to foliated submanifolds

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya, Tematicheskie Obzory. Vol. 57, Geometriya-5, 1998.

1072-3374/00/9906-1699 $25.00 ~) 2000 Kluwer Academic/Plenum Publishers 1699

has increased due to studies of some special embeddings with degenerate second fundamental form: ruled and tubular submanifolds, (strongly) parabolic, k-saddle submanifolds, and submanifolds having nonpositive extriusic curvature, small codimeusion, etc.

In the case of a curvature-invariant strongly parabolic submanifold (for example, when the ambient

manifold is a space form), the domain where the index of relative nullity is minimal has a ruled developable

structure (with constant Kmix in the case of a space form). More general parabolic submanifolds, which were introduced by A. Borisenko in 1972, constitute a wider class and contain a large number of rulings under a certain condition on the curvature tensor of the ambient space.

In Secs. 4.1, 4.3, 4.5, and 4.6, a detailed survey of studies of ruled submanifolds in space fomls and CROSS is given. In Sec. 4.2, which continues the studies carried out in Sec. 3.7, some recent results on sub- manifolds with nonpositive extrinsic sectional curvature (Otsuki's lemma and its corollaries on nonembedding

of manifolds, and extremal theorems by A. Borisenko) are extended to a more general case of extrinsic partial curvature. In Sec. 4.4, we continue the study of Toponogov's problem in the case of foliated submanifolds. We introduce a (synthetically defined) class of uniquely projectable submanifolds along generators in a Riemamfian space, which is closely connected with the class of ruled submanifolds with Kmi• _> 0. A key result of Sec. 4.4 is that a ruled submanifold in the sphere or in the complex projective space with ruling of "high" dimension and a "small" norm of its second fundamental form (i.e., in particular, /(mix >_ 0) is congruent to the Segre embedding; the latter plays the role of a "cylinder" in a space form of positive curvature. Combined with the method for the volume (or area) of an L-parallel vector field (from Sec. 3.5), we obtain the test for the Segre-type decomposition of a ruled and parabolic submanifold in a space of positive curvature.

A c k n o w l e d g m e n t s . The author express his gratitude to Professor Victor Toponogov (Novosibirsk) for his help and support during many years of work on the theme, and to his colleagves and friends.

Chapter 1 F O L I A T I O N S O N S M O O T H M A N I F O L D S

1.1. Def in i t ion and E x a m p l e s o f Fol ia t ions

Intuitively, a foliation corresponds to a decomposition of a manifold into a set of connected submanifolds of the seane dimension, called leaves, which locally look like the pages of a book.

Def in i t ion 1.1 [280]. A family 5= = {L~}~eA of connected subsets of the manifold M m is a u-dimensional foliation if

(1) UaEA Lo = M m,

(2) a # ~ =~ L ~ N L ~ = ~,

(3) fbr any point p �9 M, there exists a Cr-chart (local coordinate system) (Up, pp) such that p �9 Up and

if Up/7 La ~ g, then the components of the set ~p(Up N L,~) are the following parts of parallel affine subspaces

(see Fig. 1):

A~ = { (x , , . . . , xm) �9 ~, (Up): x~+l = c~+ , , . . . , xm = c~}.

Condition (3) does not follow from (1) and (2); this is obvious from the example Sunrise on ~t 2 in Fig. 2.

The coordinate system (Up, ~p) in Definition 1.1 is called a foliated chart. A local picture of a foliated manifold

is simply a space of the form R ~ x N, where sets of the form R ~ x {n} are leaves and N is a transversal, as in Fig. 3. The integer m - v = dim N is called the codimension of a foliation. For an equivalent definition of a foliation by means of a pseudogroup structure or integrabIe G-structure, see [51] and [227].

Every leaf of a foliation is a u-dimensional connected manifold with a countable base. A leaf of a foliation

is a component of the manifold M in the leaf topology, which is generated by all sets of the form U N ~ ; ' (At),

where U is an open subset of Up. The injection mapping of a leaf into the manifold M with the original topology is continuous, but the image is not necessarily a closed subset. A leaf L is said to be closed if it is a closed subset of M in the original topology. A leaf is proper if its two topologies coincide, as, for example,

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Fig. 2. Sunrise.

(c~,+, .... 'c~)[- ~

R ~

Fig. 1. Foliation

2

2

Fig. 3. Transversal. Fig. 4. Foliation o n T 2.

when a leaf is compact. On a 2-dimensional toms, we have vector fields that generate line foliations, each of whose leaves is everywhere dense in the whole manifold (see Fig. 4). In this example, the leaf topology on each leaf L = ll~ is different from that induced by the original manifold topology.

Foliations on smooth manifolds were studied in [51, 280, 131, 226, 227]. In [191], a foliated space that generalizes the notion of a foliated manifold is considered. In this case, M is a separable metrizable space equipped with a family of open sets {Up: p E M} with p E Up a~d homeomorphisms ~v: Up --+ R ~ x Np. The

continuous change of coordinates l~as the form t' = ~(t, n), n' = ~b(n); moreover, a set of the form lt~ ~ • {n}

in Up is smoothly mapped into a set of the form R ~ • {r The level surfaces coalesce to the form of maximally connected sets called leaves. Every leaf is a smooth u-dimensional manifold. An example of a foliated space that is not a foliated manifold is a solenoid; in this case, t, = 1 and each Np is homeomorphic

to a subspace of a Cantor set. In [188, 45, 329, 313, 227], foliations with singularities on Riemannian manifolds are studied. In this case,

the dimension of the leaves is not constant. The theory of Riemannian foliations (see Sec. 2.5) can be applied to certain infinite-dimensional situations, where foliations arise as a result of special actions of continuous groups (see [286]).

Foliations arise in connections with such topics as vector fields without singularities (when ~ = 1), lute- grable tz-dimensional distributions, submersions and fibrations, actions of Lie groups, and direct constructions of foliations such as Hopf fibrations, Reeb foliations, etc.

Def in i t i on 1.2. A v-dimensional distribution on a manifold M is a smooth field {T(p)} (p E M) of ~- dimensional tangent subspaces, i.e., a function whose value at each point p E M is a v-dimensional subspace T(p) of the tangent space TpM. A distribution is integrable if a L,-dimensional integral submanifold that is tangent to the given distribution at each of its points passes through each point p E M. A l-dimensional distribution is also called a line field.

A family of maximal integral submanifolds of an integrable distribution forms a foliation. By the Frobe- nius theorem (see [177]), a distribution T on M is integrable if and only if, for any vector fields x, y on M that

axe tangent to the distribution, the vector field Ix, y] (a bracket or a commutator, see Fig. 5) is also tangent to the distribution at every point. A line field is always integrable; it is equivalent locally, but not globally, to a vector field; see the example below (a Reeb component on a toms with one compact leaf) and the examples

in R ~ \ {0} in Fig. 6. The 2-dimensional field T on ~3: T(xl, x2, x3) = {a plane generated by X~ = (1, 0, 0)

and X2 = (0, exp ( -x l ) , exp(xl))} does not have integral surfaces: [)(1, )(2] = (0, - e x p ( - x l ) , exp(xl)) ~ T.

D e f i n i t i o n 1.3 [51]. A smooth mapping 7r: M --~ B of the differentiable manifolds M and B is called a

submersion if, for any point m E M, the differential 7r.(m): TraM --+ T~(m)B is a surjective map of tangent

spaces; M is the total space, B is the base, and {Lb = 7r-l(b)} are fibers (leaves). A C~-submersion It:

M m -+ B ~ is a (locally trivial) C~-fibration if the following conditions hold:

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Fig. 5. Bracket. Fig. 6. Foliations on R 2 \ 0 that are not vector fields.

Table 1

I n-1 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 p(n)-1 1 3 1 7 1 3 1 8 1 3 1 7 1 3 1

(I) V b E B, the set ~--1(b) is a u-dimensional submanifold that is Cr-diffeomorphic to a fixed manifold

L u ,

(2) for every b E B, there exist a neighborhood Ub and a Cr-diffeomorphism ~: ~r(Ub) ~ U x L such

that Cb0r-l(b')) ---- {b'} x L, b' E Ub (M is the total space of the fibration, B is the base, L is a fiber (leaf), and ~r is the projection).

The foliation L x {b}, where b E B, on the direct product of manifolds M = L • B is called a product bundle. Thus, locally, a foliation is a product bundle. The simplest nontrivial bundles in dimension two are the Mhbius band with fiber 11r 1 and base S 1, and the Klein bottle with S 1 a s a fiber and a base. The most well-known bundles are vector bundles, in particular, the tangent bundle p: T M --+ M of a manifold and the

normal bundle p: T M • -+ M of a submanifold M C/~I. A submersion with compact leaves or with compact total space and connected base is actually a fiber

bundle [280]. The question on conditions under which a foliated manifold is actual ly a fiber bundle, in

particular, a product of manifolds, was studied in terms of an Ehresmann connection (see Sec. 1.3).

D e f i n i t i o n 1.4 [51]. A Ck-differentiable action of a Lie group G (with identity e) on a manifold M is a

Ck-map ~: G x M --+ M such that

x) = x, o x) = g2 c , �9 e M ) .

The orbit of the point x E M under the action ~ is a subset O~(~) = {7~(g,x): g E G}. We say that ~: G x M --+ M is a foliated action if, for every x E M, the tangent space to the orbit of ~ passing through x

has a fixed dimension v; if u equals the dimension of G, then ~ is locally free. The action of the group G = ~ is called a dynamical system (flow) on the manifold M.

D e f i n i t i o n 1.5 (J. Milnor). The rank of a manifold M is the maximum r for which there exists a locally free action of R ~ on M. Equivalently, the rank of this manifold M is the maximmn number of continuous pointwise linearly independent commuting vector fields that are admitted by the manifold.

A compact 3-dimensional manifold with finite fundamental group (for example, S 3) has rank 1; moreover,

a compact 3-dimensional manifold of rank 2 is a fiber bundle over S 1 with toroidal fibers T 2 (see [51]). Another "similar" topological invariant, the maximum number of continuous pointwise linearly indepen-

dent vector fields on a manifold, also has a close relationship with foliations, in particular, with foliations of nonnegative mixed curvature.

T h e o r e m 1.1 (see [92] and [138]). (a) The maximum number of continuous pointwise linearly independent

vector fields on the sphere S n-1 is equal to p(n) - 1, where

p((odd) 24b+~) = 8b + 2 ~, (b _> 0, 0 < c < 3).

(b) I f u _< -~, '~ then a continuous u-dimensional distribution on S '~-1 (and moreover, a u-dimensional

foliation) exists i f and only if there exist u continuous pointwise linearly independent vector fields on S n-1 .

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Fig. "7. Reeb component. F ig . 8. Reeb annulus. Fig. 9. Reeb solid torus.

Note that p(n) < 2 log 2 n + 2 < n. Table 1 contains some values for the function p(n) - 1.

A solution to the algebraic problem on the existence of orthogonal multiplications on R" [138] allows us

to construct p(n) - 1 orthogonal unit vector fields {wi(x) = Bix} on the sphere S'~-t: for all v < p(n) there

exist v orthogonaI matrices {B~} of size n x n with the properties B~ = - E , B~Bj + BjB~ = O, i # j . They define a nonintegrable geodesic distribution on a sphere, i.e., every large circle that is tangent to

this distribution at one point preserves this p roper ty at all its points.

An example of the R e e b fo l i a t i on plays an impor tan t role in the development of foliation theory. The

Reeb component in the strip S = {(x, y) E R2: [xl __ 1} is the foliation {La}, (a E R t2 4-o0) (see Fig. 7)

L a = { ( t , a + f ( t ) ) " Itl < 1}, L i ~ c = { ( + l , t ) - I t [ < o c } ,

where, for example, f ( t ) = exp(~t_~t. ) - 1 leads us to a Ca-fol ia t ion [280]. Reeb components can also appear

on mtrfaces. By factoring a strip via translat ion along the Y-axis, we obta in the following Reeb components:

a MSbius band and its double covering on a cyl inder (Reeb annulus) (see Fig. 8). The Reeb annulus with pairwise glued boundary lines forms a foliation on the torus or on the Klein bottle. Each of these foliations has exactly one compact leaf, but on the torus such a foliation cannot be obtained fl'om a nonsingular vector field.

In 1979, H. Gluck posed two basic problems concerning the correlation of the structures of foliations and metrics on manifolds.

FM1. Th, e existence of a geodesibIe metric for a given foliated manifold. For a foliation on a 2-dimensional manifold, an obstruction to geodesibility consists of a Reeb component

(see Sec. 1.1). Among foliations that are genera ted by Morse-SInale vector fields without sing~flarities, only

suspensions of diffeomorphisms are geodesible [103]. For n > 4, there exist nongeodesible foliations of AI ~

even with closed curves (see Appendix A in [23]).

FM2. The e~istence (and classification) of geodesic foliations on a given Riemannian manifold. Much is known about geodesic (mid Riemannian) foliations on space forms, (see Sec. 2.4.3; resp., Sec.

2.5.3). All geodesic foliations on the round 3-sphere were catalogued in [104] (see also Sec. 2.6), but the same

question is open for S 2'~+~ with n >_ 2; for foliations on compact Lie groups, see [230]; for foliations on some

standard hypersurfaces in Euclidean n-spheres, see [217].

Rotating a strip about the Y-axis, we obta in a Reeb component on a solid cylinder D e x R (and hence,

on a solid toms, see Fig. 9), which can also be defined by some submersion f : D ~ x R --+ 1r Let f~: m a --+ R

be a similar submersion defined on R a by the formula f t ( x l , x2 , x3) = a(r 2) exp(x3), where a: R --+ R is a

C~-funct ion such tha t a(1) = 0, a(0) = 1, and if t > 0, then a'(t) < 0. Let {L} be a foliation of R 3 whose

leaves are connected components of the submanifolds f ~ ( c ) , (c E R). All leaves in the interior of the solid

cylinder C = f~-~([0, 1]) = {(x~,x2, x3) : x~ + x 2 <_ 1} are homeomorphic to the plane R;. The boundary

of C, OC = f[-t(O) = ( (x t , x2, x3) : x 2 + x 2 = 1}, is also a leaf (a cylinder). Outside C, all leaves are

homeomorphic to the cylinder (see Fig. 10). This submersion is not a fiber bundle. We can factor the Reeb component in a solid cylinder and thereby obta in a Reeb foliation in a solid torus

D 2 x S ~ filled in by copies of R -~, whose leaves accumulate only in the neighborhood of a compact leaf, the

boundary toms O(D 2 x S 1) = S t x S t. This foliation cannot be defined by a submersion.

Froin two Reeb components in D" x S 1, we obta in a C~ on the whole 3-sphere, since S 3 = {x =

1703

,: ~ ~ planes

Fig. 10. Submersion that is not a fiber bundle.

A Lo-% B

B - - H o/(L_0) = _Z~ _ _ A

Fig. 11a. Holonomy of foliations on a M6bius band. Fig. l l b . Holonomy of the Reeb foliation.

(xl, x2, x3, z4) E R 4 : ~i=,4 xi2 = 1} can be represented as the union of two solid tori S~_ = {x E $3 : x 2 + x 2 _>

� 8 9 3 _ = { x E S 3 : x 2 + x 2 2 _ < � 8 9 2 = { x E S 3 : x 2 + x 2 = ! 2 } . This

Reeb foliation on S ~ has exactly one compact leaf. Therefore, it is optima/according to the Novikov theorem

(see [51, 280]), which states that every 2-dimensional C2-foliation on a compact 3-dimensional manifold M

with a finite fundamental group has a compact leaf, which is homeomorphic to a torus in the case M = S 3.

1.2. H o l o n o m y

The notion of holonomy is a generalization of the Poincar~ first return map from flows to foliations. The mappings of transversal cross-sections associated with curves lying in a leM L allow us to obtain a representation of ~r~(L, x) (the fundamental group of the leaf L) in a group of germs of diffeomorphisms of the transversal manifold at a point x. If the holonomy group of the leaf L is smaU, then the recurrence of leaves near L is small.

We denote by F~ the group of germs of Ck-diffeomorphisms of R n leaving fixed the point O.

L e m m a 1.1 [227]. Let L be a leaf of a foliation of codimension n, x E L, and let h be an embedding of

R '~ transversal to the foliation such that h(O) = x. I f the foliation is of class C k, then h is also assumed to

be of class C ~. Then there exists a homomorphism f ( L , x , h) taking ~rl(L,x) into F k, which is defined up to conjugacy by the foliation and the leaf.

D e f i n i t i o n 1.6. The holonomy homomorphism of a leaf is the homomorphism Hol (L,x): 7rl(L, x) --+ F k

introduced in the statement of Lemma 1.1. The holonomy group of a leaf is the image F(L, x) of 7rt(L, x) under the holonomy homomorphism.

The holonomy of the core circle 3' on a flat M6bius band foliated to circles equidistant to 3' is the group

Z2 generated by the diffeomorphism f (x ) = - x (see Fig. 11 a). For the Reeb foliation of S 3, the holonomy

group of a unique compact leaf (torus T 2, see Fig. 11 b) is isomorphic to Z @Z and can be represented by two

C~162 f , g: R --+ it with f(0) = g(0) = 0 such that

f ( x ) { < x f o r x > O { = x forx>_O = x f o r x _ < 0 ' g(x) < x f o r x < 0 .

L e m m a 1.2 (see [227]). The union of all leaves with trivial holonomy is an everywhere dense set that is the intersection of no more than a countable number of open sets.

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E x a m p l e 1.1 [191]. Let K be a Cantor set of positive measure in a unit circle, and let f : S 1 --~ S 1 be

a homeomorphism that has K as its fixed point set. The associated foliation (obtained by means of the

suspension of homeomorphism construction) on a torus has closed leaves each of which corresponds to each point of K and each of these leaves has a nontrivial holonomy.

All fibers of a fiber bundle are diffeomorphic, and for a foliation, the sufficient conditions for leaves to be diffeomorphic to each other are studied in stability theorems. Holonomy is a key notion when all leaves of a foliation are diffeomorphic.

T h e o r e m 1.2 (local stability) [51]. Let M be a smooth foliated manifold and let L be a compact leaf with (a) a finite holonomy, (b) a trivial holonomy. Then there exists a saturated neighborhood U of L with one of the following properties:

(a) all leaves in U are compact with finite holonomy groups, (b) there exists a diffeomorphism ~: L • R '~ --+ U that preserves the leaves.

The global stability theorem for codlin L = 1 was obtained by Reeb. From Lemma 1.2 and Theorem 1.2, it follows that for a compact foliation (i.e., a foliation with compact leaves), the union of all leaves with trivial holonomy is an open everywhere dense set of M.

T h e o r e m 1.3 (see [227]). The following statements are equivalent for a compact foliation: (a) the holonomy

group of the leaf L is finite, (b) the space of leaves is Hausdorff in a neighborhood of L, (c) the volumes of leaves in a neighborhood of L are uniformly bounded in some (and hence, evey~j) Riemannian metric on M,

(d) the leaf L admits a fundamental system of saturated neighborhoods.

For a compact foliation, we have the following: (1) if either codirn L = 1 or codlin L = 2 and M is compact, then, according to [226] and [74], all leaves have finite holonomy groups, (2) if either codlin L > 3

or codim L = 2 and M is not compact, then the assertion in (1) is violated (see [23]).

There are two groupoids a.~ociated with a foliation, namely, the homotopy groupoid and the holonomy groupoid, which is sometimes called the graph of a fohation.

Def in i t ion 1.7. The graph (holonomy groupoid) F(M, L) of a foliated manifold M is the set of all triples

(x, y, loll), where x, y belong to the same leaf L, c~ is a piecewise-smooth path going from x to y in L, and [c~] is the holonomy equivalence class of e. Two triples (x, y, loll) and (x', y', [/3]) are equivalent iff x = x', y = y',

and the holonomy of the curve o J -~ is trivial. The homotopy groupoid of a foliation is defined in a similar way (as the holonomy groupoid). We identify leaf curves that are homotopic relative to their fixed ends in the corresponding leaf.

The graph F(M, L) has a locally Euclidean topology of dimension dim M + dim L, but, in general, this topology is not Hausdorff. The concept of the holonomy groupoid was introduced in [76]; it was later studied

in more detail by Winkelnkemper in [310], who was primarily interested in its application to Riemannian foliations (the graph is Hausdorff in this case). The study of the analytic properties of the graph of foliation was begun in [66]. Likewise, the homotopy groupoid is a manifoht, but not necessarily Hausdorff. There is

a natural submersion from the homotopy groupoid onto the holonomy groupoid of a given foliation. In [130] (see also [69]), one can find an example of a foliation for which the holonomy groupoid is Hausdorff, but this is not the case for the homotopy groupoid.

A (homotopy) vanishing cycle for a foliation {L} is a mapping c: S ~ • [0, 1] such that for any t 6

[0, 1], ct = Cls~• } is a loop in a leaf of {L} and Co is not homotopic to a constant loop in the leaf, but, for

t > 0, all loops c~ are homotopic to a trivial loop. Moreover, the curves c~ cart be chosen to be tangent to the horizontal subbundle. A holonomy vanishing cycle for a foliation {L} is a mapping c: S ~ • [0, 11 such that

for any t �9 [0, 1], ct = cls~ • is a loop in a leaf of {L} and co has a nontrivial holonomy, but for t > 0, the

loops ct have no holonomy [310]. It is demonstrated in [70] that the nonexistence of a vanishing cycle (of a homotopy) is equivalent to the Hausdorff property of the homotopy groupoid of a foliation. The graph of the

foliation {L} is Hausdorff if and only if {L} has no holonomy vanishing cycles [311].

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To any holonomy group we can associate the family of infinitesimal holonomy groups, which is obtained by taking the j-jets (j _< k) of tile elements of the holonomy group (or groupoid). In the case j = 1, we have the linear holonomy.

For a foliated Riemeamian manifold M, {L} the embeddings h: R n -+ M (see Lemma 1.1) can be chosen to be orthogonal to a given leaf L at all points of the path a. Then tile linear holonomy acts on

the orthogoual subspaces T~L • along the path a. For a Riemannian foliation, the hotonomy acts by local isometric transformations on (the germs of) transversal manifolds, and hence, is completely defined by its

1-jets at each point, i.e., by the linear holonomy, neanely, by orthogonal transformations of T~L • [285-287].

The trajectory of a vector y E TxL • under the linear holonomy displacement is called an L-parallel (or

basic) vector field along the curve a C L (see also Sec. 3.3). These vector fields are defined globally on leaves

with a trivial holonomy. An L-parallel vector field can be characterized by using the Bott connection ~7 in

TL • which is defined by

~7xs = Pz[z, Y~], (z C TL, s C TL•

where Y~ C T M is any vector field that is projectable onto s under the action of the orthogonal projection

P2: T M -+ TL • The curvature R(x~u) is zero for x, u E TL by the Jacobi identity for the bracket of vector

fields. This means that (TL • f7), restricted to each leaf, is a flat vector bundle. The ~r-parallel displacement

in TL • along a path in L is a linearized version of holouomy, and hence, depends only on the homotopy class of a path in a leaf.

1.3. E h r e s m a n n Fo l ia t ions

Ehresnmnn [76] defined a connection in a fiber bundle as a distribution complementary (transversal) to fibers with the following completeness condition: every curve in the base admits a horizontal lift to the total space. The concept of art Ehresmann connection is defined in [28-30] for a foliate([ manifold without boundary as a complementary (transversal) distribution to the foliation satisfying a certain condition. These studies

were continued, in pm'ticlflar, for a foliated manifold with boundary in [158-161]. The graph of a foliation

with Ehresmaam connection is studied in [327] and [313, 314]. Consider a foliation {L} on a smooth manifold M, together with a complementary distribution D. This

distribution is transversal, but in the case of a Riemannian manifold, it may not be orthogonal to TL. We call a piecewise-smooth curve in M whose velocity vector field lies in TL (resp., D) an L-curve (resp., D-curve).

Def in i t ion 1.8 [28]. Every D-curve /3: I --+ M uniquely defines a family of local diffeomorphisms gt: V0 -+ Vt, (t E I) from one leaf onto another one (an element of holonomy along fl) such that (1) Vt is a neighborhood of 13(t) in the leaf passing through 13(t), (2) gt(/3(O)) = 13(t) for all t, (3) for each x E V0, the curve gt (x) is tangent to D, (4) go is the identity map of V0. When the leaves of a foliation have a geometric

structure (a measure, linear connection, and Riemannian metric), we say that D preserves the geometry of the leaves if the elements of holonomy along D-curves are local isomorphisms of this geometric structure.

For example, when {L} is a totally geodesic, mnbilic, or minimal foliation on a Riemannian manifold

and D = TL • then an element of the holonomy along each horizontal curve is a local isometry, conformal mapping, or volume-preserving map of the induced metrics on the leaves, respectively.

Def in i t ion 1.9 [38]. A piecewise-smooth mapping 5: [0, 1] x [0, 1] -+ M such that for every fixed so, the curve ~(., So) is a D-curve, and for every fixed to, the curve ~(t0,-) is an L-curve is said to be a rectangle in the foliated manifold M. Also, the curves (f{0, -), ~(1, .), 5(-, 0), and 6(-, 1) are called the initial L-edge, terminal L-edge, initial D-edge, and terminal D-edge of d, respectively. The rectangle whose initial edges are an L-curve a and a D-curve t3 is called a rectangle associated with ot and ~ and is denoted by ~.~. (It is easy

to show that two rectangles with the same initial edges coincide, so the last notion is well defined.) It is easy to show that two rectangles with similar initial edges coincide, and therefore, Definition 1.9

holds. For a sufficiently small positive ~, the rectangles ~lt0.,j.z and (~,~ll0,,i always exist.

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D e f i n i t i o n 1.10 [28]. A complementary distribution D to the foliation {L} is called an Ehresmann connection

for {L} if, for every L-curve a and every D-curve/3 with the same initial point, i.e., a(0) =/3(0), the rectangle 5~,~ exists. {L} is called an Ehresmann foliation whenever such a distribution D exists.

The basic property of the Ehresmann connection is that for any rectangle 5~,~, the curve 5(1,-) depends

only on the homotopy class of tbe curve a, and the curve 5(., 1) depends only on the homotopy class of the curve /3. Moreover, for an Ehresmann foliation: (a) any two leaves can be joined by a D-curve, (b) the universal coverings of any two leaves are diffeomorphic.

T h e o r e m 1.4 (topological product) [28]. Let D be an integrable Ehresmann connection ('with leaves L •

for the foliation { L } on M. Then the universal covering ,~I is topologically the product ]_, • L• of universal

coverings of the leaves L and L •

C o r o l l a r y 1.1 [28]. A compact manifold M with a finite fundamental group does not admit an Ehresmann

foliation of codimension 1. In particular, the Reeb foliation on S 3 is not geodesible.

An Ehresmann foliation on M with dim M > 2 can be not geodesible and may not admit a bundle-like metric [28]. For a submersion with compact connected leaves, every distribution complementary to the leaves is the Ehresmann connection.

Let {L} be a foliation on a Riemannian manifold, and let a (resp. /3) be an L-curve (resp. a D-curve). We denote by Rec (a, .) the set of all rectangles whose initial L-edge is a and whose initial D-edge is a regular

curve, and by Rec (-,/3) the set of all rectangles whose initial D-edge is t3 and whose initial L-edge is a regular

curve. Let us define the function G~ on Rec (a, .) by G~(5) -- l(~.~) for 5 E Rec(a , - ) and the function

G~ on Rec(-,/3) by G~(5) = l(~,.) for 5 �9 Rec(-,/3), where l(-) is the curve length. Note that G~ - 1 for l(~o..) every D-curve/3 in the case of a totally geodesic foliation, and G~ - 1 for every L-curve a in the case of a Riemannian foliation (i.e., bundle-like metric). The following theorem generalizes the results for Riemannian foliations and for totally geodesic foliations.

T h e o r e m 1.5 [161]. Let {L} be a foliation on a Riemannian manifold M satisfying one of the folio'wing

conditions: (1) the induced metrics on the leaves are complete and sup G~ <oc for every TL• /3, (2)

M is complete and sut){G~H0,~] - s �9 [0, 1)} < oc for eveT~] L-curve a without a terminal point. Then TL •

is an Ehresmann connection.

Ehresmann foliations also appear among totally umbilic foliations [28] and foliations with the following transversal structures: the Caftan structure, G-structure of finite type, system of differential equations of arbitrary order (see [327]), Lagrange foliations under certain constraints, [312], foliations on a manifold over a local algebra [272], and in field theories [171].

T h e o r e m 1.6 [224]. I f the surjective submersion 7c: M --+ B admits an Ehresmann connection D, then it .is a fiber bundle.

Theorem 1.6 generalizes the result from [134] for the orthogonal distribution D of Riemannian submersions and the result from [224] for submersions with totally geodesic fibers.

An Ehresmann foliation has no vanishing cycles; as a consequence, the homotopy groupoid of such a foliation is a Hausdorff manifold [311].

For an Ehresmann foliation, a more special holonomy and graph can be defined in the following way.

D e f i n i t i o n 1.11 [28]. Let {L} be a foliation on a manifold M with the Ehresmann connection D, and let 12~ be the set of all horizontal curves with origin x. Then the fundamental group 7rl(L, x) of a leaf L acts on gt~ in the following natural way:

for all a E ~ , [r ~rl(L,x) : ['r = 5(1,.),

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where ~ is a rectangle associated with the curves ~b, a. Let KD(L, x) = {[~b] E zrl(L, x): [r = a Va e 12~} be the kernel of this action. The quotient group FD(L, x) = zrl(L, x)/KD(L, x) is called the D-holonomy group of the leaf L (compare with Definition 1.6).

HD(L,x) does not depend on the specific Ehresmann connection, and trance, is an invariant of the foliation. In some sense, HD(L, x) is a globalization of the germinal holonomy group F(L, x) and admits a natural surjection onto it.

Def ini t ion 1.12 [327]. Let {L} be a foliation on a malfifold M with the Ehresmann connection D. The D-graph (D-holonomy groupoid) FD(M, L) is the collection of all triples (x, y, [a]), where x, y lie in the same leaf L, a is a (piecewise) smooth path going from x to y in L, and [a] is the equivalence class of a (two paths a, 3 with similar endpoints x, y are equivalent if a3 -1 E KD(L, x)).

The graph Fv (M, L) is always a Hausdorff (dim M+dim L)-dimensional manifold, and there exists a local isomorphism FD(M, L) --+ F(M, L). Moreover, the graph F(M, L) is Hausdorff if and only if it is isomorphic (in the canonical way) to Fo(M, L).[327].

The graph FD(M, L) is useful for studying stability problems.

T h e o r e m 1.7 (global stability) [28, 327, 311]. Let M be a foliated manifold with the Ehresmann connection D, and let L be a compact leaf or a leaf with finite volume; moreover, assume that L has a finite D-holonorny group. Then all leaves have finite D-holonomy groups and are compact or have finite volume, respectively.

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Chapter 2 FOLIATION ON R I E M A N N I A N MANIFOLDS

2.1. Bas ic Fac t s f r o m t h e T h e o r y of S u b m a n i f o l d s

Recall some necessary facts and formulas from the theory of submanifolds (see [60, 61] and [155]). If M

is a submanifold of a Riemannian space lkr, then

~7~y = ~7~y + h(x, y) (Gauss formula): (2.1) ~7~ = - A c x + V{~ (Weingarten formula),

where x, y are vector fields tangent to M, ~ is a vector field orthogonal to M, h: T M x T M --+ T M • is the second fundamental form of embedding, which is symmetric with respect to both arguments: h(x, y) = h(y, x),

~7• is a (metric) normal connection in the normal vector bundle T M • and Ar is tile operator of the second quadratic form for a normal vector ~. Froin the formulas above it follows that

(Acx, y) = (h(x, y), ~). (2.2)

We denote by/~, R, and R • the curvature tensors of the connections ~ , X7, and ~7• respectively. The key role in the theory of submanifolds is played by the following equation.s of Gauss, Codazzi, and Ricci:

(R(x, y)z, u) = (R(x, y)z, u) + (h(x, u), h(y, z)) - (h(x, z), h(y, u)),

~(~, y)z • = (VT~h)(y, z) - (VT~h)(~, z), (2.3)

(/~(x, y)~, r/) = (R• y)~, 77) - ([d~, A,]x, y),

where the vectors x, y, z, u are tangent to M, the vectors ~ and T/are orthogonal to M, and the derivative ~ h is defined by (~7~h)(y, z) = U~(h(y, z)) - h(V~y, z) - h(y, ~7~z).

For a submanifold in a space form ~I(k) (and some submanifolds in CROSS, compact symmetric spaces

of rank one), we have the relation

y)z • = 0 y, �9 TM) , (2.4)

which is equivalent to (~7~h)(y, z) = (~Tyh)(x, z). A submanifold with property (2.4) is called a curvature- invariant submani.fold. A totally geodesic submanifold is always curvature-invariant; for recent results, see [141].

D e f i n i t i o n 2.1. For a submanifold M n C ~I, the mean-curvature vector field H C T M is defined by the formula H = E i ~ h(e~, ei) (note that we have deleted the usual factor 1/n), where {ei} is a local orthonormal

basis in TM. A submanifold M" C/~I having one of the conditions

h = O , h ( x , y ) = l H ( x , y ) , H = O

is totally geodesic, totally umbilic, or minimal, respectively.

D e f i n i t i o n 2.2 [60]. A vector bundle E with metric over a Riemannian,manifold M and with the second fundamental form, i.e., a smooth section h in Horn (TM~3TM, E) satisfying the symmetry condition h(y, z) = h(z, y) for all y, z �9 T M , is called a Riemannian vector bundle.

T h e o r e m 2.1 [60]. (a) Let M ~ be a simply connected Riemannian manifold with a Riemannian m-dimensional vector bundle E that satisfies Eqs. (5.3 a), (5.3 c) and (5.4). Then M ~ can isometrically be embedded into a

space form ~I'~+m(k) with normal bundle E. (b) Let f l , f2: M -+ F/I(k) be two isometric embeddings with normal bundles E1 and E2, and let there exist an isometry ~o: M --+ M such that ~ can be covered by a bundle map ~: El --+ E.2 which preserves the metrics and the second fundamental forms of the bundles E1 and E2. Then there exists a rigid motion g: IQ(k) -+ ~I(k) with the property g o f t = f2 o ~o.

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2.2. M a i n T enso r s of a Fol ia t ion

We denote by 7'1 and T2 two complementary orthogonal distributions on a Riemannian manifold M. We define the structural tensors Bx: T2 • TI --+ 2"1 and B2: Tl x 2"2 --+ T2 by the formulas

Bl(y,:r) = P I ( V ~ ) , B2(x,y) = B2(Vy5:), (2.5)

where P/: T M --+ Ti (i = 1, 2) are orthogonal projections, k C Tl and ~ C 7"2 are local vector fields containing the vectors x and y respectively. The equation B~ = 0 means that tlm distribution T~ is involutive and tangent to a totally geodesic foliation. The relation B1 = ]32 = 0 indicate that M splits along Tl and 2"2, i.e., M is locally a Riemannian product Ll • L2, where the leaves {Ll} and {L2} are tangent to T1 a~ld T2, respectively.

In the special case of a totally geodesic foliation (L} (i.e., B1 = 0), the tensor corresponding to /32 is B:

T L x TL -L --+ TL • which is defined in [86] with the opposite sign and in [78] is as follows:

B(x, y) = (Vu:~) • (2.6)

This tensor (the conullity operator) is introduced in [234, 117, 202], and [206] for a (relative) nullity foliation. Historically, the first to be de~led (in [118] and [204]) were configuration tensors, T (vertical) and 0

(horizontal), by formulas different from those in (2.5):

T~'v = P2(Vp,~PIv) + Pl(Vp, uP.2v), Our ~- Pt(Vp2uP.2v ) + P2(Vp.2,Ptv ). (2.7)

The tensor T is defined by its values T::lx2 or by its values T~y, and similarly for tlle tensor O, in view of identities

(T.:,z2, y) = -(T.~ly, x2), (Os, y,, x) = -(Ou, x, Y2), (2.8)

where x, xl, x2 E 7'1 and y, Yl, Y2 E T2. These classical tensors can be expressed in terms of the structural tensors B1 ~md Be as follows:

T~y = Bt(y,x) , O:~x = B.,(x,y). (2.9)

In the case of the foliation {L} on a Riemamlian manifold M, the configuration tensor T is the second fundamental form of the leaves, and, ttmrefore, the identity T = 0 nleans that the leaves are totally geodesic submanifolds. In the sequel, vectors and vector fields on T M taslgent (respectively, orthogonal) to the leaves

of a foliation are said to be vertical (resp., horizontal). The second fundamental forms (symmetric tensors) th: Tt • 2"1 --+ T2 and h2:T2 • T2 "+ 7'1 and the

integrability tensors (skew-symmetric) AI: 7'1 • Tt --+ T2 and A2:T2 x 7"2 --+ TI are defined by the formulas

hi(x, u) = P.z(V~'n + V,,:~)/2, At(x, u) = P.2(V~:(z - V,,Y:)/2, (2.10) h2(y, z) = P~(~7u5 + V.~))/2, A2(y, z) = Pt(~Tu5 - V~f/)/2,

where '5, ~ C 7"1 and Y, 5 C T2 are local vector fields containing the vectors u, x, y, and z, respectively. Their geometrical meaning is the following: hi is tile second fundamental form at a point p E M of the submanifold consisting of geodesics in M that are tangent to the subspace Ts(p). In view of the identities

A l(x, u) = 1 ~P2[x,u] and A2(y,z) = t ~Pt[y,z], the relation Ai = 0 means that T~ is tangent to the foliation

{ni}.

The -mean curvature vector fief& ~ = tr he C Tl and H1 = tr ht C T2 (of the distributions 2"2 and 7"1, resp.) can be calculated by the formulas

(He, x) = - t r B2(x,-), (Hi, y) = - t r Bl(y, .). (2.11)

We denote by B +, B~- (i = 1, 2) the symmetric and skew-symmetric components with respect to T~ of the structural tensors Bi.

L e m m a 2.1. B + (i = 1, 2) plays the role of self-adjoint Weingarten tensors associated with hi, and B~- (i = 1, 2) serves as an equivalent to integrability tensors of distributions Ti by virtue o/ the identities

(B+(y, x), u) = - (h i (x , u), y), (B+(x, y), z) = - (h2(y , z), x), (2.12) (BF(y, x), u) = - (Al (x , u), y), (B~(x, y), z) = - (A2(y , z), x).

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Indeed, B + is tim shape operator at a point p E M of the locally defined transversal submanifold consisting of geodesics in M that are tangent to the subspace Ti (p).

Def in i t ion 2.3. A distribution Ti on a Riemannian manifold (M,g) is (1) geodesic, (2) minimal, or (3) umbilic if its second fundamental form has the property

(1) hi = 0 , (2) Hi ---0, or (3) hi(x,y) = (HjdixnTi)(z ,y) .

The geometrical meaning of geodesic distribution is that every geodesic in M that is tangent to T~ at one point is tangent to Ti at each of its points.

Using Definition 2.3, some well-known classes of foIiations can be introduced in terms of their tangent or transversal geometry.

Def in i t ion 2.4. A foliation {L} on a Riemannian manifold M is said to be totally geodesic, minimal, or totally umbilic if its leaves are totally geodesic, minimal, or totally umbilic submanifolds, respectively. A foliation

{L} on a Riemannian mmfifold M is said to be Riemannian or conformaI if the orthogonal distribution TL • is geodesic or umbilic, respectively. A totally umbilic foliation is spherical if its mean-curvature vector field is parallel in the normal bundle along leaves (i.e., the leaves are extrinsic spheres in the total space).

Some pairs of foliations are dual in a certain sense: totally geodesic and Riemannian (see [187]), and

totally umbilic and conformal foliations. Also, a foliation whose orthogonal distribution is minimal (i.e., with a holonomically invariant transversal volulne form; see Proposition 2.1 below) is dual to a minimal foliation.

Classes of foliations on a smooth manifold can also be defined in terms of the existence of certain adapted Riemannian metrics. A foliation is said to be

Riemannian if there exists a Riemannian metric for which the leaves are locally equidistant submanifolds (an example is a homogeneous foliation obtained by an isometric action of a Lie group on a Rielnannian

manifold), umbilicalizable (resp. geodesible) if there exists a Riemannian metric for which the leaves are totally

umbilic [49] (resp. totally geodesic) subxnanifolds, taut if there exists a Rielnannian metric for which the leaves are minimal sublnanifolds, tense if there exists a Riemannian metric for which the differential mean-curvature form of tim leaves is

parallel along the leaves [146], mean-curvature invariant (MCI) if there exists a Riemannian metric for which the mean curvature vector

field is a basic vector field along the leaves (i.e., the foliation is invariant under the local flow generated by

the mean curvature vector field) [307]:

( [ H , x ] , y ) = 0 , ( x E T L , y E T L •

E x a m p l e 2.1. Assume that f : M --+ N is a smooth map of Riemannian manifolds with constant rank r(f) < d i m M (in particular, a submersion when r(f) = d i m N < dimM), f.: T M --+ T N is the differential

of f , f - I ( T N ) is the induced bundle over M, and ~ is a sum of connections of N and M in f - I ( T N ) ~B TM. Then T M = T1 ~B T2, where the distribution TI = ker f = {x E TM: f .x = 0} is always integrable and T_,

is the orthogonal distribution. The second fundamental form by: T M • T M --+ f -1 (TN) of f (a symmetric

bilinear map) is defined by the formula

hy(x, y) = fT~f.(y) - f , (Vzy) . (2.13)

H f --- tr hf is the mean-curvature vector of f . The map f is harmonic if Hf = 0; in this case, ker f defines a minimal foliation,

umbilic if h f ( x , y )= 1 ~ H I (x, y), where m = dim ker f ; in this case, ker f defines an umbilic foliation

and T2 is tangent to a totally geodesic foliation,

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geodesic (or projective) if it preserves the geodesics, i.e., there exists a 1-form 0J on M such that hf (x , y) =

~ (x ) f . y + ~ ( y ) f . x [197]; in this case, k e r f defines a totally geodesic foliation and T2 is tangent to an umbilic foliation,

affine if it is a geodesic (or projective) mapping and preserves the natural parametrization of geodesics;

in this case, hi(x , y) ---- 0 [274]. The notion of the second fundamental form of a mapping between manifolds endowed with a connection,

constructed for the s tudy of harmonic mappings (see [78]), generalizes the second fundamental form of a submanifold isometrically immersed in a Riemannian manifold; it was used in [297] to study totally geodesic mappings and Riemannian submersions, in [128] to s tudy projective mappings, in [319] to study harmonic and affine mappings, in [197] to study projective and umbilic mappings, and in [274-276] to study mappings of this kind with a nonconstant rank.

An interesting interpretation of the second fundamental form h of foliation (L} follows from the formula

O(y)g(x,,x2) -- - 2 g ( h ( x l , x 2 ) , y ) , (y e T L • xl, x.2 �9 TL) , (2.14)

i.e., for a vector field y orthogonal to {L}; the y-component of h is the Lie derivative (9 with respect to y of the metric g along the leaves. Hence a totally geodesic (or totally umbilic) foliation {L} on (M, g) is characterized by the condition that the induced metric glL along the leaves is invariant (resp., conformally

invariant) under the flows of vector fields orthogonal to the foliation, i.e., for y �9 T L • the relation O(Y)glL = 0

(resp., O(y)gln = - 2 ( H , Y)glL) holds [285]. Similarly, a Riemannian foliation is characterized by the condition

that the induced metric on T L a- is holonomicaUy invariant, that is, e(x)glTL~ = O, (x �9 TL) [285]. If all leaves L are orientable, then the foliation is said to be tangentially orientable. The characteristic

form X of the /~-dimensional tangentially oriented foliation {L} on (M, g) is a u-form on T M that takes the value 1 when it is evaluated on a locally oriented orthonormal frame {ej} of {L}. For arbitrary vectors u l , . . . ,u~ �9 T M , the form X is given by

X(Ul, . . . , u~) = det{g(u~, ej) }ij. (2.15)

From formulas (2.14) and (2.15), we obtain

~)(Y)XIL = -H(y)XIL (y �9 T L • (2.16)

(see [285]). In other words, the mean curvature is the first-order variation of the volume of the leaves for

variations in the direction of the mean curvature vector field. The transversal volume form X • which is similarly defined, satisfies the following equation, which is dual to (2.16) [285]:

O(X)XL]TL ~ ~-- - - H L ( x ) ) ( . ~ T L • (X �9 TL). (2.17)

Hence (a) minimal foliations are characterized by the condition that the restriction XIL of the volume form

to the leaves is invariant under the flows of vector fields orthogonal to the foliation, i.e., ~)(Y)XIL = 0 (y �9

TL• (b) transversally orientable foliations with a holonomically invariant transversal volume form X • are

characterized by the condition H • = 0 [285].

2.3. A R i e m a n n i a n A l m o s t - P r o d u c t S tructure

Two complementary orthogonal distributions T1 and T2 on a Riemannian manifold M (see Sec. 2.2) can be considered in analytic terms.

De f in i t i on 2.5 [227]. A Riemannian manifold M with metric ( , ) and a tensor field P of type (1, 1) on T M such that

p2 --_ Id, (Pu, Pv) -- (u, v) (2.18)

is called a Riemannian almost-product structure. The eigenspaces of P for the eigenvalues +1 and - 1 are called the vertical distribution T~ mid the horizontal distribution T2; they are orthogonally complenmntary to

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each other, i.e., T M = T~ ~ T2. Indeed, Pt = (Id + P)/2 and Pe = (Id - P)/2 are orthogonal projections of T M onto 7'i mid T~_, respectively.

The integrability tensors A1, Ae (cf. (2.10)) are given by

1 1 A~(u, v) = -~P2[e~u, P,v], A.,(u, v) = -~P,[P2u, P2v]. (2.19)

It is easy to see that Ai vanishes if and only if T/ is tangent to the foliation. In view of formulas (2.18) and (2.19), the Nijenhui.s tensor (or torsion.) N = 8(A~ + A..,) can be written as

Y(u, v) = [P, P](u, v) = [u, v] + [Pu, By] - P[Pu, v] - P[u, Pv]. (2.20)

Both distributions 7'1 and T2 are integrable whenever N = 0 holds. The second fundamental forms hi and he (cf. (2.10)) are given by

1p h,('u,v) = �89 h.,(u,v) = ~ l{P,u, P~v}, (2.21)

where {u, v} = (V,v + V.u) /2 is the Jordan bracket of two vector fields u and v. In view of forInulas (2.18) and (2.21), the Jordan tensor L = 8(th + h~) can be written as

L(u, v) = {P, P}(u, v) = {u, v} + {Pu, By} - P{Pu, v} - P{u, Pv}. (2.22)

Both distributions Tt and T2 are geodesic whenever N = 0 hohts. The vector field H = 8(H1 + / / 2 ) , where Hi = tr hi (i = 1, 2), can be written as

H = tr{P, P}. (2.23)

The covariant derivative V P can be decomposed into the sum of eight pointwise irreducible components with respect to the action of the group O(TI) ~ O(Ta). For the distribution D (T1 or T~), we consider the

corresponding eight conditions [101]: (1) (V~P)',, = 0,

(2) = O,

(3) ( d i m D ) ( V . . P ) v = (u, v)

(4) o (5) = 0, (6) (v P)v = = o,

(7) (v ,P)v = ( v . P ) u , ( d i m D ) ( V u P ) u = (u, v)

(8) (V P)v = (V P)u,

GD is a geodesic distribution:

M D is a ufinimal distribution,

UD is an umbilic distribution,

A, no conditions

GF is a totally geodesic foliation, M F is a minimal foliation,

UF is a totally umbilic foliation,

F is a foliation.

Properties (5)-(8) are similar to (1)-(4), but with an integrability condition. Combining these eight conditions and eliminating the dual situations, we obtain 36 different classes of

almost-product structures, each of which is characterized by some algebraic condition on V P [194].

For example, a (GD, A) almost-product structure (i.e., the distribution 7"1 is GD and T2 is A; a similar notation is used for the other 35 classes of ahnost-product structures) is called an almost foliated metric in [298] and an anti-foliation in [194]; in other words, it is a geodesic distribution. The (GD, F) and (UD, F) almost-product structures are actually Riemannian and conformal foliations.

Let us consider some examples of the 36 classes of almost-product structures constructed by using hyper- surfaces and products of manifolds given in [185]. There are nine classes of almost-product structures on real

(complex) hypersurfaces M in K "+~ (K = C, H). Let M be an orientable real hypersurface in C n+~, N be a

unit normal vector field on M, and let J be a canonical almost-complex structure on C "+l (see Fig. 12). The

canonical almost-product structure on M is defined by the distributions V = J N and H -- J N • Observe that J H = H. Every M of this ldnd belongs to the class (UF, MD).

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N

~ +1

Fig. 12. The structure (UF, MD) on a hypersurface in C "+l.

L e m m a 2.2 [185]. For M 2n+I C C "+1, we have the following equivalences (here x, y E H): (1) M is (UF, GF) r h(x, y) = O, (2) -hf is (GF, GO) .'. :- h(z, Jx) = O, h(x, JY ) = O, (3) M is (GF, MR) .~ :. h(x, Jy) = h(Jx, x), h(x, Jx) = 0,

(4) M is (UF, GO) -', :. h(x, Jz) = O, (5) M is (UF, MF) .~ :. h(z, Jy) = h(Jx, y), (6) M is(GF, MD) .~ ~. h(x, JN)=O.

Let M be an orientable real hypersurface in ]t~ +1, N be a unit normal vector field on M, and let {J1,

J,2, J3} be a canonical almost-complex structures on lI~ T M . These elements define the canonical almost-product

structure on M with V generated by the vector fields {J1N, J.,.N, JaN} ~md H = V • Observe that J~H = H. Every M of this kind belongs to the class (A, MD). Almost-product structures on a complex hypersurface

M in H "+l can be considered analogously. Conformal transformations of metrics on products of manifolds with a pair of distributions lead to some

additional examples of almost-product structures [185]. The rules for metric products of manifolds with Riemannian almost-product structures are as follows:

(M, P) (M', P ' ) (.hi x _hi', P + P')

(MF, GF) (GF, MF) (MF, MF) (MF, GF) (GF, GD) (MF, GD) (MF, GF) (GF, MD) (MF, MD) (GD, GF) (GF, GD) (GD, GD) (GO, GF) (GF, MD) (GO, MD) (MD, GF) (GF, MD) (MD, MD)

(2.37)

Indeed, there are interesting classes of foliations and distributions outside of the above scheme, such as, for example, the following:

1. Foliations with the property that the vector field P2(Vyz) is basic whenever y and z are basic vector fields; these foliations are characterized by the condition

((Vzh~)(y, y) - 2 (%h2)(z , y), x)

+(h2(y, y), B~(z, x)) - 2(h2(z, y), B~(y, x)) = 0 (2.24)

(see Sec. 3.8).

2. Foliations with constraints on the extrinsic curvature of the leaves, for instance, foliations of negative (nonpositive) extrinsic sectional curvature (see [39]) and some generalizations for partial Ricci curvature (see Sec. 3.7).

3. Weakly ha'rmonic distributions [233], which generalize the concept of minimal foliation.

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2.4. Constructions of Totally Geodesic and Totally Umbilic Foliations

2.4.1. Nu l l i t y fo l ia t ions . Totally geodesic and totally umbilic foliations appear naturally as null-distributi- ons (or kernels) in the study of manifolds with degenerate curvature-like tensors and certain differential forms. Examples of such nullity foliations can be divided into two classes.

1. A curvature-like tensor field R on a Riemannian manifold M is a (1, 3)-tensor field that satisfies the

first and second Bianchi identities. For a curvature-like tensor field R, the k-nullity space (the nullity space when k = 0) at m E M is defined by the formula

Nk(m) = {x E TraM: R(x, y)z = k((y, z)x - (x, z)y) for all y, z E TraM},

and its dimension is called the index of k-nullity of R at m. Let G be an open (nonempty) subset of M on which the index of k-nullity of R takes its minimum value. Then the distribution Ark is involutive and geodesic on G, and if M is complete, then the maximal integral manifolds are complete. The nullity space of the Riemannian curvature tensor was first studied in 1966 in a number of works and was later generalized to any curvature-like tensor (see [155]).

For a hypersurface M s C/VI ~+1 (k) with type number r(x) > 2 (r(x) is the rank of the second quadratic

form at x), the index of k-nullity ~k(x) of the curvature tensor is equal to n - r(x) (see [155]). Since r(x) is equal to 1 or 0 if and only if M n has constant sectional curvature k at x, the Riemannian manifolds of constant k-nullity n - 2 are of special interest. Every Riemannian manifold M of nullity n - 2 is a semisymmetric space, i.e., R(x, y) o R = 0 (x, y E TM) holds, where o denotes the derivation on tim algebra of all tensor fields on M. Since the nullity distribution defines a totally geodesic and locally Euclidean foliation, these spaces are said to be foliated semisymmetrie (see [31]). (Other semisymmetric spaces are either locally symmetric spaces or 2-dimensional surfaces, or Szabo cones; see [164].) A key problem of classifying 3-dimensional Riemannian manifolds with k-nullity 1 with respect to some constant k E R is studied in [207] and [126] (for k = 0, see [42]). In these papers, a special system of PDE's is solved and explicit fornml~ for these metrics are given;

using these formulas, interesting hypersurfaces in ~14(k) are constructed. The decomposition of a Riemzmnian manifold M whose curvature tensor has a positive index of nullity

is studied in [234] and [294]. The nullity space of the Weyl conformal curvature tensor (which does not satisfy the second Bianchi

identity) is studied in [97] and [268]: leaves of a certain foliation are totally umbilic and conformally flat.

The nullity space of the Bochner curvature tensor (which also satisfies the second Bianchi identity) on a Khhlerian manifold is studied in [150]: leaves of the corresponding foliation are K/~hlerian totally geodesic

submanifolds. Contact metric manifolds (M2~+1r u,g) satisfying the condition that the characteristic

vector field ~ belongs to the k-nullity distribution, i.e., R(X, Y)~ = kO?(Y)X - ~?(X)Y), are studied in [162] and [214]. Sasaki metrics on the unit tangent bundle S~M (see Sec. 4.1.2) with a positive index of k-nullity

((k > 0)) are studied in [318].

2. The relative nullity space of the second fundamental form h of the submanifold M C/VI at m E M is given by

ker h(m) -- {x E TraM: h(x, y) = 0 for all y E TraM}.

A curvature-invariant submanifold of a Riemannian space with a positive index of relative nullity #(M) = min{dimkerh(m): m E M} has the structure of a ruled developable submanifold (see Sec. 4.1.1). Relative nullity distributions were first studied for hypersurfaces in the Euclidean space (see [155]); for submanifolds in a projective space, see [9]. As a generalization, arbitrary principal-curvature functions on submanifolds in a space form in [211, 203, 223, 135, 32] and the corresponding spherical foliations induced by curvature surfaces are studied. For similar foliations that appear on hypersurfaces in conformally flat manifolds, see [139]. The

normal nullity space of a normal curvature tensor R • of a submanifold is considered in [186]. Relative nullity foliations on infinite-dimensional submanifolds of a Hilbert space are studied in [199].

Introducing the notion of L-parallel Jacobi fields, Dombrowski [73] developed a method for studying the completeness of a totally geodesic foliation induced by the kernel of a geodesic differential form. This

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technique is generalized in [229, 135] and [32], where variations of circles in Riemannian manifolds are studied and results on the completeness of spherical foliations induced by the kernel of a certain (spherical) differential form are given.

2 .4 .2 . D o u b l e - t w i s t e d p r o d u c t s . A popular generalization of the product of two (pseudo-)Riemannian manifolds is tile warped product of these manifolds introduced in [25]. In fact, it appeared in mathemat ica l and

physical literature long before that time; for example, polar coordinates of the Euclidean plane R 2 describe a warped product representation of ~t 2 \ {0}, and most of the space-time models are warped products. Warped products allow us to construct new examples of complete Einstein manifolds [24]. In recent studies of spaces with the Ricci curvature bounded from below, a class of warped products (a, b) x ~ M for some fixed s serves as a set of possible smooth rigid models [58, 59]. Tile result of Cartan stating that a Riemannian manifold M that satisfies the axiom of s-planes with fixed 1 < s < dim M is isometric to a space form was later generalized by geometers in two directions, namely, to more general submanifolds (totally umbilic, extrinsic spheres, submanifolds with parallel second f lmdamental form) and manifolds with additional structures (complex, quaternion, etc.) (see [172]).

The warped product of an l-dimensional space form with an arbitrary Riemannian manifold contains a large number of s-dimensional totally geodesic submanifolds for any s > l; they are inverse images of the projection onto the base of (s - l)-diinensional totally geodesic submanifolds in a given space form. In [200] this property is used as a basis of the synthetic axiom of (l, s)-planes, which is a generalization of the Cartan axiom: for every point p E M and any l-dimensional direction V C TpM, there exists a totally geodesic s-dimensional submanifold tangent to V. The conjecture (which is partially proved in [200]) states that the axiom of (l, s)-planes (for any 1 _< l < s _< dim M - 1) characterizes the warped products of l-dimensional space forms with an arbitrary Rienmnnian manifold.

For simplicity, we denote by (L~} the canonical foliations on the product manifold M1 x/1/I2 with natural

projections p~ onto M~, (i --- 1, 2). Let P~ be the projection of T(M1 x M2) onto TLi, and let p(L = Id -P~ , (i = 1, 2). The following notion is a natural generalization of warped products.

D e f i n i t i o n 2.6 ([218] and [160]). Let (M~, gO, i = 1, 2, be a Riemannian manifokl, and let A~: M~ x M:.2 --+ R be positive differentiable flmctions. The double-twisted product M1 x(~,,A2) M.~ is the differentiable manifold

J~'/1 x M2 with a Riemannian metric defined by

(x, y) -- A2gl(Plx, PLY) + A~g.2(P.,.x, P2Y)

for all vectors x and y tangent to M1 • Ms. In particular, if Ai are independent of the Mi-components, then M1 x (~,,~2) M.2 is a double-warped product.

This definition generalizes the Bishop notion of umbilic product M~ x~ M2, which in [61] is called the twisted product and which is a special case of the double-twisted product M1 xo,~ ) M2. If, in this case, A depends only on a point of M1, then A'/l x ~ M2 is a warped product. Note that the confomml change of a Riemannian metric can be interpreted as a twisted product, neanely, a product where the first factor M1 consists of only one point. Therefore, formulas and assertions concerning double-twisted products are applicable in many situations. For information on double-twisted products with more than two factors, see [160] and [98].

P r o p o s i t i o n 2.1 [160]. In the double-twisted product M~ x(~,~2) M.2, (1) the leaves {Li} form a totally umbilic foliation with mean curvature vector field

Hi = - P ~ (grad (log Ai))

(in the case of the twisted product A1 = 1, the leaves {LL} are even totally geodesic); (2) the leaves {L~} have parallel (in a normal connection) mean curvature vector if and only/fAi /s the

product of two positive functions on M1 and At2.

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L e m m a 2.3 [218]. The Levi-Civita connection V and the curvature tensor R of the double-twisted product

M l x (~1,~2)M2 with Ui = - g r ad (log Ai) and also the related Levi-Civita connection ~7 and the curvature tensor

of the ordinary product of Riemannian manifolds IV[1 and 1512 satisfy the relations

% y = Txy + E ( ( P x, - - (y, i

+ E((v u - (x , A - - (y , A P x), i

where, for all u, v �9 TpM, the wedge product u A v denotes the linear mapping w --+ (v, w)u - (u, w)v.

Double-twisted and double-warped products can be characterized in terms of the geometry of their canonical foliations.

P r o p o s i t i o n 2.2 [218]. Let g be a (pseudo-)Riemannian metric on a smooth manifold M~ x Me; assume that the canonical foliations {L~} and {L2} intersect each other at right angles everywhere. Then g is a metric of

(1) the double-twisted product 15I~ X(A,,A2) 15/'2 if and only if {L~} and {L2} are totally umbilic foliations,

(2) the twisted product 5I~ x:~ M2 if and only 'if {L~} is a totally geodesic foliation and {L2} is a totally umbilic foliation,

(3) the warped product M~ x ~ 1512 if and only if { L~ } is a totally geodesic foliation and { L.,} is a spherical foliation,

(4) the ordinary (pseudo-)Riemannian prod'act if and only if {L~} and {L2} are totally geodesic foliations.

Note that for a simply connected manifold M, the splitting (i.e., the local metric decomposition; see (4) of Proposition 2.2) is equivalent to the global product structure. On the other hand, a geodesic foliation on the Klein bottle with a fiat metric splits only locally. If a complete connected and simply connected Riemannian manifold M has two complementary orthogonal totally geodesic foliations, then the de Rham decomposition theorem (see [155] and [213]) asserts that M is isometric to the product of two leaves through any point m E M; for the case where M is not assumed to be simply connected, see [306].

Manifolds with two complementary orthogo,ml totally umbilic foliations (totally umbilic orthogonal 2- nets [203]) were considered by many authors (see [25, 98, 195]). In [160] and [218], the "double-twisted decomposition theorems" are proved. Even if a totally umbilic orthogonal 2-net M is a simply connected and complete Riemannian manifold, we cannot expect M to be globally isometric to a double-twisted product. If, however, one of the foliations is totally geodesic, then M splits globally as a twisted or warped product [218]. The case of M with bou,ldary is considered in [67]; for the case of M with additional assumptions concerning the metric, see [99, 100]; for the case of M with an additional complex structure, see [307] m~d [201]. Finally, for the cases of M with an additional symptectic-type structure, consult [64] and [308].

2.4.3. T o t a l l y geodes i c fo l ia t ions o n space fo rms . For k > 0, totally geodesic foliations on a 3-sphere are classified in [104]; in the c~e of S '~ (n > 3), there exist some interesting subclasses such as skew-Hopf fibrations and fibrations associated with a curvature tensor (see Sec. 2.6).

For k < 0, there are no totally geodesic foliations on a complete space form M(k) with a finite volume [323, 324]; the same is true for locally symmetric spaces. For k < 0, there are no totally geodesic foliations with compact leaves on the space form M(k) even locally (see [251] for Km~ < 0). Some results for totally geodesic foliations ou a hyperbolic space ~ ( k ) are given in [87, 11, 47].

For k = 0, a totally geodesic foliation, given locally on M(0) and having compact leaves, splits (see [251]

for K,~x = 0). In [13], conformal geodesic one-dimensional foliations on the 3-dimensional space form M(k) are studied.

Minimal foliations on the space forms M(k) are studied for k > 0 in [144] and [116].

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2 .5 . R i e m a n n i a n Fo l ia t ions

The study of Riemannian foliations was initiated by Reinhart [227, 228]. Recent results are systematized in [187] and [285-287].

Leaves of a Riemarmian foliation are locally given by level sets of a Riemannian submersion. Tim second fundamental form h2 of the normal distr ibut ion T2 vanishes (i.e., the operator B2(x, -) is skew-symmetric for all x E TL) . Therefore, this distribution on T M is geodesic. In particular, a geodesic tha t is orthogonal to a leaf at one point of M is orthogonal to {L} at every one of its points. This property, which characterizes Riemalmian foliations, was one of the initial observations of Reinhart [228] concerning this subject.

Since the curvature of a Riemannian foliation has stronger vanishing properties than that of an arbitrary foliation, its characteristic classes of Riemannian foliation should possess special properties. Many ordinary classes vanish, bu t there are additional classes that can be defined only for Riemannian foliations (see [227]). The idea advanced in [286] is to characterize the tautness of a Riemannian foliation by cohomological prop- erties. It is based on the following theorem (for simplicity, let T L and T M be orientable).

T h e o r e m 2.2 ([259] and [279]). A Riemannian metric gL on leaves (with the volume form coL) induces a

Riemannian metric g on M for which all leaves are minimal if and only if COL is a restriction of a v-form X to M that satisfies the condition dx (x l , . . . , x~+~) -= O, where ~, (= dim L) of the vector fields among {xi} are

sectio~z of T L C T M .

For example, the foliation Jr(G, H) of a Lie group G by a Lie subgroup H is t au t [181]. A compact orientable foliation on a compact orientable manifold M is taut iff the holonomy groups of all leaves are finite [227].

The umbilicalizable Riemannian foliation {L} on a compact connected manifold is tense, and the following conditions imposed on it are equivalent: (1) {L} is taut; (2) {L} is geodesible; (3) {L} is cohomologically

taut (i.e.; the cohomology of the basic forms, which are the differential forms that are locally pull-backs of the forms on the local quotient manifold ~r satisfies the Poincar4 duality) [49].

2.5.1. T h e bas ic c o h o m o l o g y . Let {L} be a foliation on M. A differentiM form ~ E ~t~(Ai) is basic if

i ( x )w = O, O(x)co = O, x c TL,

where i(x) denotes the inner product with respect to x. In a disting~fished chart ( x b . . . , x~; y~ , . . . , ym-~) of the foliation, this means that

w = ~ w~.. .~dy~, A . . . A dyo~, oq <,. .<(~,.

where the functions w~,...~. are independent of (x~, . . . , x~), i.e., o__~w~o, .--,~ _- 0. The set f~B(L) C f~(M) is a

subcomplex since the exterior derivative preserves the basic forms dln,(L): f~B(L) -+ ~B(L)- Introduce the

notation dlua(L ) ---- ds . By definition, the basic cohomology is H~(L) = H ' (~B(L) , dB). It plays the role of the de Rham cohomology of the leaf space of a foliated manifold. For the case of a foliation of codimension 1, there are just two groups, H~(L) and H~(L) .

The transverse orientation of the foliation {L} is defined by the orientation of T L • The transversal

volume form X • on T L • of a Riemannian foliation is holonomy invariant, i.e.,

(9 (X)x • = 0, X C TL. (2.25)

The holonomy invariance condition shows us that X • 6 ~~ Consequently, dx • : 0. It is of interest

to examine the cohomology class [](• E H~~ which plays the role of the orientation class for the leaf space of a foliation.

T h e o r e m 2.3 [287]. Let {L} be a transversally oriented foliation on a closed oriented manifold M satisflfing

one of the following conditions: (1) {L} is a minimal foliation with holonomy-invariant transversal-volume

form x• (2) {L} is a taut Riem nnian foliation. The # 0 in

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The mean-curvature form (1-form on T M ) of the foliation is given by

k ( y ) = g X ( g , y ) , y E T L •

where H C TL • is a mean-curvature vector field. By construction, i (x)k = 0 for all x E TL. Furthermore, for

every Riemannian foliation on a closed manifold, there is a well-defined cohoinology class [ks] E H~(L) (there is a bundle-like metric such that k is a basic one-form; see [286]) whose vanishing characterizes the tautness of the foliation. For a transversally orientable Riemannian foliation {L} on a closed orientable Riemmmian manifold M, the following three conditions are equivalent (see [298]):

(1) {L} is taut; (2) n ~ d i m L ( L ) ~ R; (3) [k] = 0.

The sufficient conditions for the tautness of a transversal orientable Riemannian foliation of codimension q > 2 on a closed orientable manifold can be expressed in terms of the transversal Ricci operator Ric:

T L • --4 TL • and the transversal curvature operator/~: A2(TLX) * --4 A2(TL• *, by virtue of the following vanishing results (see [286]):

(1) /f Ri'-'~ > 0, then H~(L) = 0, (2) i f R > O, then H~(L) = 0 for 0 < r < q.

An infinitesimal automorphism V (i.e., the flow of V maps leaves into leaves) acts on s C TL • as follows:

e(V)s = [V, y~].L, (2.26)

where Y.., C T M is such that (Y~)• = s. The right-hand side of Eq. (2.26) does not depend on the choice of Y~. Let V(L) be the set of all infinitesimal automorphisms of the foliation {L}, and let Y E V(L) be an infinitesimal automorphism of {L}. The transversal divergence divBY is defined by

O(Y) x • = divBY- X •

Here we use the fact that e (Y)X • E g~dimL(L) C Ar176177 Observe that d ivsY E g~~ and in

fact, depends only on Y• By using the Stokes theorem, one can prove the transversal divergence theorems for harmonic foliations

with holonomy-invariant transversal-volume form and for Riemannian foliations.

T h e o r e m 2.4 ([148] anti [287]). Let {n} be a transversally orientable minimal foliation with holonomy

invariant transversal volume form X • on a closed Riemannian manifold ( N[, g), and let Y be an infinitesimal automo~Thism of {L}. Then

divB(V ) dvol O. M

Tile value of this integral in the case of a Riemannian (not necessarily minimal) foliation is given in the following theorem.

T h e o r e m 2.5 [287]. Let {L} be a transversally orientable Riemannian foliation on a closed orientable man- ifold (M, g), and let V C V(L). Then

f divs(VX)dvol = f g• V • (H, vx ) • (2.27) M M

where _L is the global inner product of the sections H and V • in TL •

2.5.2. T ransve r sa l K i l l i ng fields. An infinitesimal automorphism V E V(L) with the property O(V)g • = 0

is said to be transversaUy metric. If this holds, V • is called a transversal Killing field [287]. For the point foliation (with dim L = 0), this is the usual definition of a Killing vector field. The (transversal) Jacob•

operator J = • - ~ : T L • -4 TL • where A is a transversal Laplacian is also associated with ~ (see [192]).

The condition 0;(Y) - 0 defines (transversal) Jacob• fields.

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The transversal divergence Theorem 2.4 is a key one for the following result.

T h e o r e m 2.6 [148]. Let {L} be a transversally orientable minimal foliation on a compact Riemannian manifold (M,g) with a bundle-like metric, and let Y be an infinitesimal automorphism of {L}. Then the following properties are equivalent:

(1) Y• is a transversal Killing field, i.e., O(Y)g • = O,

(2) Y• is a transversally divergence-free Jacob• field,

(3) Y• .is transversaIly affine, i.e., ~(Y)~7 = O. Moreover, if codlin L = 2, then the following properties are equivalent.

(4) Y• is a transversal conformal field, i.e., O(Y)g • A. gZ,

(5) Y• is a transversal Jacob• field.

By this theorem, the linear space of transversal Jacob• automorphisms of the (harmonic) Hopf fibration p: S 3 --+ 5 r is isomorphic to the linear space of infinitesimal conformal fields on $2; in particular, this space is 6-dimensional.

The operator Az: T L • -+ T L • for Y C T L z is defined by the law Ay(2) -- -~TzY, where Z C T M

with Z • = Z [320]. The following theorem generalizes the results for Killing fields on a Riemannian manifold M by Kostant and Curr~s-Bosch.

T h e o r e m 2.7 [320]. Let { L} be a minimal foliation on a connected orientable complete Riemannian manifold (M, g) with a bundle-like metric. Let Y be a transversal Killing field with a finite global norm. Then, for each

m E M, (Az)m belongs to the Lie algebra of the linear holonomy group ~fT(m), where ~7 is the transversal

Riemannian connection of {L}.

If, in Theorem 2.7, the transversal Ricci operator Ri~-~ of {L} is nonpositive everywhere and negative for at least one point of M, then every transversal Killing field with a fi,fite global norm is trivial [321]. Note that any Killing field of a bounded length (for instance, on a compact manifold) preserves a codimension-1 totally geodesic foliation [207, 208].

2.5.3. Transversa l ly s y m m e t r i c R i e m a n n i a n fo l ia t ions . There are Riemamfian foliations whose transversal geometry can be locally ,nodeled on a Riemannian symmetric space. A transversal symmetry can be chm'acterized by conditions imposed ou the canonical Levi-Civita connection on a normal bundle.

T h e o r e m 2.8 [287]. Let {L} be a Riemannian foliation on (M,g), and let g be a bundle-like metric. Then the foUowing conditions are equivalent:

(1) {L} /s transversaUy symmetric,

(2) local geodesic symmetries (geodesic reflections) on a model space are isometries,

(3) ~7:R(z, y, z, y) = O, z, y E T L •

(4) ~7.R(z, y, z, y) + 2R(z, O..y, z, y) = -6((~7.O)~y, O~.y), z, y E T L •

All these conditio,ls are purely local and are automatically satisfied for a Riemanniml codimension-1 foliation. For a totally geodesic Riemannian foliation, this characterization can be improved in the analytical case with the use of the following results (see [286, 287]): reflections in the leaves are isometric i f and only if

the geodesic reflections on the model space are (local) isometries. In a space of constant curvature, the reflections in totally geodesic submanifolds are isometries. Hence,

from the restflts xnentioned above, we obtain the following theorem.

T h e o r e m 2.9 [287]. Let (L} be a Riemannian foliation on a space (M, g) of constant curvature, and let g be a bundle-like metric. Then {L} /s transversaUy symmetric if and only i f

(Oyz, T(Oyz, y)) -- O, y, z E T L • (2.28)

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Def in i t i on 2.9 [75]. The union F~ of two half-horocycles with opposite normals at a common point is called

a broken horocycle. The trajectory D~ (containing x E l) that is orthogonal to the family N of all straight

lines intersecting the l-axis at a given angle a is called the fifth line. (Recall that a straight line, a circle, a

horocycle, and an equidistant are the first four lines in H2.)

T h e o r e m 2.11 [75]. There exist exactly three nontrivial metric fibrations (i.e., fibrations with fibers different

from one-point fibers and from the whole plane) of H 2 with connected fibers: (1) a fibration into horocyeles,

(2) a fibration into broken horocycles, (3) a fibration into the fifth lines.

2.6. L a r g e - S p h e r e Fo l i a t i ons

Consider a round sphere S 3 = {x E R4: [x[ = 1} in a Euclidean space •4 with an orthonormal basis

{ei}. A 2-dimensional subspace ~ of R4 with an orthonormal basis {a, b} intersects S 3 along the large circle

(geodesic) 7(t) = a c o s t § bsint, which plays the role of a straight line in a sphere. Two large circles in

S 3 may not intersect, but are always engaged. For every large circle 3' C S 3 and every point x0 E S 3 \ 3'

at a distance dist (X0, 3') < 1r/2, there exist exactly two large circles 3'1,3'2 C S 3 that are equidistant to 3'

(Clifford parallelness), i.e., for all x e 3'i kA 3'2, we have dist (x, 3') --- dist(x0,7). The set 3'• -- {x E $3: dist (x, 3') = ~r/2} of points that are at themaximal distance from 3' is also a large circle; it is orthogonal to

3" (see Fig. 13). In particular, a surface that is equidistant to 3', T(7, s) = {x E $3: dist(x, 3') = s} with

0 < s < Ir/2, has a locally Euclidean metric and two (orthogonal for s = lr/4) foliations by large circles;

T(3', 7r/4), is called the Clifford torus [326]. Each of two continuous partitions of S 3 by large circles that are

Clifford parallel to the given 3' are called Hopffibrations; their base is S 2. The Hopf fibration introduced in [136-137] about sixty years ago had a very considerable effect in topo l -

ogy , since it provided the first example of a homotopically nontrivial map from one sphere to another of lower dimension. It also had a great effect in g e o m e t r y because the fibers are equidistant geodesics and form a Riemannian foliation. The Hopf fibration can be defined in several equivalent ways.

1. The Hopf fibration p: S 3 -~ S 2 ~- C P 1 for complex variables zl = xl + ix2, z2 = x3 + ix4 in {C 2 = C l x C 1 } A S ~ has the form

p(z , z 2 ) =

for real variables this p: (xl, x2, x3, x4) --+ (Yl, Y2, Y3) has the form

2(x x + y 3 2( 2 3 = + - = = -

Y'+~ Then p can 2. We introduce a complex variable z on S 2 via the stereographic projection: z = 1-y3

Xl q'ia:2 be written in a more compact form z -- ~3+~4 as an ordinary map from the unit sphere in C 2 -- R 4 into the

complex projective line C P 1 = S 2 [104].

3. Consider the action ~(t)(z~, z2) = (exp(2zrit)zl, exp(2zrit)z2) of the group ~1 on S 3. The orbits of this foliated action form a Hopf fibration.

4. The Hopf fibration can be given as a collection of intersections of S 3 with all holomorphic 2-planes {a = x A Jx}, where J is any complex structure in R 4, i.e., a linear operator given in some orthonormal basis {ei} by the rule

J e l = e 2 , J e 2 = - e l , Je3=e4, J e 4 = - e 3 .

Note that the multiplication by J gives us an orientation of fibers. The projective space K P n (K ---- R, C, H) can be introduced as a set of orbits for the right action of the

group K* = K \ {0} on K '~+1 \ {0}, i.e., x ,,~ y ~ there exists a A E K* such that x = yA. This quotient space

is not defined for a (nonassociative) Cayley algebra Ca. For K = C, H, the orbits are equidistant (dimK -- 1)- dimensional large spheres. We set the distance between the points on I ( P n equal to the distance between

the corresponding orbits on a round sphere S ~ dimK-I and obtain a canonical metric on the projective spaces

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Condition (2.28) is satisfied both in the case of a totally geodesic foliation (i.e., tensor T = 0) and in the case of a Riemannian foliation with an integrable normal bundle (i.e., tensor O = 0); therefore, these foliations are necessarily transversally symmetric. Conversely, the existence of a transversally symmetric foliation imposes strong constraints on the geometry of the foliated space. A typical result is that the transversal symmet ry of a foliation defined by a Killing vector field of unit length on a complete simply connected (M, g) implies that (M, g) is a naturally reductive homogeneous space (see [298]). A Riemannian foliation can be compared with transversally symmetric foliations; for results concerning the case of a small basic mean curvature, see [285].

The following classes of Riemannian foliations defined in terms of the eigenvalues and eigenspaces of a transversal Yacobi operator generalize the notions of e-space and ~-space introduced by J. Berndt and L. Vanhecke in 1992.

Def in i t i on 2.7 [288]. A Riemannian foliation {L} on (M, g) is a transversal r if the eigenvalues

of the Jacobi operator/~(~/' , ")I" are constant along each geodesic 7 that is orthogonal to the leaves of the

foliation. {L} is a transversal ~-foliation if the eigenspaces of R(7',-)~" can be spanned by parallel fields of eigenvectors along each geodesic ~/orthogonal to the leaves of the foliation.

To give examples of transversal ~- and ~-foliations, it suffices to consider warped products B xf F over a r or ~-space B. These concepts are used for a new characterization of transversally symmetr ic foliations.

T h e o r e m 2.10 [288]. The foliation {L} is transversalIy symmetric if and only if it is simultaneously a transversally r and a transversally ~-foliation.

The following interesting subclass of transversally symmetric (Riemannian) foliations was introduced and studied in [111-114].

Def in i t i on 2.8. Locally Kilting-transversally symmetric spaces (locally KTS-spaces) are Riemannian mani- folds (M, g) endowed with the isometric flow generated by the unit Killiug vector field such that the local reflections with respect to the flow lines are isometries.

These foliated spaces were studied in [111-114] and in [110] with the use of the extrinsic and the intrinsic geometry of geodesic spheres and tubular hypersurfaces around flow lines and around geodesics orthogonal to flow lines.

2.5.4. R i e m a n n i a n fo l ia t ions on s p a c e fo rms . For k > 0, Riemannian submersions from spheres Sn(k) are Hopf fibrations, except for, possibly, the case where dim h i = 15 and dim L = 7 (see [82-83, 221-222] and [120, 121]). The class of Riemannian foliations on the spheres Sn(k) is larger than the class of Hopf fibrations (see Sec. 2.6), and for d i m L < 3, consists only of homogeneous foliations (i.e., foliations generated by an isonietric action of a Lie group) [121]. There are no homogeneous 7-dime,isional foliations on the spheres S~(k) [174].

For k = 0, a Riemannian submersion of the complete M(0) is the projection of the direct product onto

one of its factors (see [304]). Pdemannian foliations of a compact space form ~f(0) split (are locally projections

of the direct product onto one of its factors); Without compactness, this statement is false even for ~2 \ {O}: consider, for example, a l-dimensional foliation generated by glide rotations.

For k < 0, there are no Riemannian foliations (and submersions) on the compact space form M(k); the same is true for compact locally symmetric spaces M of negative curvature (see [304]). In contrast, there exist Riemannian submersions from noncompact H~(k) with various types of leaves (see [87, 47, 115]).

One-dimensional Riemannian foliations of Einstein spaces are studied in [212]. The concept of a metric fibration as a parti t ion of a metric space to isometric and locally equidistant sub-

sets is studied in [75]. For Riemannian manifolds, this leads one to a special case of Riemannian submersions. The only metric fibrations of ~n with connected fibers are the unions of parallel k-planes (k = 0, 1 , . . . , n) [75]. The hyperbolic geometry is much more interesting from this point of view.

1721

Fig. 13. Clifford parallelism.

C P n and H P n. In particular, this canonical metric on C P i transforms it into a round two-sphere of radius 1/2. Generalized Hopffibrations (see [106]) and a similar complex Hopf fibration [81]

S l C S 2n-1 ~ CP n-l, S 3 C S 4n-1 ~ H P ~-i, S T C S is ---+ S s,

C P i C C P e~-i -+ H P ~-i (2.29)

are Riemannian submersions. Moreover, the fibrations in (2.29), except, perhaps, for one (see Sec. 2.5.4), are

unique Riemannian submersions of Euclidean spheres [121]. On a round sphere, the foliation by geodesics is

like a large-circle fibration. The simplest of them, a generalized Hopf fibration (2.29 a) with fibers {S1}, can

be given as a collection of intersections of S '~n-i with all holomorphic 2-planes {a = x A Jx}, where J is a

complex structure, i.e., the linear operator in R ~-'~ given for some orthonormal basis {e~} by the rule

Je2i-i -~ e2i, Je2i = -e2i-1 (1 _< i < n). (2.30)

Let .T0(S 2n-i) denote the space of all Hopffibrations of the sphere S 2n-i. Each fiber spans the corresponding

oriented 2-plane through the origin in ~ " , and hence, defines a point in the Grassmannian manifold G(2, 2n).

Def in i t ion 2.10 ([104] for n = 2). The skew-Hopffibration is given by the intersections of S 2'~-1 with all

holomorphic 2-planes {a = x A Jx} , where J is an almost-complex structure, i.e., the linear operator on ~2,~ given by the rule (2.30) for some affine basis {e~}. In other words, the skew-Hopf fibration can be obtained

from the Hopf fibration as a result of a nondegenerate linear transformation of ~.2n and then by the central projection of the images of fibers back to S 2"-i.

Let 9ri(S 2~-1) be the space of all skew-Hopffibrations of the sphere S ~'~-i. Let ~ ( S 2~-1) be the space of all oriented large-circle fibrations of S 2n-1.

For simplicity, below we will give some facts concerning geodesic foliations on a 3-sphere. The space .~-0(S 3) is 2-dimensional and homeomorphic to a pair of disjoint 2-spheres. The 8-dimensional space ~ i (S 3)

is the disjoint union of two copies of S 2 x ~6. The following two indicated spaces are homogeneous [104]:

.T0(S 3) = 0(4)/U(2), .T~(S 3) = GL(4, ~t)/GL(2, C). (2.31)

The space 9r(S 3) is infinite-dimensional. Let V be the Hopf unit vector field, and let D -~ be a small ball

transversal to V at a point p. Small CLperturbations of V on D" that are identical in a neighborhood of the boundary OD 2 lead us to different large-circle foliations of S 3 [104]. There also exist discontinuous fillings of S 3 by large circles: one can fill the closed solid toms 2 2 2 2 $3 x~ + x 2 _> x 3 +x a on as for the Hopf fibration and then

fill the remaining open solid torus as for the Hopf fibration defined by the second continuous partition of S 3 by large circles. Using the isometry G(2, 4) - S ~ x S 2, the Hopf fibration h E :P0(S 3) can be characterized by

the fact that its orbit space Mh appears inside the Grassmannian as {point} • S 2 or S 2 x {point}. However,

not all submanifolds of G(2, 4) can serve as M I for f E 9v($3).

T h e o r e m 2.12 [104]. A submanifold in G(2, 4) - S 2 x S 2 corresponds to the fibration f E ~'(S 3) /f and only if

it is the graph of a distance-nonincreasing map ] from either of the spheres S 2 ( a factor in the Grassmannian)

1723

to the other. The fibration f is di.~erentiable if and only if the corresponding map ] is di~erentiable with Idfl < 1.

Hence the space 5v0(S 3) is a deformation retract of the space ~($3). The catalogue of large-circle fibrations of the 3-sphere (in Theorem 2.12) is one of the first nontrivial examples in which one has a clear overview of all possible geodesic foliations of a fixed Riemanniml manifold. The structure of the space of skew-Hopf fibrations ~'i (S 3) is s tudied in [94] and [237].

The interest in fibrations of an n-sphere by large u-spheres is due to the Blaschke problem and by extremal theorems in Riemannian geometry (see [82, 83, 94, 104-108, 120-122, 124, 136-137, 144, 174, 237, 244, 256, 265-266, 295, 318]).

A compact Riemannian manifold M is called a C, rmanifold if all its geodesics are closed and are of length 7r. Tlfis class includes Blaschke manifolds for which, by definition, all cut loci Cut(p) c TpM, p E M, are round spheres of cbnstant radius and dimension. Examples are CROSS: spheres or projective spaces. The B l a s c h k e c o n j e c t u r e claims that any Blaschke manifold is isometric to its model CROSS. For a Blaschke manifold, the exponential map expp: TpM --+ C(p), being restricted on the sphere Sp with radius d(p, C(p)),

defines a large-sphere foliation; for CROSS, this foliation is a Hopf fibration. Since every large-sphere foliation of S N is homeomorphic to the Hopf fibration [266] (for some special cases, see [105]; some results for differential ill)ration by large spheres can be found in [124]), a simply connected Blaschke manifold is homeomorphic to its model CROSS.

In cases where the extremal value for the curvature, diameter, or volume of a manifold is considered (under other given conditions), it is often possible to prove that the manifold is isometric to a model space from a finite list. A beautiful example of these extremal theorems is the following.

T h e o r e m 2.13 (on minimum diameter) [21]. Let M be a complete connected, simply connected Riemannian manifold urith sectional curvature 1 <- KM ~_ 4 and diameter ~. Then M is isometric to CROSS: a sphere of

curvature 4 or one of the projective spaces C P ~, HP", and CaP 2, with its canonical metric. Gtuck et aL II07] gave a constructive proof of this theorem; by using Berger's geometric arguments, they

showed tha t the exponential map from the tangent cut locus to tlm cut locus of a manifold is a fibration of a round sphere by parallel large spheres; hence, it is a Hopf fibration. Then they checked how this fibration encodes tim curvature tensor and used it to construct an isometry between M and a round Sphere or the projective space. Later on, the Berger theorem was generalized in several directions (sce [21,309, 122]). One of the key points in these results comes from the study of Riemannian foliations on a round sphere. For 1- and 3-dimensional leaves, they are always Hopf fibrations [120, 121]; a partial classification of Riemannian foliations on S 15 with 7-dimensional leaves is given in [309, 221,222, 174].

Toponogov has studied V m ( - c e , 4), the set of complete simply connected Riemannian manifolds with sectional curvature <_4 and injectivity radius >7r/2. Since the diameter of these manifolds is bounded from below, the case of the extremal value 1r/2 of the diameter is especially interesting.

T h e o r e m 2.14 [291, 292]. A manifold M E V2'~+~(-c~,4) with diameter ~/2 is isometric to a sphere of

curvature 4. A manifold M E V2~(-c~, 4) with diameter 1r/2 either is isometric to a sphere of curvature 4

or its geodesics are grouped into the following families (as for projective spaces): (F1) for every point p E M and any vector )~ E TpM, there exists an a-dimensional (a = 2, 4, or 8, and

ira = 8, then d i m M = 16) subspace d(A) C TpM such that all geodesics 7 C M (~/(0) = p, ~/(0) E d(A)), form a totally geodesic submanifold F(p, )~), which is isometric to the sphere S"(4);

(F2) for all nonzero vectors )~, A2 E TpM, the submanifolds F(p, )~) and F(p, A2) either coincide or their intersection consists of only one point p.

Manifolds with properties (F1) and (F2) constitute a special case of Blaschke manifolds. Toponogov [54-56] conjectured that the manifolds in V2n(-oc, 4) with extremal diameter equal to 7r/2 are isometric to CROSS.

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Note that tangent a-planes to the submanifolds { F(p, .)} (in Theorem 2.14) induce an (a-1)-dimensional large-sphere foliation of the round sphere Sp in the tangent space TpM. In the case KM > 0, for almost every point p E M, such a fibration fp is related to the c l ~ s ~ n of analytic geodesic fibrations on CROSS by means of the function of sectional curvature in TpM [290-292]. For simplicity, it is defined below for the sphere S 2n-1.

Def in i t i on 2.11 [290]. The fibration f E ~ '(S ~ - I ) belongs to the class ~'~(S 2'~-~) if there exists a multilinear function ("curvature tensor") R: R '" x R2, x 1r 2n • iir 2" --+ R l with the following properties:

(Rt) symmetries and tile first Bianchi identity:

n(x , y, z, w) = R(z , w, x, y) = - R ( x , y, w, z),

R(x, ~, z,'w) + n (z , ~, .V, ~) + n(y, z, ~, w) = 0,

(R2) for almost evetbr vector x E S 2'~-t, there exists a unique 2-plane a 9 x with R(x, y, x, y) = 1 (y E

y • M= 1), (R~) if a 2-plane c~ contains a fiber of f , then R(x , y, x, y) = 1 for all x, y E a (y • x, [x[ = lyl = 1), i.e.,

the sectional curvature of R is extremal on a.

The surprising fact is that :FR(S ~-~) = 9v~($2'~-1), since 5vn(S :~-~) --- ~0(S 2'~-l) was expected (see [237, 244]).

The natural strategy to prove that a Riemannian manifold with properties (R~)-(Rs) is isometric to CROSS (and hence, to prove t h e Toponogov conjecture) is to deduce first that the induced large-sphere foliations in the tangent spheres {S~}, p E M, are Hopf fibrations. This claim was conjectured in [54, Lemma 6] for a foliation with the properties (R2) and (Rs) and the curvature symmetries (R~). By considering a curvature tensor at only one point p of M, one can only show that such a foliation on the tangent sphere Sp, is skew-Hopf fibration (the proof of Lemma 6 in [290] is incorrect). Later, the Toponogov conjecture was proved in a different way [257], nmnely~ by using the global integral geometric methods of Berger and Kazdan (see [23]) and recent topological results for manifolds with closed geodesics.

T h e o r e m 2.15 [257]. A Riemannian manifold M E V'~( -oc ,4 ) with extremaI diameter ~r/2 is isometric to CROSS.

We will use Theorem 2.15 below in Theorem 4.10 of Sec. 4.2.2. From Theorem 2.15, we obtain a gener- alization of Theorem 2.13 by Berger.

Coro l l a ry 2.1 (on the diameter rigidity). A complete connected, simply connected Riemannian manifold M 2~ '~ith sectional curvature 0 < KM <_ 4 and diameter dimn (M) -- ~/2 is isometric to CROSS.

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Chapter 3 F O L I A T I O N S A N D A M I X E D C U R V A T U R E

3.1. Fol ia t ions a n d a C u r v a t u r e

The relations between curvature and topology are fundamental for Riemannian geometry "in the large" (see [96, 48, 156]), where two points of view can roughly be distinguished [96], namely

- - the traditional point of view: by assuming some rather strong properties of tile curvature (e.g., sign

definiteness and pinching), we obtain sharp structure theorems, - - the rough point of view: under weaker assumptions we obtain, e.g., the finiteness results for topological-

type manifolds. Examples of this kind of result can be found in [215] and [300], respectively. In 1979, Gluck posed the following question: what prevents a foliation on M from being geodesible?

In the last few years, the interest of geometers in similar problems on the existence of adapted metrics and foliations (or distributions) on a manifold with additional curvature restrictions has become stronger.

FC1. Is there a complete Riemannian metric g for a foliated manifold M, {L} which induces given geometrical and curvature properties of a foliation?

FC2. Is there a foliation {L} with given metric and curvature properties for a fixed complete Riemannian manifold (M, g)? If these foliations exist, then what is their classification?

There are three types of sectional curvature for a foliation, namely, tangential, mixed (denoted b y / ( ~ • a plane that contains a tangent vector to the foliation and a vector orthogonal to the foliation is said to be mixed), and transversal.

We will concentrate our attention in this paper on a mixed curvature such as a sectional, Ricci, or scalar curvature (s.ee [304, 302, 147, 324]) whose geometrical sense follows from the fact that for a totally geodesic foliation, certain components of the curvature tensor along any geodesic "7 C L are contained in the Jacobi equation Y " + R( t ) Y = 0 for the Jacobi tensor induced by the foliation and the Riccati equation B' + B2 + R( t ) =

0 for the structural tensor B: T L • T L • -+ T L • defined by B(x, y) = (U~y) z, (x E TL, y e TL• The

tensors Y and B are related as Y ' ~- B(~/', Y); see below. In the case of constant curvature, the solutions to the above Jacobi and Riccati equations (and therefore, the relative behavior of geodesics on nearby leaves)

are well known (see Fig. 14). For a given foliated manifold, a metric of constant tangential, mixed, or transversal curvature can be

regarded as canonical. Examples of foliations with Kmix = const are Hopf fibrations and the following foliations (see Sec. 2.4.1):

(1) k-nullity foliations on a manifold with degenerate curvature tensor; in [123], these metrics are called partially hyperbolic, parabolic, or elliptic,

(2) relative nullity foiiations on a submanifold M of the space form IVI(k). It is easy to find the following upper bound for the dimension of leaves of a compact totally geodesic

foliation with Kmi~ > 0:

dim L < codim L (i.e., dim L < �89 dim M)

(for the generalization for a partial Ricci curvature, see Sec. 3.7). The idea is that two compact totally geodesic submanifolds L, and L2 with dim L, + dim L._, _> dim M

in a Riemannian space M of positive sectional curvature (for example, two large spheres in a round sphere) intersect each other. For dual results, for K ~ = const < 0, on a foliated manifold with finite volume, see [323, 324]. For/(mix = const > 0, Ferus proved the following theorem in 1970; it has a number of corollaries.

T h e o r e m 3.1 (local in transversal directions) [86] If, for a totally geodesic foliation {L}, h'~i~ = k > 0 along a complete leaf Lo, then dim L < p (codim L). (For the definition of p(n), see Sec. 1.1.)

C o r o l l a r y 3.1. Let M "~ be a complete submanifold in the sphere S '~+p. (1) I f the index of relative nulh:ty #(M) > u(n), then M" is a totally geodesic submanifold [86[. (2) / f the extrinsic sectional curvature is

nonpositive and 219 < n - u(n), then M " is a totally geodesic submanifold [91[.

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j j J

K = 0 K = I K = - I

Fig. 14- Behavior of geodesics and Jacobi Kelds for K~i~ = coast.

The sequence u(n) is defined as u(n) = max{t: t < p(n - t)}. Some values of u(n) are given by

u(n) = n - (higher power of 2 _< n); for n _< 24, in particular, u(n) <_ 8d - 1 for n < 16 d, and u(2 ~) = 0. For

example, u(1) = 0, u(2) = 0, u(3) = 1, u(4) = 0, u(5) = 1, u(6) = 2, u(7) = 3, u(8) = 0, etc.

T h e o r e m 3.2 [235]. The estimate dim L < p (codimL) in Theorem 3.1 is

(a) sharp : for all u < p(n), there ezists a tube around a u-dimensional large sphere in the round sphere S "+~ with a foliation by u-dimensional large spheres (dim L < 2 for the Kfhlerian case),

(b) false for nonconstant K~ix > 0: there exists a tube around a closed geodesic in any one of the projective spaces K P n (n >_ 2; K = C, r4, Ca) which can be foliated by closed geodesics,

(c) t r u e for ruled submanifolds M n+~, {L ~} 'with/(mix > 0 in the sphere SN(k).

Thus, the following problems are natural: (1) when, for a given foliated smooth manifold M, does there

exist a complete geodesible metric of nonnegative (or positive) mixed curvature, (2) what is the structure of these foliations having large dimension, in particular, when do they split, (3) how can we effectively apply the technique of foliations to submani[olds with generators (ruled, parabolic, etc.) in a Riemannian space with nonnegative curvature. The first problem is interesting even locally for a foliation by closed geodesics, i.e., when dim L = 1 and codlin L --= 2m + 1,

1.44 P r o b l e m , Let M = S t x B ~-m+L be die product of the circle and the odd-dimensional bali with the product foliation {S t • b}. For what maximal ~ ~ [1/4, 1) does there exist a geodesible metric on M with positive and 5-pinched iimi~ ?

m ') (m >_ 2), Theorem 3.2 (b), (3.25), and Sec. 3_4.2. Toponogov For examples with O k (,~7~) see

conjectured that for a totally geodesic foliation on a compact Riemannian manifold with K~,,~ > 0, the Ferus inequality dim L < p (codlin L) holds.

3.2. Jacobi Vector Fields and the Riccati Equation

The study of the second variation of the length or energy of a curve in a Riemannian manifold leads one to the Jacobi equation. Sometimes, the simple use of the second variation allows us to establish relationships between curvature properties and the structure of a manifold in general. We will give here some basic material from the theory of variations o] geodesics; for more details, see [155, 48, 96].

For smooth normal vector fields y and z along a geodesic 7 (i.e., y, z _L~/), the index bilinear form is defined by the well-known formula

l(7) z) = (y', z),~) - / (z, y" + P~,y) ds, (3.1) I(y,

o

f~(~) [7'(s)l 2 ds in the direction of y for the case where obtained from the second variation of energy E(7) = ~ 0

z = y. Here P~, := R(-,7')7' is the Jacobi operator (see, e.g., [22]) and y' := V~,y, y" := Vz,~Zz, y.

Definit ion 3.1. A vector field y along a geodesic "7 in a Riemannian manifold M is called the Jacobi vector field if it satisfies the Jacobi differential equation

y " + P~,y = 0. (3.2)

With each rectangle 6: Q -+ M, we associate two vector fields: x = d f (~ ) and y = d f ( ~ ) . A smooth

variation (rectangle) 5: Q ~ M is called a geodesic variation if all T-edges 5(7, .) are geodesics (see Fig. 15).

1727

~r

= 7 o

Fig. 15. Variations of geodesics and Jacobi vector fields.

The geometrical sense of a Jacobi vector field is as follows.

L e m m a 3.1 [96]. f f S: Q --~ M is a geodesic variation, then y = dS( ~ ) is the Jacobi vector field along every r-edge. Conversely, every Jacobi vector field can be generated by some geodesic variation.

The tangent and the orthogonal component of the Jacobi vector field y along the geodesic 7 are also Jacobi vector fields. The Jacobi vector fields orthogonal to the geodesic 7 are considered in many situations. A conjugate point and a focal point along a geodesic are defined in terms of Jacobi vector fields.

The Jacobi equation (3.3) in the space form M(k) can be decomposed, and its solutions (Jacobi vector

fields) can be written in the following simple form:

cos V/kt ~ + sin x/kt ~ for k > 0 y(t) = tfl + 2 for k = 0

cosh ~L--kt ~ + sinh yrL--kt ~ for k < 0,

where y, z are parallel vector fields along 7. Consequently, conjugate and focal points are absent when k __ 0. There exist comparison theorems for the lengths of Jacobi vector fields when the curvature of one manifold

is a majorant for the curvature of another manifold. In most cases, the model for comparison is a space of constant curvature (see [48]).

Def in i t ion 3.2. A tensor field Y(t): 71 --+ 7 • is called the Jacobi tensor if

(1) the Jacobi equation

Y " + R~,Y = 0 (3.3)

holds; (2) Y(t) is nondegenerate in the sense that ker Y(t)Nker Y'(t) = {0} for all t (this condition holds precisely

when the action of Y on linearly independent parallel sections of 7 • gives rise to linearly independent Jacobi vector fields).

A Jacobi tensor Y is called a Lagrange tensor if (3) the Wronskian W(Y, Y) = (Y ' )*Y - Y*Y ' is zero (which, in particular, is true when Y(t) = 0 for

some t), or what is equivalent, if the tensor B := y t y - 1 is self-adjoint at all points where Y is invertible.

For example, the following is a version of the Meyer theorem in terms of n-dimensional Jacobi tensors [81]. I f tr P~, _> c > 0 in (3.3) with Y(O) -- O, then the first positive value of t at which Y(t) is not invertible

satisfies t < ~ r v ~ and equality occurs if and only if P~, - (c/n) Id.

When Y is invertible, we can set B(t) := Y'( t )Y-~( t ) . If we differentiate B covariantly and substitute the result into Eq. (3.3), then we find that

B' + B 2 + / ~ , = 0. (3.4)

L e m m a 3.2. Let the matrix-valued function Y(t) (det Y(0) ~ 0) and the matrix-valued function B(t) satisfy the equation

Y'(t) = B(t)Y(t) . (3.5)

Then Y(t) satisfies the Jacobi equation (3.3) for a certain matrix-valued function P~, if and only if B(t) satisfies the matrix Riccati equation (3.4).

1728

The invertible Jacobi tensor Y is a Lagrange tensor if and only if the corresponding tensor B is self- adjoint for some t. Moreover, every Lagrange tensor can be obtained by a normal geodesic variation of ~/along some hypersurface [81]. Important examples are Jacobi tensors, which vanish at some point of a geodesic, and stable Lagrange tensors (when they exist) with the condition Y = lims-~+oo Y~, where the Jacobi tensors Y~ satisfy the conditions Y~(0) = Id and Y.~(-s) -- 0. In the latter case, B( t ) is the second fundamental forin

of the (stable) horosphere through ~/(t) and tr B( t ) is its mean curvature.

According to the decomposition B = B + + B - , the matrix Riccati equation (3.3) can be polarized into symmetric and skew-symmetric parts:

(B+) + (B+) 2 + ( B - ) 2 + R ( t ) = 0, (3.6) ( B - ) ' + B + B - + B - B + = O.

The expansion 8, the sheaf tensor a, and the vorticity tensor w can be calculated from B as follows: 8 :=

1 if 7 is a Riemannian or time-like geodesic and tr(B) = t r B +, c ~ : = B + - a o I d , a ; = B - , w h e r e a - dim~•

a = ~ * if 7 is null. Then, at points where Y ( t ) ~ 0, we have O = tr(Y'Y -1) - (detV)'det Y " Note that cr --- 0

exactly for any conformal operator B; moreover, ~r 2 > 0, but a; 2 < 0. For Lagrange tensors in Riemannian or Lorentzian manifolds, the parameter values t for which det Y ( t ) = 0 are isolated [81].

From Eq. (3.6 (a)), it follows that 0 satisfies the following Riccati equation (which is called the Ray-

chaudhuri equation in relativity theory; see [165, 77]):

O' + aO 2 + {tr/~, + t r (a 2) + tr(w2)} = 0. (3.7)

In particular, if Y is a Lagrange tensor, then, since w -- 0, 8 satisfies

O' + a8 ~- + {t rR v, + tr(a~)} = 0. (3.8)

In global Riemannian geometry, it is often better to use the Riccati equation (or inequality) (3.4) than the Jacobi equation (3.3). In [149], a Riccati equation (or inequality) for the shape operator is employed to the s tudy of local differential geometry of level sets of the local distance function. The volume comparison and extremal theorems (for balls, horospheres, and tubes) are also obtained by using the Riccati equation (see [119, 296, 80]); for the case of semi-Riemannian nmnifolds, see [7]. The Lagrange tensors and the Raychaudhuri equation are employed to the study of hypersurface geometry, geodesic congruences, and the boundary rigidity problem (see [81]).

L e m m a 3.3. Let the Riccati matr ix equation B ' + a B e + R(t) = O, where a = const > 0 and R( t ) is a

symmetr ic continuous (n x n)-matr ix , admits a solution B( t ) defined for - ~ < t < c~ 'with a symmetr ic

initial value B(O). Then 8

l iminf f { t r R ( t ) + tr(~r2)} dt <_ 0 (3.9) s--~+oo J

- - 8

and the equality is attained i f and only i f t rB(t) = trR(t) + tr(a 2) =- 0. Moreover, i f trR(t) _> 0, then

B( t ) = R( t ) -- O.

The application of Lemma 3.3 to manifolds without conjugate points (when a stable Lagrange tensor

exists along every geodesic) is given in [77]. Indeed, inequality (3.9) can be replaced by a more traditional inequality:

l iminf f tr R( t )d t < O. (3.10) s--++z~ J

- - 8

We cannot omit the requirement that B(0) be a symmetric matrix in Lemma 3.3, since, for every n > 1,

there exists a symmetric constant (n • n)-matrix R~ > 0 such that the Riccati equation (3.4) with R( t ) = R~

1729

has a continuous solution B(t) for t E R. For example, if n = 2m, then R~ -- E,, and

B(t) = : ". : whereB2 = 0 1 . (3.11) " ' - 1 0 '

0 .. B2 m blocks

this corresponds to the Hopf fibration p: S 2m+1 -+ CP m with the fiber {$1}, and if n = 2m + 1, then

p ~ = [1, 1 1 ~ , . . . , ~] is a diagonal matrix, B2 is as in (3.11), and

B(t) = : "'" : : 1B where B3 = 0 sin(t) cos(t) . (3.12) 0 . . . ~ ~ 0 ' s in (~ ) - c o s ( ~ ) - s i n ( ~ ) 0 . . . 0 B3

rn blocks

The Jacobi tensor Y(t) along a geodesic 7 that corresponds to the matrix B(t) in (3.12) (by virtue of the

equations Y' = B(t). Y, Y(O) = En) can be induced by an n-parameter family of geodesics. Since this Jacobi

tensor is nondegenerate and geodesics in CP m+l are simple closed curves, this family forms a foliation near 7 (see Theorem 3.2 (b)). The solution (3.12) of the Riccati equation (3.4) allows us to infer that the result

in [50] (Lemma 3.4 and, hence, Theorem 3.2 stating that Ferus' result is true for nonconstant Kmix > 0) is incorrect.

3.3. L-Para l le l Vec to r Fields

De f in i t i on 3.3 [118]. The horizontal connection X7 and the horizontal curvature operator R of an almost- product structure are givenl by

Vvz = P2(Vvz), R(y, z)u = Vp.~ty,.]u - [Vv, ~7=]u, (y, z, u ~ T2). (3.13)

The vertical connection X7 and the vertical curvature operator R can be defined in a similar way.

The operators/~ and/~ satisfy all curvature symmetries and the Bianchi identities.

T h e o r e m 3.3 [118]. The following Gauss, Codazzi, and Ricci equations for two complementaw orthogonal distributions T1 and T2 on T M hold:

(a) (R(y, z),~, ,w) = ([~(v, z)a, w)

-(O~u, o~w) + (O~w, o ~ ) - (O~z, O~w) + (o..y, O~w),

(b) (R(y, z)w, x) = ((V=O)yw - (VyO)~'w,x) + (Ovz - O~y, T~w), (3.14)

(C) (R(y, xt)z, x2) = ((Vx, O)yz + O(OvXl, z), x2) + ((VyT)x,X2 + T(TxiY, x2), z),

where y, z, u, w E T2 and Xl, x2 E T1.

The Gauss and Codazzi equations have dual equations. The Ricci equation and the second Bianchi identity yield a formula for (R(Xl, x2)y, z). All of these equations, taken together, solve the problem of computing the curvature of the distributions T1 and T2 by using the curvature of M as well as the problem of computing the curvature of M through the curvature of the distributions T1 and T2. There are formulas for the covariant derivatives of R corresponding to (3.14) [118], but we omit them.

L e m m a 3.4 [252] The following differential equation holds:

((V,~B2) (x2, y), z) + (B2(x2, B2(x~, y), z) (3.15) +((VyB1)(z , Xl) , x2) + ( B l ( z , Bl(y, Xl)), X2) + (R(y, xl)x2, z) = O.

1730

P r o o f . We have

((VvT)x, x2, z) = ((VvB1)(z, x2, ), xl), T(T~y, x2), z) = (Bl(z, Bl(y, xl)), u)

for the tensors T and B1, and similarly, for the tensors O and B2. Therefore, (3.15) follows from (3.14 (c)). If 7: I --+ M is a unit speed Tl-geodesic, i.e., 7' C T1 and Vr, 7' C T2, then the endomorphism field

B2(t) = B2(7'(t), .) of T2 along 7 satisfies the following Riccati differential equation:

+ + F (t) = 0, (t e / ) , (3.16)

where (F2(t)y, z) = (n(y, r')r', z) + ((VvB1)(z, r '), r') + (Bl(z, BI(y, r ')), r'). In the case of a totally geodesic foliation, Eq. (3.15) is given in [73],

(VxlB2)(x2, y) + B2(x2, B2(xl, y)) + R(y, xl)x2 = 0, (3.17)

and the Riccati equation (3.16) takes the form

B~' + B~ + R ( t ) = 0, (3.18)

where (R(t)y, z) = (R(y, 7')7', z). For the distribution T~ C TM, we define the concept of a T~-parallel vector field similar to that of a basic

vector field along a leaf of a foliation.

D e f i n i t i o n 3.4. A vector field y: 7 -+ T2(7) along a unit speed Tl-geodesic 7 C M is said to be Tl-parallel if the following first-order ODE holds:

P2(Vr, y) = B2(7', y). (3.19)

Let {e~(0)} be an orthonormal basis of T2 at m E 7(0). We continue it along 7 as a solution to the differential equation P2(Vr, e~ ) = 0 and obtain an or thonormal frame field {ei} in T2 along 7. Let {y~} with yi(0) -- ei(0) be Tl-parallel vector fields along 7- This yields the Tl-parallel tensor (endomorphism) field Y: T2(7) --+ T2(7) along 7 given by Yi = Ye+ Clearly, P2Vr, yi = P2(Vr, Y)ei and

B2(Vr, Y ) = B=(7', Y) (Y(0) = Id). (3.20)

A T~-parallel vector field y is L-parallel if the distr ibution T~ is tangent to the foliation {L}. (By analogy we can define a T2-paraUel vector (tensor) field.)

Since the ODE (3~ 19) is homogeneous, it follows that a TL-parallel vector field has no zeros or is identically zero. For a foliation {L} (i.e., for TL = T~), L-parallel fields coincide with the horizontal basic vector fields {y} and are locally defined on the leaves (globally along the leaves with trivial holonomy or for a submersion ~: M --+ B with ker. ~r = T1) and satisfy the equat ion

P2(Vxy) --- B2(z,y), x E TL. (3.21)

For the totally geodesic foliation {L} (i.e., T L = T~), the concept of an L-parallel Jacobi field (i.e., induced by a foliation) is introduced in [73]. Let {L} be a total ly geodesic foliation, and let 7 be a geodesic of M tangent to L. Then the foliation generates a (dim M - d im L)-dimensional linear subspace Jr(L) in the (2 dim M)- dimensional space of all Jacobi vector fields of M along 7- This subspace can be parametrized by elements of T.y(0)L • and the initial values of these Jacobi vector fields are

y(O) E Tr(o)L • Vr,(0)y = B2(7'(0), y(0)). (3.22)

Therefore, J~(L) coincides with the space of L-parallel vector fields along 7. The Jacobi vector fields y from Jr(L) are generated by one-parameter families of geodesics of M each of

which is a geodesic in a leaf L. Note that a nontrivial Jacobi vector field y from Jr (L) has no zeros, since the

ODE is homogeneous (see (3.21)):

v ,v(t) = B,(7',

1731

This fact implies numerous global results about totally geodesic foliations (for instance, Corollary 3.1 !). In particular, many authors prove the completeness of leaves of certain foliations on open submanifolds of complete manifolds in the case where the foliation is given by integral manifolds of the null spaces of certain tensor fields or differential forms (see Sec. 2.4.1).

For an u,nbilic or a spherical foliation, the variational interpretation of L-parallel vector fields is given in [223] and [135]. The case of an arbitrary foliation {L} is considered in [303], where certain ergodic properties of a geodesic flow of foliation are studied. L-Jacobi fields occur as variation fields of a varying leaf geodesic c among leaf geodesics. The second-order differential operator

J~y = - y " - R(y, c')c' + (fYyhl)(c', c') + 2h~(P~y', c')

is similar to the operator - P 2 V r (P.,_Ur - A d y (see (3.24) below) acting on the space of vector fields along c. Of course, the L-Jacobi operator J~ depends on the curvature of M as well as on the second fundamental form of {L}. The operator J~ plays the role of the second variation formula for the arc length s and the energy C of the leaf curves, but leaf geodesics appear as critical geodesics for s and E only for certain so-called admissible variations [309].

For the structural tensor Be(t) and the T~-parallel vector field y(t) (the tensor Y(t)) along a Tr-geodesic, we have Eqs. (3.19) and (3.16) (see Lemma 3.4). The "missing" Jacobi-type equation for these tensors can be obtained from the following lemma.

L e m m a 3.5 [251]. Let M be a Riemannian manifold with two complementary orthogonal distributions T~ and T., and let y: 7 -+ T2(7) (y(0) # 0) be a T~-parallel vector field along a T~-geodesic 7 C M. Then

P2V.y,(P.2V~,y) + A~,y = O, (3.23)

where the tensor A~,: T2(7) --+ T2(7) is given by the formula

(A~,y, z) = (R(y, 7')7', z) + ((~YyB1)(z, 7'), 7') + (B, (z, BI (y, 7'), 7'). (3.24)

For a geodesic distribution T~ (in pm'ticular, for the case of a Riemannian foliation {L} with T L = T2, where the geodesic 3' is orthogonal to the leaves of the foliatiou), we have

(A~,y, z) = (R(y, 7')7', z) + (Al(z, d , ( y , 7'), ~/) (3.25)

and the tensor A~, is symmetric. For a totally geodesic foliation {L} with T L = T~, A~, coincides with the Jacobi operator restricted to T2.

For the geodesic distribution T1 and any geodesic 7 C M, (7' C Tl), all Tl-parallel vector fields are Jacobi vector fields. Moreover, for the Riemannian foliation {L} (with T L = T~), all T2-parallel vector fields can be induced by variations of T2-geodesics [287] and [303].

T h e o r e m 3.4 (see [287]). Let 7(t) C M be a geodesic tangent to a geodesic distribution T1, and let the orthogonal distribution T2 be integrable. Then the following conclusions hold:

(a) an ordinary Jacobi vector field y along 7 is orthogonal to TI i f and only i f it satisfies the following initial conditions at m --= 7(0):

y(O) = Yo 6 Te(m), V~,(o)y = B2(?'(0), Y0) + B, (Y0, 7'(0)); (3.26)

(b) a T.,.-paraIlel vector field y along ? is an ordinary Jacobi vector field orthogonal to T1 if and only if it satisfies the following initial conditions at m = 7(0):

y(0) = Yo 6 T2(~.~), P2V~,(0)y = B2(3/(0), Y0). (3.27)

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3.4. F o l i a t i o n s b y G e o d e s i c s w i t h Kmix > 0

From the properties of the Riccati equation (see Lemma 3.3), it follows that a totally geodesic foliation with complete leaves satisfying Km~• > 0 has the nowhere integrable orthogonal distribution. Moreover, a

totally geodesic foliation with complete leaves and integrable orthogonal distribution T L l satisfying Rict _> 0 (i.e., the mixed Ricci curvature is nonnegative) splits locally ([i, 46] for h'~x > 0).

The known rigidity and splitting resu]ts for foliations of nonnegative curvature in mixed directions are not as systematic as the facts in the theory of Riemannian manifolds of nonnegative curvature; results for foliations usually contain strong additional assumptions, for example, the normal distribution is integrable or totally geodesic, the (co)dimension is equal to one, the curvature is constant, or the results are for a specific

class of foliations such as (UF, UF) or (GF, F).

The proof of Theorem 3.1 with the best estimate dim L < p (codim L) for the dimension of a totally geodesic foliation with/(,nix > 0 is based on the following scheme, which was used later by other geometers such as K. Abe, M. Magid, S. Tanno, P. Dombrowski, A. Borisenko, and the author.

3.4.1. Ferus scheme. 1. O'Neill and Stiel [206] have studied submanifolds with a positive index of

relative nullity in space forms ~l(k) and have used the following property of the matrix Riccati equation with Kmix = k = COIISt:

B' + B 2 + k E = 0. (3.28)

This is true for the structural tensor B( t ) = B('y'(t), .) of a relative nullity foliation (L} along a geodesic

7 C L (see Sec. 2.4.1): the eigenvectors of the solution B(t) do not depend on t; every eigen.function A(t)

satisfies the scalar Riccati equation A(t)' + A 2 (t) + k = O, and hence, for k > O, a solution cannot be ear, ended to the whole real line - o c < t < oc.

2. The key fact for the existence of an eigenvector for the structural tensor B: T L x T L • --+ T L • of a totally geodesic foliation (L} (see Lemma 3.6 below) is the following: in the case dim L > codim L, for any

7)oint m E L, there exist nonzero vectors x E TmL and y E T,~L • and also a real number A with the property B(x, y) = Ay.

The first property (see Eq_ 3.28)) is not true if kE is replaced by a nonconstant symmetric matrix

R(t) > 0. Moreover, the inequality dim L < codlin L without additional conditions is not true for geodesic

foliations with /(mix > 0 (see Sec_ 3.4.2).

As a rule, a linear operator defined on an even-dimensional vector space R ~ has no eigenvectors. The situation changes drastically when we consider a u-parameter faanily of linear operators, and it becomes even more complicated when linear operators commute with an additional complex structure.

L e m m a 3.6 [86]. Let V1 and V2 be real vector spaces of dimension dim Vl = u and dim 1/:2 - n, respectively,

and let B : VI x V2 -+ V2 be a bilinear operator. If, for any nonzero vector x E V1, the endomorphism B(x, .): V2 --+ ~ has no real eigenvalues, then v < p(n).

Proof . We can assume that V1 and V2 are Euclidean spaces. Let {ei} C V1 be an orthonormal basis. Then

the z~ continuous vector fields w~(y) = B(e~, y) - (B(e~, y), y)y, 1 < i < z~, are tangent to the unit sphere

S ~-1 C I/2. If these vector fields are linearly dependent at some point y E S '~-1, i.e., ~ i Aiwi(y) = 0, then

B ( ~ Aiei, y) = [ ~ A~(B(e~, y), y)]y; this is impossible by the hypothesis of the lemma. Therefore, u < p(n).

L e m m a 3.7 [1]. Let J be a complex structure on real even-dimensional vector spaces Vt and V2. Let B :

V1 x V2 -+ V2 be a bilinear operator with the property

S ( J x , y) = B(z , Jy) = J B ( x , y). (3.29)

TT~en, for some nonzero vector x E Vi, the endomorphism B(z , .) : V2 -+ V2 has an eigenvector.

Lemmas 3.6 and 3.7 are used as estimates of the dimension of a foliation (or distribution) with A%i~ =

coast > 0 (B is a part of the structural tensor (2.5)), and also as estimates of the index of relative nullity of

submanifolds in S N and C P N. We complete these results with the following lemma.

1733

L e m m a 3.10. entries

L e m m a 3.8 [242]. Let J be a complex structure on real vector spaces Vt and V2 of even dimension dim V1 = u and dim V2 = n, respectively. Let B: V1 x 1/'2 -+ V2 be a bilinear operator with the property

S ( J x , y) = J S ( x , y). (3.30)

If, for any nonzero vector x E V1, the endomorphism B(x,-): V2 --+ V2 has no real eigenvalues, then r, = 2 and n is divisible by 4.

Lemma 3.8 allows us to prove, for example, tha t (1) a complete K~lflerian manifold M en with k-null index (k > 0)

2, n is odd, #k(M) > 0, n is even,

is isometric to C P n, (2) there are no (Riemannian)submers ions from C P 7 and Hp3; this is assumed in [83].

3.4.2. Fo l i a t ions on c lo sed geodes ics w i t h Kmix =~ 1. The Hopf fibrations serve as an example of foliations of odd-dimensional Riemannian manifolds by closed geodesics {L = S 1} with Kmix = const > 0. In Sec. 3.1., the problem on the existence of an even-dimensional Riemannian manifold foliated by closed geodesics with (5 ~ 1)-pinched positive Kmi x is fornmlated.

T h e o r e m 3.5 [253] For any 5 E (0, 1), there exists a Rieman~tian manifold M 2'~+2, where n > vq with

a fibration by closed geodesics {L V- $1} and with a positive 5-pinched mixed sectional curvature.

We assume that Theorem 3.5 holds as 5 --+ 1 when the dimension n is fixed. In proving Theorem 3.5, on M 2~+2 we construct a metric with mixed curvature '~ 2 (h--+i) --~ Kmix ~ 1 and with the length of geodesics

l(L) = 2~-(n + 1) tending to: infinity as 5 --+ 1. Can one obtain, for any ~ E (0, 1), an example with the length l(L) < c (for instance, with c = 27r) under the s tandard asstunption tha t ~ ~ Kmi~ _< 1?

Theorem 3.5 directly follows from Lemmas 3.9 and 3.10.

L e m m a 3.9. Assume that a symmetric matrix R( t ) and a nondegenerate matrix Y( t ) are of order n x n and are T-periodic and satisfy the Jacobi equation

Y(t) + R ( t ) . Y ( t ) = 0, (0 < t < T).

Then there exists a Riemannian metric on the product M n+l = S i x B'~(r) of the circle S 1 by the n-dimensional bail B~(r) of radius r with the following properties:

(a) the closed curves {~,..(t) = (t, Y(t)z)}._eB~(~) are geodesics;

(b) the components of the mixed curvature R(. , "7o)7o along %( t ) are expressed by the formula R(z, ~/o)7o = R(t)z , z

(c) the Jacobi tensor of the foliation {%} has the form Y( t ) for some parallel orthonormal basis along %.

For any n E N, there exists a square matrix Zn,n+1 (t, s) (see Table 2) of order 2n + 1 with

ajk cOS(n)+ bj} s i n ( ) , j _< n, zjk

. S

ajk cos (~ - -~ ) + bjk s m ( ~ - ~ ) , n + 1 ~ j < 2n + 1,

satisfying the condition det Zn,n+ l ( t, s) = cos(t - s ) . In particular, the determinant of the matrix Y2n+ l ( t ) :=

Zn,n+l (t, t) is identically equal to 1.

In fact, the matr ix Z1,2(t, s) with 5 = 0.25 is given in (3.12). Consider some more examples, where,

for simplicity, we set Cn := cos ( t ) and Sn := sin ( t ) . The matrix Ys(t) :--- Z2,3(t, t) satisfies the condition

1734

4 6---- ~ = 0.44 and

Tab le 2. The structure of the matrix Z~,,+I (t, s)

Cn

Z2,1 . . .

Zn, 1

[ o r o ~ d ~, f:r pve, l l rt 0 c~ 0

A1

0 Zn+l'n

Z2n+ l,n

--Sn

0

A2

0 Z2,1

A1 Zn, 1

0 Zn+2'n+l...

Z2n+ 1 ,n+ 1

--Sn

0 0

A2 �84

0 83 -C3 0 0 ) S 3 C 3 0 3C3 0

Y ~ ( t ) = C3 3S3 o S~ o �9 0 o - 2 S 2 C2 S2

o o 0 S2 C2

The matr ix YT(t) := Z3,4(t, t) satisfies the condition 6 = ~ ~ 0.56 and

z - ( t ) =

C 3 o o - S 3 o o o

$3 C3 383 0 0 0 o 3C3 $3 C3 0 0 0 0

0 0 0 C4 + $4 -224 0 0

0 0 0 C4 C 4 - $ 4 0 2 C 4

0 0 0 0 0 C 4 § 2 S 4

0 0 C 4 - S4 O, 0 C4 C 4 - S4

16 25 r,J We do not give here the matrices Yg(t) := Z4,~(t, t) with 6 = ~ ~ 0.64, YH(t) :---- Z~,6(t, t) with 6 ---- = 0.7,

etc. to save room [253]. Neither do we give here the matrices B2~+l(t) = Y2n+l(t)'' ( ~ + l ( t ) ) -~, where n = 2,3,4 . . . .

3.5. Sp l i t t i ng Fol ia t ions w i t h N o n n e g a t i v e M i x e d C u r v a t u r e

3.5.1. T h e a r e a of an L-para l l e l v e c t o r field. The solution Y(t) C ~g of the Jacobi matrix equation y"+

R(t)y -- 0 with constant curvature matrix R(t) -- k E > 0 can be written in the form .y(t) = y(0) cos(v~t) +

:/(o) sin(x/~t). If the initial values y(0) and y'(0) are linearly independent, then the curve y(t) is an ellipse in v~ the 2-plane y(0) A y'(O) and the area of the parallelogram y(t) A y'(t) is constant. We introduce and study a similar function (called the area of an L-parallel vector field) along the geodesics of a Riemannian manifold with a foliation (or distribution).

The following comparison lemma is used in Theorem 3.7 below for estimation of the length of L-parallel vector fields in terms of mixed-sectional-curvature pinching.

L e m m a 3.11 [253]. Let a solution y(t) C R ~ to the Jacobi ODE

y"+ R(t)y = 0, 0 < t < ~ /v~ , (3.31)

y'(o) be written in the form y(t) = Y( t ) + u(t) C ~t n, where Y(t) = y(0) cos (v/-kt) + ~ sin (v/kt). / f the norm

l lR(t) - h e l l <_ ~ < ~, then

lu(t)l < ~ f v/k~lY(s)l sin(x/k(t - s)) ds . (3.32) - k - ( i - o

D e f i n i t i o n 3.5. For a Riemannian manifold with complementary orthogonal distributions T1 and T2, the turbulence along a Tl-geodesic 7 (a rotation component of the tensor B2) is defined by

a(T~,~/)--sup{(B2(x,y) ,z) : x e TI(~/), y, z-LTI(7), y_l_z, Ix] = lY]---Izl = 1}.

1735

[~ (t) l =kx

x

y (t o ) =O

Y

lY(OI

rain < /g-ff

t �9 7

01 ~ ~ t l I

Fig. 16. The behavior of the L-parallel field y(t) along the "extremal" geodesic.

For a foliation {L}, the turbulence along a leaf Lo is defined by

a(Lo) =sup{ (B2(x ,y ) , z ) : x E TLo, y, z E T L ~ , y_Lz, Ix[ = lY[ = [z[ = 1}

(see [205] and [147] for Riemannian submersions or foliations).

It follows from the relation a(L) = 0 for all leaves tha t the orthogonal distr ibution T L • is tangent to a totally umbilic foliation.

T h e o r e m 3.6 (local). Let {L ~} be a totally geodesic foliation on a Riemannian manifold M n+~, and let there

exist a point m E M such that along any leaf geodesic 7: [0, ~-/v/k] --+ M, 7(0) = m, we have

k2 >_ K(7', y) > kl > O, y E TL• (3.33)

and one of the following inequalities holds:

(1 - kl/k2), max{a(TL, ~/)2/k, 1} < 0.337, (3.34a)

{2(IVB+[ + [B~12(a(T L, 7) + [B2+[)) + (k2 - k~)[B+]}

x max{a(TL,7)2/k , 1} 2 < 0.0082k.~ (u~ith ki >_ 0.583k2), (3.34b)

where k = (kl + k2)/2 and B+y is a component of B+y that is orthogonal to the vector y. Then v < p(n), and i f T L is invariant with respect to the K6hlerian structure on Af, then z, : 2 and n is divisible by 4.

Theorem 3.6 is not true without conditions (3.34), but their coefficients 0.337 and 0.0082, respectively, can be obtained by the method for proving the theorem and can presumably be increased. Coudition (3.34 a) holds if kl = k2, and condition (3.34b) holds if the foliation is conformal. The assumption in Theorem 3.6 that the distribution T L is geodesic and integrable is made to simplify the formulation.

T h e i d e a o f t h e p r o o f of T h e o r e m 3.6. S t e p 1. Let us assume the contrary, namely, that there exist unit vectors x0 E Tl(m), Y0 E T2(m) and a real A < 0 with the property B(x0, Yo) = Ay0 (see Lemmas 3.6

and 3.8). The "extremal" leaf geodesic 7: [0, ~-/v/k] --+ M (O"(0) = x0) and the L-parallel Jacobi field y(t) along 7 with the initial value y(0) = Y0 play a crucial role. We decompose this vector field into a s tandard

and a "small" term y(t) = (cos(v/kt) + :~k sin(v/kt)) yo + u(t), where u(0) = u'(0) = 0. (For k2 = k,, we

have u(t) -- 0, and hence, the L-parallel Jacobi vector field y(t) vanishes at the point to = cot -1 ( - ~ ) / v ~ .

This contradiction completes the proof of Theorem 3.1 by Ferus. For conformal foliation, i.e., a foliation with B + = 0 and K~x > 0, see [235].) In the general case k2 > k~ with assumption (3.34) (actually, for 6 _> 0.5821), we prove (using Lemma 3.11) that the function ]y(t)l , "the length of the L-parallel vector field

y(t)," has a local minimum at tl e (0, ~r/v~] (see Fig. 16). S t e p 2. We note that the function V(t), "the area of a parallelogram on the vectors y(t), y'(t)," changes

"slowly" along the geodesic % that is

1 k ~ ~ (a) IY(t)'l _< -kdly(t)P, _ _ (3.35) (b) I(V(t)2)'l __ [2(I(B+)'I + IB+I2(a(TL, + I B r IB+l(k2 - k l ) ]y ( t ) 4

1736

(this flmction is constant for k2 = kL or for a conformal foliation). We obtain a contradiction here, since the

function V(t) cannot increase from the zero value V(0) to the "large" value V(t~) on the given interval of

length <_ ~r/v~.

In [253l, Theorem 3.6 is proved by a different method.

L e m m a 3.12. Let M be a Riemannian manifold ~ith a compact totally geodesic foliation {L} and Km'~ ~_ O.

Then M splits along L and L •

The condition of Lemma 3.12 imposed on Kmix of M can be replaced by a weaker condition of nonnega- tivity of the mixed Ricci curvature.

Theorem 3.6 and Lemma 3.12 imply the following theorem.

T h e o r e m 3.7 (decomposition). Assume that M '~+~ is a Riemannian manifold with a compact totally geodesic foliation {L ~} of curvature kz >_/X~mix ~ kl ~ 0 and one of the following inequalities hold:

(k2 - k~)-max{a(L0) 2, k} < 0.337 k2 k, (3.36)

{ (IVB I + IB+I2(a(Lo) + [B+I) ) + (k., -

x max{a(L0) 4, k 2} _~ 0.0082 k 2 k 3, (with ki >_ 0.583k2), (3.37)

where k = (kl + k2)/2, Lo is a leaf, and B+y is a component of B+y orthogonal to the vector y. I f u > p(n) (u > 2 for n divisible by 4 in the case of Kdhlerian M and {L}), then kl = k2 = 0 and "At splits along

L and L •

The assumption in Theorem 3.7 that the leaves are totally geodesic is given for simplicity. In Chap. 4, Theorem 3.7 is used to obtain results concerning the metric decomposition of ruled and parabolic submanifolds.

3.5.2. T h e v o l u m e o f a n L-paral lel v e c t o r field. Here we develop the method used in Sec. 3.5.1. Assume that x E TmL is a unit vector, u E [2, dim L] is an integer, 3' C L, 7'(0) = x, is a unit speed

geodesic, and Y: L -+ T L • is an L-parallel (Jacobi) vector field in a neighborhood of 7- We define the following quantities:

a-r(Y) = sup{ lV, ,Y[ / lYI : v. E T-rL, lul = 1}, (3.38)

t (Y) = sup{lR(u,Y)vl/JY[: u • = Ivl = 1}, (3 .39)

where ~ is a component of the given vector that is orthogonal to Y. Obviously, av(Y) <_ a(L).

Defini t ion 3.6. For any V-parallel u-dimensional subbundle Z of TvL that contains 7', the nonnegative

flmction V~+I(Y) is defined as a (u + 1)-dimensional volume of the parallelepiped Y A Very A .. . A Vr in

TzL • where {er is an orthonormal basis in Z. (The values V~+I (Y) do not depend on the choice of the

basis in Z. For brevity, we will call V~+~(Y) a (u + 1)-dimensional volume of Y.)

Lem_ma 3.13 [251]. / f the mixed curvature k2 >__ K(u, Y) >_ kt, u E Z, then the derivative of the (u + 1)- dimensional volume satisfies the inequality

1 (3.40)

Note that V~+I (Y) = const if M is a real space form. tn the proof of Theorem 3.6, we considered a

special case of (3.40) for the function �89 which is the area of the parallelogram. Y A Y ' in T~L • (i.e., for

u = 1), and obtained inequality (3.35). Lemma 3.13 is used in Theorem 3.8 (and Theorems 4.24-4.25) in the case /2 = 2.

1737

For a foliated Riemannian manifold M, {L} we denote

5R = sup{(R(x, y)u, z) : x, u E TL , y, z E T L • x • u, y • z , Ixl = lyl = lul = Izl = 1 } .

(3.41)

T h e o r e m 3.8. Let M ~+~ be a Riemannian manifold with a compact totally geodesic foliation {L v} satisfying k2 >_ Kmix >_ kl >_ O, and let one of the following conditions hold:

(k2 - kl + 25R) dL exp(31B+l dn) a(n) < 2kl, (3.42)

I k (3.43) (k2 - kl + 25R) a(L) max{a(L) 2, k} 3/2 _< 0.004 k2k2, with kl >_ 5 2,

where dL is the maximal diameter of the leaves. I f ~ >_ p(n), then kl = ke = 0 and M splits along L and L •

Conditions (3.42) and (3.43) hold automatical ly in the case 5R = 0 (say, for a conformally flat metric) and k, = k2. In contrast to (3.36) and (3.37), condit ion (3.42) does not contain an a priori lower bound for the mixed-curvature pinching.

P r o o f . S t e p 1. By Lemma 1.2, there exists a leaf L0 with a trivial holonomy. All L-parallel fields are globally defined on L0.

Let (3.42) hold. By Lemma 3.6, for any point m E L0, there exist unit vectors Xo E TmLo and Yo E TmLo ~

and a real immber A _< 0 with the property B(xo, Y0) = Ay0. Let Y: Lo --+ TLo ~, (Y(m) = yo) be an L-parallel (globally defined) vector field, m' E Lo be a m i n i m u m of the length function IYI: Lo --+ R+ (m' exists o,1 a compact L0), and let "),: [0, l] -+ L0 (~/(0) = m, q,(l) = m', l < diam L) be the minimal geodesic between the points m and m'. Let x~(t) = "y'(t) be the velocity vector field a~ld x2(t) e T~(t)L be a unit vector field that is parallel along "), and has the properties

x= (0) _L x1(0) and the vectors (x2(0), xl(0), x0} are linearly dependent. (3.44)

Let y(t) = Y(~/(t)) be the restriction of Y to % and let y~(t) = V~(t)Y.

S t e p 2. Let V3(t), 0 < t < l, be the volume of the parallelepiped y(t) Aye(t) Aye(t) in T~L • By Lemma 3.13 (for ~, = 2) and the inequality R~(Y) < 6R, we obtain

Iy (t)'l _< ((k= - k )/2 + a ( n ) l y ( t ) l 3. (3 .45)

Let {0~, 02.} be an or thononnal basis of the plane xz(l)Ax2(l) with the property Vo~Y_LV~Y. By vir tue of the

well-known formula for the second derivative of the length of a Jacobi vector field, we have I Vo, Y I -> x/~lY (1)l, and, therefore,

V~(l) = ly(l)l- IVo, r l - IVo~YI > k, ly(1)l ~. (3.46)

(Since "~(1) is a point of min imum of the function IYI, the vector y~ = "Coy is orthogonal to the vector y(1).) Therefore, using the condition V3(0) = 0, we obta in

l l

k~ly(l)l ~ < _< f [�89 <_ a(L)((ke - k,)12 + 5R) -- f ly(t)13dt. (3.47) o o

S t e p 3. The relation (y(t)2) ' = 2(y(t), y1(t)) = 2(B+(xl(t) , y(t)), y(t)) implies I(y(t)2)'l <_ 21B+l.ly(t)[ 2 , and, therefore,

ly(t)l _< [y(l)l exp([B+]l). (3.48)

From (3.48) and (3.47), we obtain an inequality t ha t is opposite to (3.42). S t e p 4. The extremal situation above implies kl = k2 = 0 or a(L) = O. In the latter case, the structural

tensor B is conformal, in particular, the dis t r ibut ion T L • is integrable. From the inequality Kmix __> 0, by using Lemma 3.3, we obtain the relation B = 0, and again, k~ -- k2 = 0.

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Step 5. Since {L} is a compact totally geodesic foliation with the condition /(mix = 0, Lemma 3.12 completes the proof.

3.5.3. Fol iat ions with/(mix > 0 and with condit ions on L-Jacobi fields.

T h e o r e m 3.9. Let {L} be a foliation into closed geodesics on a Riemannian manifold M n+l with k2 >_ /(mix ~ ~1 > 0, and let the following conditions hold:

(1) k l /k 2 > A where A = max{a(L) / v~ , 1} and k = (kl + k2)/2; - - A + 1 . 5 1 ~

(2) the sets of local minima of the functions {y2} (where {y} are L-parallel Jacobi fields along the leaves) are connected;

(3) the 1-dimensional subspaces in T L • are invariant under the complete parallel displacement along every leaf L;

(4) the length of every leaf L does not exceed ~r/v/k. Then the dimension n is even.

Theorem 3.9 can be generalized to compact foliations of dimension ~ > 1. Conditions (1)-(4) of Theorem

3.9 hold for geodesics and Jacobi fields in the projective space ]~(k). In the case kl = k2, the curvature tensor

has a standard form and every function y2(t) (where y(t) is a Jacobi vector field) can be written as

al cos(2v~t) + a2 sin(2v/k~t) + a0 (0 < t < ~r/v/k ~)

and hence, (2) holds. If kl/k~ is close to 1, then the "implicit" property (2) can be expressed in terms of the curvature tensor, but, in this paper, we do not calculate this expression. In view of A > 1, condition (1) implies kl/k2 >_ 0.399; the constant 0.399 is obtained by the method used for proving Theorem 3.9 and can be improved.

Idea o f t h e proof. Let m E L, and let S "-~ C TmL -L be the unit sphere centered at the point m E L. Assume that the length of every nonzero L-Jacobi field y(t) along L has a unique point of global minimum. Since the mixed curvature is positive, the derivative y'(t) at a point of minimum t is a nonzero vector that is

orthogonat to y(t). We transport all these vectors y'(t) along L to tim initial point m and obtain a continuous

line field on the sphere S ~-l. Hence n is even. (If the length of some nonzero L-Jacobi field y(t) along L h ~ more than one point of global minimum, then, by using conditions (1) and (4), we can change this line field

in a neighborhood of its singularity locus in order to obtain a continuous line field on the whole sphere S n-~.)

3.6. Integral Formulas for M i x e d Curvature

3.6.1. Integral formulas for mixed c u r v a t u r e . The first integral formula for the curvature of a foliated manifold was obtained by G. Reeb.

T h e o r e m 3.10. Let H be the mean-curvature function of a transversally orientable foliation of codimension 1 on a closed oriented Riemannian manifold M. Then fM H d v o l -- 0, where dvol is the volume forum of M.

If a foliation has a basic projectable mean-curvature function, then, according to Theorem 3.10, the mean- curvature is zero along one of its leaves, and this leaf is minimal [299]. The following problem is studied in [209, 210] and [301]: what functions on M can serve as the mean curvature functions of a given codimension-1 foliation?

For a given Riemannian manifold M with a codimension-1 foliation, l e t / ( ( m ) be the determinant of the second fundamental form of the leaf through the point m E M. A foliation minimizing the total curvature fM ]/(I is said to be tight. The integral formulas of these foliations are studied with the intensive use of integral geometry in [166] and [168]. For example, the following problem is open: prove (directly !) that S '~ does not admit a tight foliation; for n -- 3 [210], the corresponding result was obtained by using the fact of existence of the Reeb component.

Let {x~} C T~ and {yj} C T2 be local orthonormal bases. The mixed scalar curvature is defined by

= yj) . (3 .49) i , j

1739

For the distributions on a space form M(k), we have s,ux = k(dimT1)(dimT2). Note that

E, ,~(B~(x~ ,B2(x i , y j ) ) , y ~ ) = IIB2+II " - IIB~II ~ = IIh2Jl ~ -IlAull2:12 (3.50) ~i,j(B,(yj, BI(yj, xi)), xi) -- ][BI+][ 2 - JlBI-[] 2 -- [[h,JJ 2 - []A~]

The contraction of Eq. (3.15)

( ( % , S 2 ) ( x i , yj), y~) + (S . . (x . B2(xi, yj), y~)

+((Vy~B1)(yi, xi), xi) + (B,(yj, B~(yj, x,)), x~) + (n(yj, x,)x,, yj) = 0 (3.51)

with respect to i and j, by virtue of the formulas

div,H2 = divH2 + ]H212, div2g~ = divH~ + IH~I 2, (3.52)

yields the equation

div (H1 -]-H2) = Sm~ + [[h,[[2 + [[h'2[[ ~ -[[A~[[ 2 -[[A~}[ 2 - [//1[ 2 -[H._,[ '~. (3.53)

For similar formulas with more than two orthogonal distributions, see [14]. In certain special cases of a Riemannian almost-product manifold, smi~ in foiTnula (3.53) has a simpler

form, for example,

v2-1 Lr 12 n2--1 H, 2 (UD, UD) : -d iv(H~ + H2) + [[A~][2 + [[A2[[2 + % 7 ~ +---~- ~ , (GD, GD) : -div(H~ + ~.~)+ IIA, II~+ IIA~lr-, (GD, MD) : div (HI + H2) + IIA,[I 2 + IIA2]I 2 - Ih.,I e, (3.54) (F, F ) : - d i v ( g t + He) - I Ih l ] t "~ -[Ih21] 2 + [H~I 2 + IH_,I 2, (MF, M F ) : - d i v ( g t + / - / 2 ) - IIh~l[ 2 - IIh.~l[ 2.

Tlm integration of Eq. (3.53) along a compact manifold M leads one (by the Green's theorem) to integral formulas that have many interesting global consequences.

T h e o r e m 3.11. Let Tl and T2 be complementary orthogonal distributions on a closed orientable Riemannian manifold M. Then

f{Sm~• + IIB~+]l 2 * IlB:ll 2 -IIBi-l l ~ - I I B ; [ I 2 - IH~[ ~ - I H ? [ 2} -- o. (3.55) M

Integral fornmlas for foliations that have a form similar to formula (3.55) are introduced in [220]. In [232], Eq. (3.55) is deduced for a Riemmmian almost-product manifold in terms of similarly defined differentials VP, 5P, and dP of the tensor P (see Definition 2.5):

/ {4Smix + IvPI ~ - IdP[ ? - JaPI 2} = o. (3.56) M

In [302], Eq. (3.55) is derived by a direct computation. In [275], Eq. (3.55) has the form

[ { 6 4 5 . ~ x + IlLIf - I lNII ~ - ]HI 2} 0, (3.5~) M

where N, L, and H are given by formulas (2.20), (2.22), and (2.23), respectively, and

1 ~ l lL l l =[Ih~[I ~+[[h2112, (3.58)

~I INII 2 = IIA~II 2 + IIA2I[ 2, ~ I H [ 2 = [H,I 2 + I//212. (3.59) 6,.,,

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This method of integral formulas is applied in [274, 276] to degenerate maps f : M -+ N of nonconstant rank

r ( f ) < d i m M (see Example 2.1). Note that Eq. (3.53) can be compared with the following formula for a Riemannian foliation [189]:

[ht[ 2 = [A212 + (APt, P~) - s.~• (3.60)

where AP1 is the LapIacian of the orthogonai projection PI- See also [95, 195, 319]. Equation (3.55) gives us decomposition criteria for compact manifolds with Riemamlian almost-product

structure under the constraints on the sign of Smix-

C o r o l l a r y 3.2 [274, 276]. (1) Let T~ and T2 be complementary orthogonaI umbilic distributions on a closed oriented Riemannian manifold M with Smix _< 0. Then M splits along T1 and T2.

(2) Let 7"1 and T2 be complementary orthogonal geodesic and umbilic distributions on a closed orientable Riemannian manifold M with Smix >_ O. If both T1 and T2 are integrable, then M splits along T1 and T2.

Some global results are connected with the integration of Eq. (3.53) over a compact leaf L0:

f{SmJx + div.,H~ + [IB~+II ~- + lIB+It ~ -IIBTII 2} = o. (3.61) Lo

C o r o l l a r y 3.3. A compact minimal foliation {L} on a Riemannian manifold M with an integrable orthogonal

distribution and Smix > 0 splits along L and L• if the manifold M is simply connected, then it is the product

M = L x L • (In particular, a minimal foliation {L} on a Riemannian manifold M with the integrable orthogonal distribution and Smix > 0 has no compact leaves [302].)

3.6.2. T h e in tegra l of a m i x e d scalar c u r v a t u r e a long a leaf. Using the method of matrix Riccati ODE's, we can deduce an integral inequality for Smix along a complete leaf; the consequences are rigidity and splitting results of Riemannian manifolds with a foliation (distribution) whose mixed Ricci or scalar curvature is nonnegative and whose norms of integrable tensors are bounded from above.

Let M be a Riemannian manifold with complementary orthogonal distributiolas T1 and T2. Let n = dim T2 and p = diinT~; denote by {yj} a local orthonormal basis of T2 and by {x~} a local orthonormal b~is of 7'1 with x~ = e. Let

IA~l(e) = sup{IAi(e,v)l : Ivl = 1},

JAil(p) = sup{JAil(e) : e E Ti(p), [e] = 1}.

Note that for e E TI(p) we have

r j tB~(yy ,e ) [ 2 = ~j,~(B~(ys, e),x~) '2= E~[A~(e,x~)]"- < ( u - 1)[A,]2(e), (3 .62 ) Ej [B;(e, yj)[2 = E~,j IA2(y~, ydl <__ n(n - 1)lA212(p).

The mixed Ricci curvatures are defined as follows:

mc~(x) = ~ K(x, ys), Ric~_(y) = ~ t, '(x,,y). j i

T h e o r e m 3.12 (rigidity). Let M be a complete Riemannian manifold equipped with two complementary orthogonal distributions T1 and T~, (dimTt = u and dimT2 = n), and let Tl be geodesic. Then

(a) for every Tl-geodesic "~(t) C M, we have

l iminf f {Ric , (7 ' ) - ( u - 1)1A 12( )- n ( n - 1)[A212(7)}dt < 0, s--~+oo J

- - 8

(3.63)

and equality holds for every "7 if and only if T2 is a geodesic distribution and the skew-symmetric operators B2 and B1 have "constant torsion" in the following sense:

B=(x, B2(x, y)) = -[A.,12(p)x2y, BI(y, BI(y, x)) = -l&12(p)y2x; (3.64)

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(b) i f the equality holds in (3.63) for every % and in addition, v >_ p(n) ( if T~ is invariant with respect to the K~hlerian structure on M, then, for v > 2, we assume that n is divisible by 4), then T2 is also integrable, and, hence, is tangent to a Riemannian totally geodesic foliation; moreover, /Xmi x > 0.

In [273] the hyperdistribution T1 such that property (3.64) is satisfied by BI is referred to as having constant torsion.

Proceeding from [125], where manifolds without conjugate points are studied, we obtain an integral formula for s~ix along a complete leaf.

Theo rem 3.13 (rigidity and splitting). Let {LV} be a foliation on a Riemannian manifold M n+" with complete minimal leaves. Assume that the quadratic f o rm

Rict (x) + trace{B2(x, B; (x , .)) - h i (x , B+( ", x) ) - (V h,)(x, x)}

on a unit tangent bundle SLo of every leaf Lo has an integrable positive or negative part. Then

/ {Snlix -- n(n -- 1)In2] 2} dvol ~_~ 0, (3.65) Lo

and equality holds for every leaf` i f and only if {L} is a Riemannian totally geodesic foliation. Moreover, if equality holds in (3.65) for all leaves and v > p(n) ( i f T L is invariant with respect to the Kiihlerian structure

on M, then for v > 2, we assume that n is divisible by 4), then M splits along L and L •

In the following corollary that generalizes Theorem 3.19 in [46], the distribution T L • is integrable and the foliation splits without any additional asstunption on the dimension. Moreover, the integrability of T L • can be assumed only along some submanifold that is transversal to TL. For a compact leaf L0, see Corollary 3.3.

Corollary 3.4 (splitting). Let M be a foliated Riemannian manifold with complete minimal leaves {L} and

integrable orthogonal distribution T L • Assume that the:function Ric~ ( x ) - t r { h~ (x, B+ ( ., x) )+ tr( V.h~)(x, x)} on S Lo has an mtegrable positive or negative part along ever~j leaf Lo. Then

f Smix dvol __ 0 (3.66) L0

and equality holds (for every leaf) i f and only if M splits along L and L •

From Theorem 3.12 and independently of Theorem 3.13, we obtain the following corollaries.

Corollary 3.5 (splitting) [243]. Let M be a foliated Riemannian manifold with complete totally geodesic leaves {L}, and let

Pdct(x) _> n ( n - 1)[A212(p), x E TvM.

If v >_ p(n) (if`TL is invariant with respect to the Kiihlerian structure on M, then, for v > 2, we assume that

n is diuisible by 4), then M splits along L and L •

Corollary 3.6 (rigidity) [147]. Let {L} be a Riemannian foliation on a complete Riemannian manifold M. If. for every geodesic "/ orthogonal to the leaves, we have

Ric2(7') _> ( n - 1)]A212(7'),

then {L} is a totally geodesic Riemannian rogation.

The inequality in Corollary 3.6 is actually sharp: all assumptions are met; for instance, see [129] for a foliation {L} with ( T L = T2) of codimension-1 with Rio1 _> 0.

3.6.3. Foliations wi th an a lmost-geodesic o r thogona l distr ibution. UsirLg the method of matrix Riccati ODE's, we obtain rigidity result for foliations of "large" dimension whose mixed sectional curvature is nonnegative and whose norms of h2 are bounded from above.

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Let {L} be a Riemannian foliation with totally geodesic leaves. Then BL = B + = 0, and it follows from (3.15) that

{V~B~ = 0 (B f ( x , B 2 ( x , y)) = (R(x, y)y, x). (3.67)

Hence Kmix :> 0. If we assume that /(mix > 0, then we find from (3.67b) that the skew-symmetric operator

B2 is nondegenerate, and therefore, has 11o eigenvectors, i.e., in this case, the Ferns inequality ~ < p(n) holds. This local property of a Riemannian foliation (and its modification for a conformal foliation with complete totally geodesic leaves) plays a crucial role in this section.

Let [hil = sup{Ih~(v,w)] Ivl = Iwl = 1, v, w E Ti}.

T h e o r e m 3.14 (rigidity). Let M be a complete Riemannian manifold with a geodesic distribution T1 of dimension dim T1 : ~,. Assume that its orthogonal distribution T2 is of dimension dim T2 : n, and let the following inequality hold:

g,~x > Ih212 + Idll 2. (3.68)

I f ~ > p(n) (if T1 is invariant with respect to the Kiihlerian structure on M, then, for ~ > 2, we assume that n is divisible by 4), then the distribution T2 is geodesic.

Coro l l a ry 3.7. Let M be a complete Riemannian manifold with a geodesic distribution T1 of dimension dim Tt = ~. Assume that its orthogonal distribution T2 is of dimension dim T2 = n, and let there exist a point p E M such that for every T1-geodesic 7: R -+ M with 7(0) = p, the following inequality holds:

8

liminf f{IC('y',yt) -IB~+(,~', y,)l + > o, (3.69) ,s--++oo j

- 8

where Yt E Tz(7') is parallel along 7. Then u < p(n), and if T~ is invariant with respect to the Kiihlerian strntcture on M, then 1., = 2 and n is divisible by 4.

A special case of Theorem 3.14 (and Corollary 3.7) ~ises when T1 is tangent to a totally geodesic foliation. Corollary 3.7 (and Theorem 3.14) can be formulated in a more general foml for the distribution T2 that is

"close" to umbilic. In this case, we replace the expression B+(7')y in (3.69) by B+(7')y - fl(~/)y, where Z: T1 --+ R is a regular linear functional.

3.7. Fo l ia t ions w i t h P o s i t i v e Part ia l M i x e d C u r v a t u r e

Some authors (in [315, 270, 271], and later in [195, 196]) studied curvature functions on a Rienmnnian manifold that fill a gap between the sectional curvature and the Ricci curvature. For q + 1 orthonormal vectors V = {x0, x t , . . , xq}, the (partial) q-dimensional Ricci curvature of M is, by definition,

q

Rice(V) = K(x0, i = 1

The 1-dimensional Ricci curvature is the same as the sectional curvature, and the (M - 1)-dimensional partial curvature of M is the ordinary Ricci curvature. For a foliated Riema,mian manifold M, {L}, there exist

two mixed q-dimensional Ricci curvatures Ricq(y0, x~, . . . , Xq) when q < dim L and Ricq(x0, y~ , . . . , yq) when

q < codimL, where xi E TL, yi E TL • If Ric q is nonnegative (positive) and q' > q, then Ric q' is also

nonnegative (positive) by virtue of the identity

q' q, k K(x,z~) - - kC~ ~ ~ K(x,z,~), (3.70)

i=l I<_ii <...<ik<_q' j:l

or by induction, by using x--q+* K(x, zi) = * x-.q+1 ~i=t ~ z...i=l ~j#~ K(x , zj). By the Meyer theorem, a complete Riemann-

inn manifold with Ric~ > c > 0 is compact. Moreover, if Ric~ > qc (resp., Ric~ t < qc) for M", then

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.E,~) > 0

Fig. 17. Distance between two submanifolds.

Ri ~ > ( q + l ) c and RiCM > ( n - - 1 ) c (resp., Ric~ 1 < ( q + l ) c and Rioq+l < ( n - 1 ) c ) . The class of " ' ~ M . . . . M - -

Riemannian maififolds with positive curvature Ric q is wider than the class of manifolds with positive sectional

curvature. For example, since M :n = Sn(1) x Sn(1), n > 2, has Ricci curvature Ricu = Ric2~ -1 = n -- 1 and

sectional curvature KM ---- Ric~s E [0, 1], its (n + 1)-dimensional Ricci curvature is positive: R i c ~ t ~ 1.

If the radius of the circle of S 2 is "small," then there exists a large circle of S 2 that is "far" from it. Morvml [193] started from this elementary fact and generalized the result of [93] by giving an upper bound for the distance between two submanifolds of a Riemannian space with positive sectional curvature. The generalization of the result in [193I for the case Ric q > 0 and the version for foliations are given below (see

[255]).

Let ]]hLII be the supremum of the norms of the second flmdamental forms of the leaves, and let diam • L be the maximal distance between the leaves of the foliation {L}.

T h e o r e m 3.15. Let {L ~} be a compact foliation on a Riemannian manifold M ~+~ with Ric~1 _> c > 0 for some q < v. Then

(diam• L)2 <- ~llhml] + (q-"F -t) ~4 if v < n - 1 , �9 if n - l < u < n - l + q , 'if v k n - l + q .

For the Riemannian foliation in Theorem 3.15, the condition on curvature can be weakened up to Ric q > c > 0, since the minimal geodesics between ally two of the leaves is orthogonal to all leaves.

The following corollary for q = 1 is proved in [281] by using the ideas of [93].

C o r o l l a r y 3.8. Let M n+~ be a Riemannian manifold with compact totally geodesic foliation {L~}, and let Ric~(L) be positive along some leaf. Then v < n - 1 + q.

Let us consider classes of submanifolds with additional conditions imposed oil the second fundamental fomn.

D e f i n i t i o n 3.7 [269]. A submanifold M in a Riemannian manifold 2fI is k-saddle if, for every normal vector

E T M • the second quadratic form A~, which is reduced to a diagonal form, has < (k - 1) coefficients of the

same sign. A foliation {L} on a Riemannian manifold M is k-saddle if every leaf is a k-saddle submanifold.

Totally geodesic submanifolds are 1-saddle. A classical saddle surface M 2 in R 3 (for instance, a surface

of negative Gaussian curvature) is a 2-saddle submanifold. In [34], another definition (the value of k differs

by 1) of a k-saddle submanifold is given. Since the maximal linear subspace of the cone

]c r

~-~a~x 2 - ~ a jx2=O, a s > 0 , i=1 j = k + l

is of dimension min{k, r - k}, [34], we have from Theorem 3.15 that the following assertion holds.

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C o r o l l a r y 3.9. Let M n+~ be a R iemann ian manifold with a compact k-saddle foliation {L~}, and let

Ric~(L) > 0 for some q <_ u - k + 1. Then u < q + n + 2k - 3. Moreover, /f Ricq(M) _> c > 0, then

diam • ~ i f L , < _ n + 2 k - 3 ,

- -V/q-~+~ +2k-3 i f n + 2 k - 3 < u < q + n + 2 k - 3 .

Def in i t ion 3.8 ([38] for s = 1). A foliation {L} on a Riemannian manifold M is of nonpositive (resp.,

negative) extrinsic s-dimensional Ricci curvature if the s-dimensional-Ricci curvature of its leaves is not greater (resp., less) than the s-dimensional Ricci curvature of the ambient space along the leaves.

L e m m a 3.14 ([102] for s = 1). A submanifold M n C ~I n+p with nonpositive extrinsic s-dimensional Ricci curvature is k-saddle with k = p + s.

By virtue of Le,nma 3.14 and Corollary 3.9, we have

C o r o l l a r y 3.10. I f {L ~} is a compact foliation on M ~+n of nonpositive extrinsic s-dimensional Ricci curva-

ture and Ric~(L) > O for some q <_ u - n - s + l, then u < 3n + q + 2 s - 3.

Similar results hold for K~hlerian manifolds of positive part ial bisectional curvature (see [151, 152] for totally geodesic submanifolds).

3.8. T r a n s v e r s a l T o t a l l y G e o d e s i c F o l i a t i o n s a n d S u b m e r s i o n s

Let us consider some generalizations of classes of Riemannian foliations and submersions. Let V and /~ be a"horizontal connection and the curvature of a foliation, respectively.

Def in i t ion 3.9. A foliation {L} is transversal totally geodesic if, for any basic vector fields y and z, the

derivative ~yz is also a basic field. A foliation {L} is transversal curvature invariant if, for any basic vector

fields y, z and w, the vector field R(y , z ) w is also basic.

The condition that the right-hand sides of (3.71) and (3.72) vanish characterizes transversal totally geodesic and curvature-invariant foliations, respectively. The class of transversal curvature invariant foliations includes the class of transversal totally geodesic foliations, and the latter includes the class of Riemann- ian foliations. The characteristics of these classes of foliations (and submersions) in terms of their second fundamental form h2 and the tensor T can be obtained from the following lemma.

L e m m a 3.15. For a foliation {L}, we have

- w ) : y ) -

-(V~he)(y, w), z) + (h2(y, z), T ,w) - (h2(y, w), T~z) - (he(z, w) , T~y). (3.71)

-

= ((VyV~h2)(z, w) - (VyV~h2) (w , u) - (VyV~h2)(z, u)

-(V=V~h2)(y, w) + (V~Vyh2)(w, u) + (V=V~h2)(y, u), x)

+( (Vyh2) ( z , w) - (Vzh2)(y , w), T=u) - ( (Vyh2)(z , u)

+(V~he)(y, u), T=w) - (Vyh2)(w, u), T~z) + (V..h2)('w, u), T=y)

- ( h 2 ( z , u), (V~T)~w) + (h2(z, w), VyT)~u) - (he(w, u), (VyT)~Z

- (V~T) ,y ) - (h2(y, w), ( U Y ) ~ u ) + (h2(y, u), (V~T)~w) . (3.72)

Note that the condition that the right-hand sides of relations (3.71) and (3.72) vanish characterizes transversal and totally geodesic and curvature-invariant foliations, respectively. A foliation {L} with a parallel

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second fundamental form of the distribution T L • i.e., (Vwh2)(y, z) = 0, is transversal totally geodesic.

Symmetric submanifolds (i.e., submanifolds having a parallel second fimdamental form) are classified in some cases (see [176]). A foliation with the condition (~7yVzh2)(v,w) = 0 is transversal curvature invariant. Hypersurfaces in ~t'* with the condition ~TVh = 0 include symmetric submanifolds and also cylinders over a plane curve - - the clothoid k(s) = as + b (see [176]).

E x a m p l e 3.1 (existence). (1) For a twisted product B x f P, the condition that the right-hand side of (3.71)

vanishes if all submanifolds {Tr[-l(b)} have a parallel second fundamental form (i.e., their mean-curvature

vector H is parallel); or equivalently, f (b ,p) = A(b)#(p). In this case, 7r2 is a transversal totally geodesic

submersion, which is not Riemannian for A(b) ~ const. (2) Let {e~}i>l be a local basis of the vertical distribution for the twisted product B x / P . The mean-

curvature vector H of the leaves 7r~ -t (b) is given by the formula H = ~i>1 (ei log f)ei. For any horizontal vectors

y and z, we have (Vz((VyH)T)) -r = - ~ i > 1 z(y(ei logf))e i . Therefore, the property (~TyVfl~2)(v,w) = 0

follows from the system z(y(ei log f ) ) -- 0, which has the solution f (b,p) = p0(p)A(b)exp(~i> ~ bi#~(p)). Here

bi are local coordinates on B, and we assume that the support of nonconstant functions ~(p) lies in the chart.

We consider a submersion 7r: M ~ ~I of smooth manifolds M and/~I with compact connected leaves

(Lb = 7r-~(b)}be~7~. Let M be equipped with a Riemannian metric, and let dim h-I > 1. Along the leaf Lb, the horizontal vector fields with the Lebesgue integrable function of the square of length form the vector space

~ib with inner product

<st, S2}b -=- / ( S t , s2) d vol. (3.73) Lb

The basic vector fields along Lb form a (dim ~l)-(timensional linear subspace Tbl~l C ~Ib, and hence, ~I with

metric ( , ) is a Riemannian manifold (see [23] for the case where M is a unit tangent bundle of Ct-manifold

and .~I is a manifold of geodesics with metric < , }).

Let pL: l~f b __+ T ~ I and pL,• Mb --+ T l~/Ij" be orthogonal projections. The second flmda,nental form

hL: T~I x TI~I -+ T M • of a submersion ~r (i.e., of a subbundle TI~I C ~I) is defined by the formula

hL(y , Z) = PL'• (3.74)

The following two indices of the nullity of a submersion can be considered.

Def in i t ion 3.10. (1) The nullity-space of h L at a point b E ATI is defined by Nb = {Y E Tbt7[: hL(y , 2) =

0 V 2 e Tb~I}. The integer #(h L) = min{dim N~ : b ~ A-I} is called the index of nullity of h L.

(2) The nullity-space of h: at a point b ~/~I is defined by Nb = {Y ~ Tb~I: h~(Y, Z) = 0 V 2 ~ T~/I}.

The integer #(h2) -- min{dimNb : b ~ 57/'} is called the index of relative nullity of h~ for the submersion ~r.

The relation #(h L) = dim ~I (resp., /z(h:) = dim A?/) indicates the transversal totally geodesic (resp., a

Riemannian) submersion ~-.

T h e o r e m 3.16. Let 7r: M -+ ~I ~ be a transversal curvature-invariant submersion of a compact Riemannian 1 manifold M with compact connected leaves {L} that satisfies the following conditions: t ( > 0 and #(h L) > ~ p.

Then h L = 0 and the submersion ~r is transversal totally geodesic.

T h e o r e m 3.17. Let 7r: M --+ 1~I ~ be a transversal totally geodesic submersion of a compact Riemannian

manifold M with compact connected leaves {L}. We assume that the transversal partial curvature Ricq > 0

and the condition #(h2) _> �89 + q - 1) holds. Then #(h2) = p and 7r is a Riemannian submersion.

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Chapter 4 FOLIATIONS A N D S U B M A N I F O L D S

4.1. Submanifolds with Generators in Riemannian Spaces

4.1.1. Ru led and ( s t rong ly ) pa rabo l i c submani fo lds . Ruled submanifohts in real space forms are

studied in [15, 16, 12, 281-283, 175, 3, 71, 157, 182-183, 239; 241, 245, 260]; for 1~I = CP n or/17/-- H/TM, see

[8, 180, 178, 153, 216, 278, 219, 325, 90]; for the Minkowski space, see [65] and [72].

De f in i t i on 4.1. A ruled submanifold is a C2-smooth submanifold M ~+n (u, n > 0) of a Riemannian space ~I

endowed with a Cl-foliation {L} having u-dimensional complete leaves (generators or rulings) that are totally

geodesic in/VI. If ~I is a Kiihlerian manifold, M and {L} are invariant, then M is a complex ruled submanifold. A ruled submanifold with stationary normal space T M • along the rulings is said to be developable. If a

developable ruled submanifold is locally isometric to the product L x L • then it is called cylindrical.

It follows from (2.1) that vectors tangent to the rulings of a ruled submanifold are asymptotic, that is,

h(x,x)=O, x e T L , (4.1)

and the sectional curvature along rulings K L ---- /(IL- For a developable ruled submanifold, we have

h ( x , y ) = 0 , x � 9 y � 9 (4.2)

Hence the curvature hmix coincides with the curvature of the ambient space and the rulings are tangent to relative nullity subspaces (see Sec. 2.4.1). Moreover, a Riemannian space of positive or negative curvature does not contain cylindrical ruled submanifolds (in the sense of Definition 4.1).

Submanifolds of constant nullity were introduced by Chern and Kuiper in 1952 and are also known as strongly parabolic (A. Borisenko), with constant rank (N. Janenko), tangentially degenerate (V. Ryzhkov,

M. Akivis), and k-developable (A. Pogorelov, V. Toponogov, and S. Shefel). In 1972, Borisenko introduced a larger class of parabolic submanifolds.

Def in i t i on 4.2. Tim rank at the point m �9 M of a submanifokl M C ~I is the integer r(m) = max{r(~):

�9 TraM• where r(~) is the rank of the second quadratic form A~ for the normal vector ~. We call

r(M) = max{r(m): m �9 M} the rank of M.

A submanifold M C _~I is (a) strongly parabolic if #(M) > 0 [63],

(b) parabolic if r(M) < d i m M [34].

Since kerh = f'l~ker Ae for any point m �9 M (we can choose c o d i m M normal vectors that form a

basis in TM• it follows that #(M) _ dim M - r(M). For a totally geodesic submanifold, the relations

#(M) = dim M and r(M) = 0 hold. For a hypersurface, we have # (M) = dim M - r(M). Obviously, a

k-saddle subnmnifold M C /17/ (see Definition 3.7) is parabolic and r(M) <_ 2(k - 1). For a submanifold

M C 57[, we have [63]

#(M) <_ u(M) <__ #(M) + codim M, (4.3)

where u(M) (see Sec. 2.4.1) is the index of nullity of the curvature tensor. Strongly parabolic submanifolds have a ruled structure under certain conditions imposed on the curvature

tensor of the ambient space (see Sec. 2.4.1).

T h e o r e m 4.1. [179]. Let M C ~I be a (complete) submanifold satisfying condition (2.4) and #(M) > O. Then the regularity domain G --- {m �9 M: #(m) = #(M)} is foliated by (complete) totally geodesic submanifolds (rulings) in ~I and the normal space to M is stationary along the leaves.

From Tlmorem 4.1 and Corollary 3.8, we obtain the following corollary.

1747

C o r o l l a r y 4.1 ([34] for q = 1). Let M n be a complete curvature-invariant submanifold in a Riemannian

space 1~1 N satisfying the condition #(M) < n. Then

{ ~ ( n + q - 1) i fPdcq(M) > 0 and M is compact, #(M) < ~ ( g + q - 1) i fRicq(ffi)lM > 0 and ~I is compact.

4.1.2. T h e Sasaki m e t r i c on a t a n g e n t a n d a n o r m a l bundle . A metric on a tangent bundle of a Riemannian manifold induced by a translation of tangent vectors was constructed in [263, 264].

Let (x i) be local coordinates on a Riemannian manifold M with the metric tensor ds 2 -- gljdxidx j.

Then, for the induced local coordinates (x i, ~i) on the tangent bundle T M , the Sasaki metric is da 2 =

gijdxi dx ~ + gijD~i D~ j, where D~ ~ = d~ + F}k~J dx k is a covariant derivative of the tangent vector field. In

other words, if dO is the angle between the tangent vector ~ at the initial point x and the vector ~ + dE at the point x + dx after the translation to the initial point along a minimal geodesic, then da 2 = ds 2 + [~[2d~2 (see

[44]). This construction can be applied to an arbitrary vector bundles over a manifold with connection. For

more general constructions of the metrics on fiber bundles, see [84, 85].

T h e o r e m 4.2 [44]. The index of nullity v(TM) of the curvature tensor of a Sasaki metric is even. The

manifold M n is the metric product M1 -v(TM)/2 X ]~v(T/)/2, and T M ~ is the metric product TM~ -v(TM)/2 • Rv(TM).

The p r o o f is based on the existence of v ( T M ) / 2 parallel linearly independent vector fields on 5I. The converse of this theorem is not tree, i.e., the strong parabolicity of M does not necessarily imply the strong parabolicity of a Sasaki metric on T M . The following hypothesis is studied in [317]: for a Sasaki metric on a tangent bundle T M , the index of (spherical) k-nullity (where k > 0) ~k(M) is zero if dim M > 3. For a

submanifold M C 571, the following two (usually different) metrics on T M can be considered: a Sasaki metric

and a metric induced by a Sasaki metric on T f I . For example, for a cylinder M 2 C 1~ 3, a Sasaki metric on

T ( M s) is fiat, but the induced metric on T(M: ) from T(~ 3) = 1r 6 has a nonzero curvature.

L e m m a 4.1 [44]. A Sasaki metric on T~/[ induces a Sasaki metric on T M i f and only if one of the following two equivalent properties holds:

(1) T M is a totally geodesic submanifold in T~[;

(2) M is a totally geodesic submanifold in 571.

The equivalence of properties (1) and (2) follows from the fact that the projection p: T M --+ M is a Riemannian submersion.

For M c R m, the tangent bundle T M can be regarded as a submanifold in the Euclidean space Tllr m = IIr 2m, which has an index of relative nullity #(TM) .

T h e o r e m 4.3 [44]. For M C ~m, we have 2#(M) _> # ( T M ) >_ #(M). Moreover, M is a cylindrical subman- ifold with ( # ( T M ) - #( M) )-dimensional ruling, and T M is cylindrical with 2(#(TM) -# (M)) -d imens iona l ruling.

The p r o o f of Theorem 4.3 is based on the analysis of the second quadratic forms of T M in TST/. A vector field ~ on a manifold M ~ can be identified with an n-dimensional submanifold ~(M) in T M .

This submanifold ~(M) is totally geodesic in T M if and only if ~ is parallel; moreover, for a compact M, the

submanifold ~(M) is minimal in T M if and only if ~ is parallel (see [44]). The volume of a unit vector field is an n-volume of the submanifotd ~(M) calculated for a Sasaki metric on the unit tangent bundle S1M, i.e.,

Vol (~) = / v/det(Xd + (V~) T ( ~ ) d v o l , i

where a covariant derivative V~ is interpreted as a linear operator on T M . For example, a unit vector field of minimal volume on a round 3-sphere is tangent to leaves of the Hopf fibrations [108]. For more general

1748

results concerning Vol(~), see [229]. Note that S~M has a constant mean curvature in T M and its volume

Vol(S~M '~) = Vol(M'~). Vol(S-.-I).

Tile problem of classifying the totally geodesic (tmlbilic) submanifolds of T M and S t M when M is a space form or CROSS is very important. Another important problem is to estimate tile codimension of an isometric embedding of T M with a Sasaki metric into a Euclidean space [44].

A Sasaki metric on the normal bundle T M • of the submanifold M C 117I was introduced by Borisenko (see [43] and [223]); it was used to study the extrinsic geometry of submanifoids in a Riemannian space. For

induced local coordinates (x i, (i) on a normal bundle T M • of a submanifold M C ~I, a Sasaki metric is

defined by dcr 2 = gijdxidx j + goD•177 j, where D• i is a normal component of the covariant derivative in

of a normal vector field (r in a tangent direction [44].

Let UH( T M z) and ~v( T M • be the horizontal and the vertical (intrinsic) index of nullity of the curvature tensor of a Sasaki metric.

T h e o r e m 4.4 [43]. (a) There exist uv (TM • linearly independent normal vector fields on the submanifold M,

which are parallel in a normal connection. (b) The submanifold M in gt m is foliated by UH(TM• )-dimensional intrinsically fiat submanifolds, which are totally geodesic in M with fiat normal connection.

A normal vector field [ on a submanifold M n C 2fI can be regarded as the n-dimensional submanifold

[ (M) in T M • with Sasaki metric. If ~ C T M • is parallel in a normal connection, then the submanifold [ (M)

is totally geodesic in T M • if and 0nly if M is a totally geodesic submanifold [316]. From the geometrical point of view, it is interesting to study not only the spherical tangent bundle SpM

but also the spherical normal bundle SpM • which contains vectors of fixed length p and is a hypersurf~e in a normal vector bundle. The formulas for the Riemannian, Ricci, or scalar curvature of Sasaki metrics on

T M and T M • (resp., SpM and SpM • are similar [44]. If KM > 0, then the sectional curvature of SpM • is positive for sufficiently small p.

The (strong) parabolicity of a given submanifold M C 11-f ensures the ruled structure of a Sasaki metric

on T M • Let ~0 be a normal vector at a point q0 with maximal rank r(~0) = r(M). Then the rank is

constant for normal vectors ~ that are close to ~0- The kernel of .~ satisfies ~=lP ~Ai j x ~ i = 0, where A O~ are

the coefficients of the second quadratic forms with respect to the orthogonal basis of normal space. Since the rank of the above system is constant, the solution space L(q, ~) depends regulm'ly on the point and the

normal vectors. Therefore, a horizontal lifting L({) of the plane L(q, ~) to the point t~ = (q, ~) of the nomml

bundle forms a different• distribution on the neighborhood of the point q0 = (q0, ~0).

T h e o r e m 4.5 [37]. The horizontal distribution L(~) on the normal bundle of the parabolic submanifold

M in the Riemannian space ~I is integrable if, at points of M, the curvature operator R of ~I satisfies the condition

l~(x, y)( J(, (x, y E TM, ( E T M • for the Kiihlerian case.

The fibers are totally geodesic submanifolds of the normal bundle with a Sasaki metric, and their projections on M are totally geodesic submanifolds of t~I.

The integrability of the lift of a distribution from a Riemannian submanifold to its tube with a not necessarily constant radius is studied in [159].

C o r o l l a r y 4.2 [37]. Let M ~ C ikf be a (complete) submanifold satisfying conditions (4.4), and let 0 < r(M) < n. Then, for any normal vector ~ E TvM with maximal rank r(~) = r(M), there exists a (complete) ruling L(~) C M (TpL(~) = kerAe) along which the normal vector ~ is stationary, i.e., ~ remains normal under

V-parallel displacement along any geodesic in L(~).

1749

A parabolic submanifold of small codimension is strongly parabolic [33], tha t is,

# ( M ) _> d i m M - l r ( M ) ( c o d i m M + 1).

The following result completes Corollary 4.1.

C o r o l l a r y 4.3 ([37] for q = 1). Let M ~ be a complete not totally geodesic submanifold in a Riemannian space

1~I g satisfying conditions (4.4) and r(M) > O.

{ ~ ( n - q + l ) / f R i c q ( M ) > 0 and M is compact, Then r(M) > 2 ( N - q + 1) if Ricq(l~/I)lM > 0 and ~I is compact.

4.2. S u b m a n i f o l d s w i t h N o n p o s i t i v e E x t r i n s i c q - D i m e n s i o n a l R i c c i C u r v a t u r e

4 .2 .1 . R a d i u s o f a s u b m a n i f o l d a n d n o n e x i s t e n c e o f i m m e r s i o n s . For a symmetr ic bi l inear map h: R '~ x R n -4 R p and an orthonormal system of q + 1 vec tors {xi}0<~<q C R ~', we define the extrinsic q-dimensional Ricci curvature by

q

Ricq(z0; x~ , . . . , xq) = ~--~[(h(x0, x0), h(x,, xi)) - h2(xo, z,)], i=1

where ( , ) is the inner product in Rv. Ric 1 is called an extrinsic sectional curvature (see [33, 34]). In view of the relation

q

mcT,(z0; E i=1

the inductive formula similar to (3.70),

l~ic q+l' 1 �9 h ~z0; x l , . . . , z ~ + 1 ) = q

q + l

Ric~(x0; x l , . . . , 2 , , . . . , xz+l), i=1

holds, where tile symbol A means the absence of the corresponding vector. Therefore, for q E [1, n - 1), it

follows from RicT~ _< 0 (RicT, --- 0) that Ric~ +I _< 0 (resp., Ric~ - 0).

L e m m a 4.2 ([211] for q = 1). Let h: ~'~ • R'* --4 ~ be a symmetric bilinear map, and let the following inequalities hold for some real c >_ 0:

Ric q_<qc 2, ]h(x,x)[ > cx 2 (x # O).

Then p > n - q .

From Lemma 4.2, it follows that the sylimmtric bilinear map h: R" x R n --~ R p satisfying the condit ions p _< n - q and Ric~ < 0 has an asymptotic vector. L e m m a 4.2 can be ex t ended to the Hilbert case n = oc;

see [199] for q = 1.

For a submanifold M C _ATI with second fundamenta l form h and or thonorlnal system of q + 1 vectors

{xi}0<i<q C T, nM, we have

Ricq(x0; x l , . . . , xq) - R-~q(x0; x l , . . . , xq) = aicT,(z0; x l , . . . , Xq). (4.5)

In particular, for a submanlfold M in a Euclidean space, we have Ric q = Ric~.

It is proved in [190] that if ffI is a complete s imply connected Riemalmian space with sectional curvature

a _</~ < b < 0 and M is a compact Riemalmian manifold with K < a - b , then M does not admit an isometric

immersion in ffI unless dim ~I > 2 dim M. On the o ther hand, an isometric immersion of all n-dimensional

compac t Riemannian manifold with sectional cu rva tu re less than ~ into R 2n-1 can never be conta ined in a

ball of radius d [142]. In [141], the Otsuki lenlma (i.e., Lemma 4.2 for q = 1) is used and a nonembedding

1750

theorem that generalizes these results is proved. Theorem 4.6 below generalizes the result of [141] to the case of partial Ricci curvature.

The positive continuous function C(b, d), b <_ 0, d > 0, is defined as in [141]:

! i f b = O , C(b, d) = d

vfL-b coth (dv/-L--b) if b < 0.

This function is monotonically decreasing with respect to both b and d; obviously, C2(b, d) > -b. Let M be a

compact submanifold in/~-;I with distance function • We denote by B(x, r) = {y �9 M: d(x, y) _< r} a closed ball and assume that

r(M) = inf{r: M C B(x , r)} = inf{max{d(x, y) : y �9 M}, x �9 AI}.

There exists a point x0 �9 A:/such that M C B(xo, r (M) ) . Moreover, there exists a point y0 �9 M such that

d(x0, Y0) = r (M) . r (M) is called the radius of M in _hT/, and B(xo, r (M)) is the minimum, ball containing

M [141]. Generally speaking, there can be several minimal balls containing M. For example, there are two

balls (hemispheres) when M is a large circle in a 2-sphere _~I. However, there is only one minimal ball for a

compact manifold that is immersed in a Euclidean space [141]

T h e o r e m 4.6 ([141] for q = 1). Let ~[n+p-q be a complete simply connected Riemannian space with sectional

curvature a ~_ f ( < b < O, and let d > 0 be a constant. Let M n be a compact Riemannian manifold such that at every point of M ~ there exists d p-dimensional subspace in the tangent space for which the inequality

1 Ric~ t < a + C2(b, d) (4.6) q

holds. Then M '~ does not admit an isometric immersion into ~/I "+p-q in a ball of radius <d.

C o r o l l a r y 4 .4 ([63] for q = 1). (a) Let .~I n be a compact Riemannian manifold with Ric~ ~ 0. Then M '~

cannot be isometrically immersed into R 2n-g. (b) Let M n be a Riemannian manifold with Ricqt < 0. Then

M '~ cannot be isometT~ically immersed into ~2~-q-1.

The product M z~ = M ~ ( - 1 ) • M'~(-1) of hyperbolic space forms has Ric] +~ _< -1: By Corollary

4.4, M ~ cannot be isometrically locally immersed into ~r Corollary 4.4 can be improved in the case of embeddings into a cylinder of a Euclidean space.

X-~s+l ( ~ ~2 _ r 2 Def in i t ion 4.3. A hypersurface C(s, r) in ~N+I that is congruent to ~i=1 ~ i j = 0, where s _< N,

is a circular cylinder of radius r with an s-dimensional parallel and an (N - s)-dimensional ruling (or axis).

(For s = N, we obtain a hypersphere of radius r > 0.) A hypersurface C(r) in 11r N+~ that is congruent to N 2 ~'~.i=i(Xi) -- r 2 ( X y + l ) 2 ---- 0, Z N + I ~_ O, i s a n n - d i m e n s i o n a l c i r c u l a r cone.

g 2 ~ d, defines a parallel of the cone C(r) at the level d. The projection The system ~i=l(xi) = r - ~ , XN+I =

of the point p �9 ~g+~ onto the axis of the cone C(r) (the ( N + 1)th coordinate o fp in the canonical coordinate

system) is called a level of the point p with respect to the cone C(r). We say that the submanifold M n in

C(r) has the level d if d has minimum value along all the levels (with respect to C(r)) of points m �9 M '~.

Note that the submanifold M ~ in C(r) does not contain straight lines of a Euclidean space; hence, the rank

of its second quadratic forms r ( M ) = n. Hypersurfaces in a cone of a Euclidean space are studied in [189].

T h e o r e m 4.7. Let M '~ be a compact manifold satisfying the condition Ric~ _ qc ~ for some constant c > O.

Then an isometric immersion of M '~ into R'~+P (p <__ n - q) cannot be inside a cylinder of radius r = ! with C

a (2p + q - 1)-dimensional parallel.

The following theorem is dual to Corollary 4.4.

1751

Fig . 18. Minimum ball�9 t - -

I R2"*~

~ ( 2 p + q I r l )

Fig . 19. Submanifolds in a cylinder.

T h e o r e m 4.8. (a) Let a complete Riemannian manifold M " satisfying the condition Ric~/ ~ qc 2, where

q E [1, n - 1] and c = const > 0, be isometrically embedded in R 2~-q inside the cone C(r) . Then the level of

M ~ in C(r) is not less than (vF~ 4- 1/r -- 1)/c. (b) Let M n C R 2n-q-1 be a complete submanifold, where

q E [1, n - 2], in a cone C(r), and let for some constant c > 0 and any vectors in T M n, which are orthogonal

to the axis of the cone, the condition Ric~ < qc 2 hold. Then the level of M ~ in the cone is greater than 1__. - - r c

(c) A complete Riemannian manifold M '~ satisfying Ric~1 ~ 0 for some q E [1, n - 1] cannot be isometrically

immersed into the cone C(r) of the Euclidean space R 2n-q.

Another generalization ~ of the notion of extrinsic sectional curvature (as the extrinsic analog of the s-dimensional sectional curvature %, in pa~'ticular, the Lipschitz-Kitling curvature of Riemannian manifold) is given in [33].

Def in i t ion 4.4 [33]. For an s-dimensional subspace V C TpM with an orthonormal basis {u~,. . . u~}, the

number ~8(V) is defined by the formula

1 ~/~ (V) - 2~/2k, ( ~ ~,~j [(h(ui,, uj~), h(ui2, uj2)) - (h(ui,, uj2), h(ui2, uj,))]

�9 i , j E S s

• • [(h(uis_l, ujs_l), h(uis, ujs)) - (h(uis_,, ujs ), h(uis, uj~_l))]),

where $8 is tile set of permutations of order s and ~i is the sign of the pemnutation i = ( i l , . . . , i8). The immber ~8(M) = sup(%(V): V C T M , dim V = s} is called the extrinsic s-dimensional curvature of M.

Obviously, the curvature ~2 is equal to Ric~. If the ambient space is Euclidean, then ~2 coincides with the intrinsic curvature %.

Below, we give a generalization of the Otsuki leinma in a way different from that of Lemma 4.2.

L e m m a 4.3 [33]�9 Let h: R '~ • R ~ ~ R p be a symmetric bilinear map of Euclidean spaces satisfying the property n - - 1 zig(h) <_ O. I f p < ~-1, then h has an asymptotic vector.

The p r o o f of Lemma 4.3 is similar to that of Lemma 4.2. As a corollary, we find that (a) a compact

Riemannian manifold M '~ with %(M) _< 0 cannot be isometrically immersed into R n+p for p -< ~----i-,n-l" (b) a

Riemannian manifold M ~ with %(M) < 0 cannot be isometrically immersed into R ~+p for p < ~-~ _ ~ - 1 [33].

Submanifolds M '~ C 2t7/n+p with ~ ( M ) < 0 and small codimension are (strongly) parabolic: #(M) >

n - (s - 1)p(p 4- 1), r (M) <_ 2(s - 1)p, [33].

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Fig. 20. Submanifolds in a cone

4.2.2. T h e i n d e x o f r e l a t i v e nu l l i t y a n d e x t r e m a l t h e o r e m s .

L e m m a 4.4 ([91] for q = 1). Let h: ~'~ • R n --+ ~P be a symmetric bilinear map of Euclidean spaces with

Ric~ < 0. Then #(h) > n - 2p - q + 51q and there exists an asymptotic subspace T C R ~ of h such that

dim T ~ n - p - q § 51q.

Lemma 4.4 is a bridge from submanifolds with nonpositive extrinsic curvature to ruled and parabolic submanifolds, since it gives us an estimate of the index of relative nullity # ( M ) of the submanifold M '~ C/17I ~+~

satisfying Ric~ < 0 and with "small" codimension (for q -- 1, see [91] and [33]): # (M) > n - 2p - q + 51q-

It is shown in [69] that any isometric immers ion of a Khhler manifold into a complex space form

l~Ig (c), c # O, with a positive index of relative nulli ty should be holomorphic. From the proof of this theorem and Lemma 4.4, we obtain the following:

C o r o l l a r y 4.5 ([91] for q = 1). Let M ~''~ be a Khhler manifold, and let a point mo E M be such that

Ricq(m0) _< qc, where c > 0 is a constant. Then there is no isometric immers ion of M 2n into the real space

f o rm ~I2n+P(c) for p < n - q + 51q.

From [1] and [69], it follows that any isometric immers ion of a complete Khhler manifold into C P g with

a positive index of relative nullity should be totally geodesic. From the above results and Lemma 4.4, we obtain

C o r o l l a r y 4.6 ([86] and [91] for q = 1). Let f : M ~ --+ S ~+p (or f : 2YI 2"~ -+ C P '~+p) be an isometric

immers ion of a complete Riemannian (resp., Khhler) manifold with Ric~ < O. I f 2p < n - v (n ) - q § 5~q

( resp., 2p < n - q + 1), then f is a totally geodesic embedding.

The next theorem follows from Lemma 4.4, Theo rem 4.1, and Corollary 4.1 (see also Theorem 3.1).

T h e o r e m 4.9. Let M ~ C 1~I ~+p be a complete curvature-invariant submanifold with Ric~ <_ 0. Then M is a

totally geodesic submanifold if any one of the following conditions holds:

(1) M is compact with Ric~(M) > 0 and 4p <_ n - s - 2q + 251q,

(2) ~I is compact with Ric~(217l)l M > 0 and 5p <_ n - s - 2q + 2(51q,

(3) 2p < n -- l ] (n) -- q § (~lq and for some k ---- const > 0

y)x = - k y ( x , x), x, y e TM. (4.7)

See Corollary 3.1 (b) for a submanifold M in the sphere S~+P(k) for which condit ion (3) holds wi th q = 1; for a submanifold satisfying the stronger ( than (2.4)) condition (4.4) with q -- s -- 1, see [37]. A similar result in the case where (3) is assumed for submanifolds in a Hilbert hypersphere is t rue without any additional assumptions on their finite codimension; for q -- 1, see [199].

Theorem 4.10 directly follows from Theorem 4.9.

T h e o r e m 4.10. Let M n C 1~i ~+p be a compact, s imply connected, curvature-invariant submanifold. Then

M '~ is a totally geodesic submanifold isometric to a uni t sphere if one o f the following properties holds:

(1) Ric~(M) < s < Ric~(/l?l)lM for some s < n -- 1 and 2p < n - v (n ) - s + 518 - 1,

(2) K(~I) tM --- 1, KM <_ 1, inj(M) > ~-, and 2p < n - 1.

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The property inj(M) > ~r in the case (2) follows from the inequality KM > 0 if n is even and from the

1 when n is odd (see Sec. 2.6). inequality KM >__

If the curvature of the manifold ~7I is subjected to stronger restrictions, then we obtain the following extremal theorem.

T h e o r e m 4.11. Let M" be a compact curvature-invariant submanifold in a simply connected Riemannian

space IVI n+p satisfying the conditions

9 > K ~ > l , R i c ~ ( M ) < s f o r s o m e s < n _ l a n d 2 p < n - s - 2 + ~ l ~ . 4 . . . .

Then ~I "+p is isometric to a unit sphere. The next theorem follows directly from Theorems 4.10 and 4.11.

T h e o r e m 4.12. Let M '~ be a compact curvature-invariant submanifold in a simply connected Riemannian

9 > K ~ > 1, and let one of the following conditions hold: space ~I ~+~ with ~ _ _

(1) Ric~(M) <_ s for some s < n - 1, and 2p < n - u(n) - s + 61~,

(2) KM < 1, inj(M) > it, and 2p < n - 1.

Then ~I ~+p is isometric to a unit sphere and M '~ is totally geodesic.

Theorems 4.9-4.12 generalize Theorems 3-6 in [37], which were first obtained in [34] for submanifolds in CROSS.

4.3. S u b m a n i f o l d s w i t h G e n e r a t o r s in E u c l i d e a n a n d L o b a c h e v s k y Spaces

4.3.1. Submani fo lds w i t h g e n e r a t o r s in a E u c l i d e a n space. It is important to study the relationships between classes of submanifolds with degenerate second fundamental form (strongly parabolic, cylindrical)

and metrics with degenerate curvature tensor (for example, with a positive index of nullity, products of manifolds). There is a natural comlnutative diagram, where the arrows indicate (exact) inclusion maps:

strongly parabolic submanifolds (2)

cylindrical subnlanifolds

(1)> strongly parabolic metrics (3) 1"

(4)> cylindrical metrics

(1). For a sublnanifold M '~ C ]I{ N, tile inequality v (M) > #(M) holds with equality for hypersurfaces.

The problem on conditions under which a Rieinannian manifold M" admits an isometric immersion into •g with the equality v(M) = # ( M ) attained is studied in [42]; the following results have been obtained for the basic case of 3-diinensional manifolds with nullity 1.

(a) There exist 3-dimensional analytic metrics with u(M) = 1 that do not admit an isometric immersion

with It(M) = 1 into ]~g, for instance,

gl~ = exp(2x2) + x 2 + x32, g12 = x3, g13 = --x2, g22 = g33 = 1, g23 = 0.

For the above metric, the sectional curvature on planes that are orthogonal to a 1-dimensional nullity foliation is -1 .

(b) The class of 3-dimensional analytic metrics with v(x) - 1 that admit isometric immersions with

#(M) = 1 into R N for N = 4 (and for N > 4) depends on two functions of two variables. (The class of 3- dimensional analytic metrics with u(x) - 1 depends on three functions of two variables. Also recall the results of A.N. Kohnogorov, V.I. Arnold, and others on the possibility of representing functions as compositions of functions of a fewer number of variables.)

(2). The distinction of the congruence classes of isometric immersions f of a connected Riemannian

manifold M ~ into the Euclidean space R ~+1 for n _> 3 is the classical rigidity problem for hypersurfaces. In this case, the type number of M at x, denoted by t(x), coincides with the rank of the Gaussian map of f and the rank of the second quadratic form r(x). It is well known from the Beez-Killing theorem that f is rigid if

r (M) > 3 and highly deformable in the flat case r ( M ) _< 1. The situation for the constant r (M) = 2 (in this

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case M has a developable ruled structure with (n - 2)-dimensional ruling) is much more complicated. In [267]

and [57], a detailed local analysis is given; the deformations constitute either a discrete set or a 1-parameter family or an infinite-dimensional family in the ruled case n _> 4. Tile compact case was studied in [262]. In

[68], Sacksteder's theorem is extended as follows.

T h e o r e m 4.13 [68]. Let f : M '~ --+ R TM, n >>_ 3, be an isometric immersion of a complete Riemannian

manifold that does not contain an open set M~ • R n-3 with an unbounded M a. Then f admits (nondiscrete)

isometric deformations only along ruled strips (i.e., ruled domains in M n with (n - 1)-dimensional rulings). Furthermore, if f is nowhere completely ruled and the set of totally geodesic points does not disconnect M, then f is rigid.

If the ambient space is the round sphere S '~+I, then the local rigidity problem is reduced to a special aspect of the Euclidean case by considering cones (see [267]). Complete submanifolds M ~ are always rigid for

n > 4 but not for n = 3 (see [68]). For similar problems in terms of the (degenerate) Grassmannian image of a submanifold in a Euclidean space, see [40].

E x a m p l e 4.1 [261]. A hypersurface M 3 C ~4 given by the equation x4 = x3cos(x~) + x2sin(x~) has an index of relative nullity #(x) - 1 but is not a cylinder.

T h e o r e m 4.14 [3]. Let M ~+1 be a hypersurface in the Euclidean space ~n+2 satisfying the following condition:

for all m E M, there exists an n-dimensional ruling lying in M. (4.9)

(a) Then M is foIiated by (n - 1)-dimensional rulings. (b) Let m be a point in M through which at least two n-dimensional rulings pass. Then there exists an

open neighborhood U of m in M that is the Riemannian product of the ruled surface N 2 by R n-2. Moreover, if two n-dimensional rulings pass through every point of M, then M is the product of R ~-2 by a double-ruled surface N 2 in ~3.

It follows from condition (4.9) that #(M) > n - 2. Note that a minimal hypersurface M ~ in I~ '~+1 that

a(hnits a foliation by Euclidean (n - 1)-planes is either totally geodesic or is the product M~ • l~ '~-~, where

the surface/l'/12 is the standard helicoid in R 3 [27].

E x a m p l e 4.2 [3]. The hypersurface M 3 of l~ 4 given by

~(u, v, w) = (u cos w, u sin w + cos w, w + v sinw, w) (4.10)

is foliated by planes (~ is linear with respect to u, v), but is not a product of a surface by R 1 (the leaves

tangent to ker h); the 2-dimensional distribution (kerh) • is not integrable.

(3). The c.ylinder theorems for strongly parabolic embeddings into a Euclidean space with additional

conditions on the sign of the curvature were obtained first in [202] for KM = 0 and then in [127] for KM :> 0, and later in [6] for RicM > 0. For more general results, see Sec. 4.3.1.

T h e o r e m 4.15 [36] Let M n C Rg be a complete submanifold satisfying the conditions RicM _> 0, and let 0 < r(M) < n. Then M is cylindrical with an (n - r(M))-dimensional ruling.

Cylinder theorems for parabolic submanifoIds of "small" rank r(M) <_ 3 and with additional conditions for an asymptotic subspace, but without any assumption concerning the sign of the curvature, were also obtained in [36].

Def in i t ion 4.5 [36]. The subspace A,~ C TraM of maximal dimension that consists of asymptotic vectors is called asymptotic. The dimension of an asymptotic subspace is called the order of planarity at the point m.

T h e o r e m 4.16 [36]. Let M n C ]t~ N be a complete submanifold of constant index of relative nullity, and let rank r(M) be equal to (a) 2, (b) 3; assume that the order of planarity at each point is equal to (a) n - 2, (b)

n - 3, respectively. Then M '~ is cylindrical with an (a) (n - 2)-, (b) (n - 3)-dimensional ruling, respectively.

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(4). The cylindrici ty of embeddings of metric products -~I1 x hi2 into R N of small codimension are

considered in [189, 10, 7].

Let M n = M[ ~' x / V ~ 2, nL, n2 >_ 2, be a product of two connected complete Riemannian manifolds. If

none of the M~ is everywhere flat or contains a Euclidean strip (i.e., an open submanifold tha t is isometric to

the product I x tI~ "~-~, where I C R is an open interval), then any isometric immersion f : M '~ -+ R '~+~ is a

product of hypersurface immersions [10, 189]. This means that there exist an orthogonat factorizat ion ~,~+2 =

R 'u+~ x R ~+~ and isometric immersions s AC ~ --+ R ~+~ (i = 1, 2) such that f (x~, x.,) -~ (f~(z~), s This remarkable global theorem has been proved for any nmnber of factors whenever the codimension equals

tha t number. In [17], the problem on conditions under which f : M " = . ~ x M~ ~ -+ R "+-" is not a product

of hypersurface immersions is studied.

T h e o r e m 4 .17 [17]. Let f : M " = .~I~' x / V ~ 2 --+ R '~+2 (ni _> 2) be an isometric immersion of a complete connected Riemannian manifold, where none of the factors is everywhere fiat. Then there exists a dense open subset each of whose points lies in the neighborhood of the product U = U1 x U2 with open U~ C Mi such that

fu: UI x U2 --+ ~r is one of the following types: (i) fu is a product of immersions;

(ii) each U~ is isometric to I~ • R "~-~ (i = 1, 2) and fu = g • Id, where g: I~ • I2 -+ ~4 is an isometric

immersion and Id: ]1r '~-2 --+ R '~-2 is the identity map;

(iii) only one of the Uj is isometric to Ij X R '~-1 and fu = ] • Id: (Ui x I j ) • R " J - ' --+ R n+2 (i # j ) ,

where Id: R ~-~ --~ R '~ -~ is the identity map and ]: U~ x Ij -+ ]R '~'+3 /s the composition ] = h a g of isometric

immersions g: Ui • Ij --+ V and h: V --+ R '~+3, where the set V C IR "~+2 is open.

For type (i), e i ther we have a product of hypersurfaces or one of the factors is total ly geodesic. Types

(ii) and (iii) are not necessarily different, since the immersion g in (ii) may, in fact, be a composition. A

complete local classification of fiat surfaces in R 4 that are nowhere compositions of isometric immersions is

given in [56], The example in [10] re]ates imnmrsions of types (i) and (ii). The case where one of the factors of M is everywhere flat is also studied.

D e f i n i t i o n 4.6 [17]. An isometric immersion f : N " + " -+ R N is an m-cylinder if there exists a Riemannian

manifold M " such tha t N '*+m, R g and f have the orthogonal factorizations N '~+m -- M '~ • R m, ~ v _-

R g-m x 1r m, and f = ] • Id, where ] : M '~ -+ R N-m is an isometric immersion and Id: R m -+ ~'~ is the identi ty map.

Recall the characterization of complete cylinders due to Hm' tman [127]:

L e m m a 4.5. Let f : M " --+ R N be an isometric immersion of a connected complete Riemannian manifold

sati~fing the condition RiCM _> 0 such that f ( M) contains m linearly independent lines passing through one point. Then f is an m-cylinder.

T h e o r e m 4.18 [17]. Let It{" be a complete connected Riemannian manifold satisfying the condition RicM > 0,

without fiat points, and let f : M '~ x R m --+ R ~+'~+2 be a CZ-regular isometric immersion. Then f is either an

m-cylinder or an (m - 1)-cylinder f = ] x Id: (M '~ x R) x R m-~ -+ R "+m+2, and there exist a .fiat Riemannian

manifold N n+2 and isometries g: M n x R --+ N "+2 and h: N n+2 --+ R n+3 such that ] = h o g is a composition.

Furthermore, i f M "~ is simply connected, in the latter case we can consider N '~+2 as an open subset of R "+2,

aT~d then g = 9 x Id, where Id: R ---+ R is the identity map and ~: M " -+ ~'+~ is an embedding whose image is a convex hypersurface.

Theorem 4.18 is not true without the assumption on the 1-regularity (see [17] and [133]). For m = 1, a

weaker result, bu t wi thout the 1-regularity assumption, is given in [198].

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A smooth ruled submanifold M ~+s C RN can be given (locally) in parametric form by

v

r ( u l , . . . , u s ; V l , . . . , vg) = p ( ' l~ l , . . . , Us) "Jr E viai(ul,'", Us)' (4 .11) i---1

where {~i(u: , . . . ,u~)} are bases of rulings and f i (u: , . . . ,u,~) is the base submanifold. In pea'ticular, for a

cylindrical submanifold, the vectors ( ~ } are constant. A one-parameter family of rulings corresponds to a

submanifold M ~+: C 37I of the form

r ( u l , . . . , Un; t) = r ( u l ( t ) , . . . , Un(t); V l , . . . , Vy), (4 .12)

where {ui(t)} is a smooth curve in ir '~ [41]. The Gauss equation (2.3a) for a ruled submanifold M ~+n C RN

yields Kmix _< 0. Therefore, if M ~+: is a ruled submanifold in the Euclidean space ~N with zero scalar curvature, then M is developable.

T h e o r e m 4.19 [41]. Let M ~+s be a ruled submanifold in ~t N. Then M is cylindrical i f and only i f all 1-parameter families of rulings have zero scalar curvature.

Note that the developable ruled submanifold M "+~ C IZ N with the integ:able distribution T L • is cylin- drical.

4.3.2. S u b m a n i f o l d s w i t h g e n e r a t o r s in L o b a c h e v s k y space. Foliations (fibrations) on the Lobachevsky space H '~ were studied in [47, 154, 75, 115, 323, 19]. Ruled submanifolds in H s were studied in [6, 196, 87,

283, 39, 173, 16, 15, 11, 52, 47, 18]. There are many strongly parabolic submanifolds in ~-P; recall that they have the ruled structure. Any

strongly parabolic submanifold in the Euclidean space ~s with singular points outside a unit ball corresponds (by using the Cayley-Klein interpretation) to a complete strongly parabolic submanifold in H ~. In [283], the

nondeveloped ruled submanifolds M ~+: C ~s with u-dimensional ruling are studied. In [6], the problem on

classification of all isometric immersions of a hyperbolic space form H n-: into another hyperbolic space form is treated. Since the rank of the shape operator A is at most one, the subset U = {x E H'~: rank(A) = 1}

consists of at most a countable number of open connected components in H ~ each of which is C~176 by complete totally geodesic hypersurfaces (a relative nullity foliation) {L~ }~A-

Def in i t ion 4.7 [6]. Let U be an open subset of ~ whose connected components are U~, i -- 1, 2, . . . , each of

which is endowed with C~176 foliation {L~}~eA~ complete totally geodesic hypersurfaces. Let {L~}~eA be the disjoint union of the sets {L~}aeA~, where A = UA~. The triple (U, { L ~ } c ~ A , H n -- U) is called a C~176 on H n.

T h e o r e m 4.20 [6]. Given a differentiable lamination on a hyperbolic plane H 2, there exists a family of

isometric immersions of H 2 into a hyperbolic space H 3 such that the induced relative nullity foliations are defined by the lamination.

For any smooth curve orthogonal to the rulings of a strongly parabolic submanifold M C ~ ( - 1 ) with

#(M) = dim M - 1, the normal curvature is <1 [86].

There exist strongly parabolic submanifolds in H g that are similar to cylinders in R N.

Def in i t ion 4.8 [39]. Let :4 z+p-~ be either a horosphere or an equidistant submanifold or a totally geodesic

submanifold in H z+p-~+:, and let FZ, ~ be a complete submanifold in H l+p-~. The union of the complete

geodesics orthogonal to BI l+p-~ in H l+v-~+: forms a submanifold F ~-~+:. The union of the subspaces H ~-~

orthogonal to H t+p-~+~ and passing through all points of F z-~+: is a complete ruled submanifold M l C l~ l+p with u-dimensional ruling, called a cylindrical submanifold.

Cylindrical isometric immersions of H 2 into a hyperbolic space H 3 are studied in [196]; see also [11].

A cylindrical submanifold M l in H z+p is strongly parabolic with index of relative nullity #(M) _> u; the

orthogonal distribution (to rulings) is integrable, and every base F ~-~ is totally umbilic. If the base F ~-~

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belongs to an equidistant or a totally geodesic submanifold (the first case), then M l is a cylinder with a

(v - 1)-dimensional ruling over a conical submanifold F ~-~+1 with the vertex outside the absolute (a cylinder

if the vertex is ~ ) . If the base belongs to a horosphere (the second case), then M s is a cylinder with a

(v - 1)-dimensional ruling over the conic submanifold F 1-~+1 with the vertex on the absolute. The normal curvature k,~ of any geodesic on the base is constant and 0 _< k,~ < 1 in the first case or

kn = 1 in the second case, if F t-~ lies on a horosphere. The metric of a cylindrical submanifold M t in H l+p is semireducible and can be written in the form

ds 2 = ds~ + ~2(xl,... ,x~)ds~, (4.13)

where ds~ is a metric in H~ that depends on the variables x l , . . . , x ~ ; ds~ is a metric oll some base F l-~

dependent on the variables x~+l , . . . ,xl; the function T has the form e x p ( - x l ) or cosh(xl) in a special

coordinate system [39]. Next we consider the relationship between cylindrical submanifolds and metrics.

De f in i t i on 4.9 [39], [173]. A complete metric on a Riemannian manifold M with a constant index of

(intrinsic) k-nullity ~k(M) > 0 (see Sec. 2.4.1) is k-cylindrical if there exists a ( d i m M - L,k(M))-dimensional

submanifold orthogonal to the distribution of k-nullity that is totally umbilic for ~k(M) < dim M - 1, and

for ~'k(M) ---- dim M - 1 is a curve of constant curvature.

T h e o r e m 4.21 [39]. A k-cylindrical metric with k < 0 is foliated by v~(M)-dimensional complete totally

geodesic submanifolds (k-nullity foliation) and can be urritten in the form (4.13) with the function ~ equal to

e x p ( - x l ) or cosh(xt).

T h e o r e m 4.22 [39]. Let M l C ~ ( - 1 ) be a strongly parabolic submanifold with l > # ( M ) > O, and let there

exist a submanifold orthogonal ~to the relative nullity distribution that is totally umbilic for #( M) < l - 1, and

for # (M) --- l - 1 is a curve of constant curvature. Then M is a cylindrical submanifold that is represented

(in the Cayley-Klein interpretation) by a cylinder with a #( M)-dimensional ruling over a conical submanifold

with vertex on the absolute or outside it.

In [173], a subclass of strongly parabolic metrics that is in a one-to-one correspondence with some subclass of strongly parabolic submanifolds in a Lobachevsky space is studied.

T h e o r e m 4.23 [173]. Let M t have a k-cylindrical metric with k < O, and let one of the following conditions

hold: (1) ~ = e x p ( - x l ) and the base ds 2 adm, its an isometric immersion into ~t t-~+p with zero index of relative

nullity,

(2) ~ = cosh(x~) and the base ds~ admits an isometric immersion into H t-~+p with zero index of relative

nullity.

Then M admits an isometric immersion into H l+p as a cylindrical submanifold (with a ~-dimensional

ruling and the same codimension as for an immersion of a base).

4.3.3. C y l i n d r i c i t y of s u b m a n i f o l d s in a R i e m a n n i a n s p a c e o f n o n n e g a t i v e c u r v a t u r e . We can generalize Theorem 4.15 in different ways.

For a submanifold M C ~I, the expression R-~T(x) with x E T M is defined by the same formula as

Ric (x) for M but by using the sectional curvature of ~I:

RicT(x) = y~/~(x , Yi), i

where {Yi} is an orthonormal basis in T M at a given point.

L e m m a 4.6. Let M n C ffl be a complete submanifoId, and let

aic (x) _> R-i-cT(x), x e TraM, (4.14)

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for some point m �9 M with r (m) = r (M) . Then i t (M) = n - r (M) , and the second quadratic form AH of

the mean-curvature vector H at m is positive definite on the subspace (ker h) •

Def in i t ion 4.10. Let M C/ITI be a complete curvature-invariant submanifoid, and let k > 0 be a constant. For any point m �9 M with #(m) = it(M) > 0, we consider the set F(m, h), which is the sheaf of geodesics

7: [0,Tr/v~] -4 M, (7(0) = rn, ~'(0) �9 kerh) (where h is the second fundamental form of M) and define a nonnegative number

a (m,h)=sup{(V~z ,~ / ' ) : y , z • y • l y l= l z ] = l , 7 � 9

This c~(m, h) is an analog of the turbulence c~(L) for a relative nullity foliation on the regularity domain C M and is well defined by virtue of Theorem 4.1. Similar to 5R (see Sec. 3.5), let 5~ be a nonnegative

number: 5~ = s u p { ( R ( x , y ) u , z ) : x, u E kerh, y, z �9 T M A (kerh) •

x , l , u , y • Jxl = lyl = luf = Izl = 1 } .

For example, 5~ = 0 holds for a conformally fiat metric on/~I along M. Theorem 4.24 (and Theorem 4.25 beloW) is based on the method developed in Sec. 3.5.2 by using the

volume of L-parallel vector fields.

T h e o r e m 4.24. Let M '~ C Y/I be a complete analytic curvature-invariant submanifold with 0 < r (M) < n satisfying the condition

k'2 >_/((x, y) > kl >_ O, x, y �9 T M , (4.15)

and let there exist a point m �9 M of maximal rank satisfying (4.14) and one of the following inequalities:

(k~ - k~) ma~{a(m, h) 2, k} < 0.3k~k, (4.16 a)

(k.2 - k, + 25~ a(m, h)max{a(rn , h) '~, k}:~ <_ 0.004 k2k2 (with 2k~ >_ k2), (4.16 b)

where k = (kt + k~)/2. Then k~ = k.,_ = 0 and M is a cylindrical submanifold with a flat #(M)-dimensional generator.

The sequence ~(n) is defined in Sec. 3.1.

T h e o r e m 4.25. Let M n C 1~I be a complete analytic curvature-invariant submanifold, and let either M ~ or

~I be compact, and n > # ( M ) > ~, (n) (n > # ( M ) > 0 for the KShlerian case),

k2 _>/~(x, y) > k~ _> 0, x, y �9 T21~/, (4.17)

and one of the inequalities (4.16 a) or (4.16 b) hold for a certain point m with #(m) = #(M): Then kt = k2 = 0

and M is a cylindrical submanifold with a fiat #( M)-dimensional generator.

Corol lary 4.7. Let M n C ~I '~+p be a complete curvature-invariant submanifold satisfying condition (4.15) for kl > 0 and condition (4.16a) or (4.16b) for a certain point m �9 M with i t (m) = i t (M) . Also assume that one of the following properties holds:

(a) # (M) > L, (n), (# (M) > 0 for the Kiihlerian case),

(b) (6.2) for the point m �9 M,

(c) a i 4 < 0 f o r some q e [1, - 2] and

n - v ( n ) - q § 2 p <

n - q + 51q for the Kiihlerian case.

Then M n is a totally geodesic submanifold.

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Theorems 4.24 and 4.25 and Corollary 4.7 generalize the results of [36] and [1, 2], (see also Sec. 4.3.1);

moreover, for the Kiihlerian case, the holomorphic bisectional curvature of .~/ in (4.15) or (4.17) can be considered. Case (c) of Corollary 4.7 follows from (a), since Ric~ _< 0 implies the inequality for # ( M ) given in (a); see Lemma 4.4. The analytic condition in Theorems 4.24 and 4.25 can be replaced by the requirement that the function #(m) be constant. Note that the sectional curvature in Theorems 4.24 and 4.25 for (4.16b) is 0.5-pinched, in contrast to (4.16a) with weaker requirements for the curvature tensor but with greater (equal to 0.7) pinching of curvature.

The following theorem is based on the procedure of the Riccati ODE from Sec. 3.2.

T h e o r e m 4.26. Let M '~ C M be a complete analytic curvature-invariant submanifold satisflfing the following conditions: 0 < r( M) < n, KIM >-- O, and

Ric (x) > ~-dT(x) , x e TM. (4.18)

Then M is a cylindrical submanifold with an ( n - r( M) )-dimensional generator.

C o r o l l a r y 4.8. Let M n C M be a complete curvature-invariant submanifold satisfying the conditions (4.18)

and I([M > O, and let r (M) < n. Then M is a totally geodesic submanifold.

Theorem 4.26 and its Corollary 4.8 complete the results of [2] (where the cases 2fI = I~ g and /~ r = S N are studied) and also the results from [36]. From Theorems 4.24, 4.25, and 4.26 and the results from Sec. 4.2.2, we can deduce cylinder theorems for submanifolds with nonpositive extrinsic curvature with "small" codimension and some additional conditions as well as for k-saddle submanifolds.

4.4. D e c o m p o s i t i o n of R u l e d a n d P a r a b o l i c S u b m a n i f o l d s

4.4.1. R u l e d a n d p a r a b o l i c s u b m a n i f o l d s in C R O S S and t h e Segre e m b e d d i n g . The ruled sub- manifolds in S m and R m are related by a geodesic m a p of the hemisphere S m = {Xo > 0}, +

- + c

L e m m a 4 .7 (see [34]). Let M be a submanifold in the hemisphere S T = {x0 > 0}, and let 1~71 be its image in

~'~ under a geodesic mapping. Assume that ~ is a normal vector to M at a point q and ~ is a normal vector

to ~I at a point ~. Then (1) the orthogonal projection of ~ onto the hyperplane Xo = 1 is a normal vector ~, i.e.,

- +

(2) Aij (~, ~) = cAij (~, q), where Aij (~, ~) and Aij (~, q) are coefficients of the second quadratic forms of M and

1~I for the normal vectors ~ and ~ respectively.

Therefore, locally (strongly) parabolic submanifolds in S m and I~ m are in one-to-one correspondence; see also [9]. In contrast to the sphere S m, where there is only one type of totally geodesic submanifold (large spheres with all possible dimensions), other CROSS (except for R P ~) have different types of totally geodesic submanifolds:

C P m: { R P "~, CP~}, n < m;

H P m : {RP ~, C P "~, H P ~, $3}, n < m;

C a P 2: { R P ~, C P ~, H P '~, Si} , n < 2, i = 3 , 5 , 6 , 7 , 8 .

These submanifolds in K P m (K = C, H) are related by the mapping exp to some subspaces ~:n+l contained

in K '~+1 [23]: (1) the type of the numbers ~: C K can be restricted, (2) the dimension can be decreased, f i_<n.

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In [34], orthogonal projections of the Riemannian symmetric spaces S m, C P m, and H P m onto the totally

geodesic submanifolds S ~, CP~, and HP~, respectively, are studied; submanifolds M ~ with regular projection axe said to be uniquely projectable.

Def in i t ion 4.11 [34]. A submanifold M s in CROSS is t-uniquely projectable if there exists a totally geodesic submanifold

(1) $3 +' C S", (2) CP(o n+')/2 C C P '~, or (3) HP(on+')/4 C H P n

such that the tangent space to the totally geodesic submanifold

(1) Sn-(~+t)(q), (2) CP"-('~+t)/2(q),

which contains the point q E M and is orthogonal to (1) S~ +t, has zero intersection with the tangent space TqM.

or (3) HP'~-(n+t)/4(q),

(2) C P ("+t)/2, or (3) H P ('~+t)14, respectively,

The image ~I of M under the above mapping q E M --+ ~ E S~ +t, CP(o ~+t)/2, and H P (n+t)/4, resp.,

is a smooth submanifold of the same rank; as in Lemma 4.7, the second quadratic forms of M and/~i are proportional.

The estimates of dim L for a ruled submanifold and tests for h = 0 for a parabolic submanifold in

CROSS can be obtained in terms of the dimension of FI or the codimension of M. But a key role is played by the number t from the definition of the unique projectivity and the following lemma.

L e m m a 4.8 [34]. For submanifblds ~1 and M = ~r-l(~i), where zr are the Hopf fibrations with an s-

dimensional fiber, the inequality r(M) <_ r(~I) + 2s holds.

The standard totally geodesic foliations on CROSS are the generalized Hopf fibrations (1.12). For a

submanifold/~I in the base of the Hopf fibration, the inverse image 7r -1 (~I) is a ruled submanifold in a sphere

or in C P 2'~+1 with equidistant 1-, 3-, 7-, or 2-dimensional rulings (fibers).

Def in i t ion 4.12. Consider the following classes of ruled submanifolds in S TM and C P m with complex structure J:

1. M ~+~ C S m with the ruling L = S ~, 2. M ~+~ C C P ~ with a ruling L = R P ~ that satisfies J (TL) 2_ T M ,

3. M "+~ C C P m with a ruling L = C P ~/2 that satisfies J ( T M ) = T M .

We caa also consider some classes of ruled submanifolds in H P m and CaP 2. In [236, 242], the following classes of ruled submanifolds in CROSS are introduced:

RS1 that are uniquely projectable, RS2 with positive/(mix. They generalize the well-known class RSo of ruled developable (or strongly parabolic) submanifolds and

are connected with the following classes of parabolic submanifolds: PSt that are uniquely projectable, PS2 with positive sectional curvature.

T h e o r e m 4.27 [242]. Assume that a complete submanifoId M in S N or C P N belongs to RS2 (resp., PS2). Then any (complete) ruling L C M has a neighborhood that belongs to RS1 resp., PS1). The dimension of a ruling of such a submanifold is estimated by

p ( c o d i m L ) - 1, dim L < 2 for the Kghlerian case.

For a submanifold M from RS1 or PS1, the above inequality can be proved by considering the projection of M onto a totally geodesic subspace of the same dimension, and then, by using Theorem 3.1. Moreover,

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for a ruled submanifold in Theorem 4.27, the structural tensor of the foliation {L} has no eigenvectors [242]

(compare with Lemma 3.6 and Lemma 3.8).

Coro l l a ry 4.9 [238]. Let M ~ be a complete submanifold in Sin(l), or cpm(1) with J ( T M ) = T M , and let

RiCM not exceed n - 2 or 4, respectively. I f

then M '~ is totally geodesic.

< f, for ?"( ~ f ) -- [ n - 1 for CP'~(1),

In the above statement, u(n) is defined as in Theorem 3.1.

The simplest ruled submanifold in a Euclidean space is a cylinder f ( M 1 x ~ ) C R g. There are no cylinders in CROSS; in these spaces, the Segre embedding serves as a standard example of a ruled subrnanifold that is a metric product.

Def in i t ion 4.13. The Segre embedding M ~'+n -~ f ( S ~ x S n) C S ~+n+~ is given by f: (ul, . . . , u~, vl, . . . , v~) --+

(u l v~ , . . . , u i v j , . . . , u,vn), where {u~} and {v j } are the coordinates of points on the unit spheres S ~ and S '~.

The complex Segre embedding M € = f ( C P~ x C P ~) C C P ~+n+~ can be similarly defined.

The rulings of the Segl"e embedding can be chosen in two canonical ways. The simplest Segre embedding M 2 = f ( S 1 x S 1) C S 3 (Clifford torus) is a flat surface.

R e m a r k 4.1. Any smooth spherical f ibration (in the topological sense) S k C E _5~ B with compact base

B admits a smooth ruled embedding into a sphere S y with large N, i.e., an embedding whose fibers are large k-dimensional spheres [217]; such embeddings are called large-sphere fibrations. There are infinitely

many topologically nonequivalent smooth 3-sphere fibrations of S z (whose bases are not diffeomo1~hic to $4).

These fibrations can be realized as ruled submanifolds in S N+r with large N. However, for N = 0 any smooth fibration of S 7 by large 3-spheres is topologically equivalent to the Hopf fibration [105], i.e., the topological structure of a compact ruled submanifold depends on its codimension.

If M C S g is foliated by large spheres, then there is a hierarchy of three questions stated here in an increasing order of difficulty [217]: (1) given two foliations of this kind, are they topologically equivalent ?

(2) if they are topologically equivalent, is it possible to deform one into the other via a one-parameter family

of such foliations ? (3) what is the homotopy type of the space of all such foliations ?

These problems are completely solved for large-circle foliations of S 3 (see Sec. 2.6). A more complicated

case where M = S p x S q is embedded into S v+q+l is considered in [217] in the standard way, wtmre some special cases are also studied.

The curvature tensor of the ambient Riemannian manifold (CROSS) along ruled submanifolds from Defi~fition 4.12 has the same simple form

=0, x E T L , y E T M , z L ( x A y ) , (4.19) R(x , y)z I I Jz, x E T L , y E T M , z • (x A y A J x A J y ) for the K:,ihlerian case

(4.20) /~(y, x ) x = kx2y, x E TL, y E T L •

We will also use the following inequality:

(A~x) 2 + (Ar ~" <_ k (x, u E TL, x W u, [x[ = [u] = I~1 = 1). (4.21)

Let x l , . . . , x~, Yl, . . . , yn be an orthonormal basis in TraM such that xi E TmL, Yi E TmL • Let A~ denote

the symmetric matrix of the.second quadrat ic form for a normal vector ~ E T,~M • 1~1 = 1, in this basis. If,

for example, A~xi = x/~yj for some numbers i , j (an extremal situation), then the matr ix A~ satisfying the

condition (4.21) has only two nonzero elements; its rank is equal to two.

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Note that from (4.21) and (4.20), by virtue of the Gauss equation (2.3a), we obtain Km~, > 0. Moreover,

(6.33) follows from the "compact" inequality

]IAr ~ := tr(A~) _< 2k, I~l = 1. (4.22)

We define the nondecreasing integer sequence fi(n) -- max{p(t): t <_ n}. From the definition of p(n) (see Sec. 1.1), we obtain the estimates

2[log 2 n] _< fi(n) < 2[Iog2 n] + 2 ([x] denotes the integer part of x).

The following synthetic concept is effective for local (i.e., in a neighborhood of a generator) research of ruled and parabolic submanifolds.

D e f i n i t i o n 4.14. A smooth submanifold M C ATl is t-uniquely projectable along a generator L C M if t

is a minimum integer such that for some point m E L, there exists a subspace V C TraM • of dimension

codim M - t that remains transversal to M under the V-translation along any path in L. For t = 0, such M is uniquely projectable along L.

For a submanifold in a space fo rm/~(k ) or CROSS, Definition 4.14 generalizes Definition 4.11. A key result of this section is the following theorem.

T h e o r e m 4.28. Let M ~+~ be a ruled ( Kiihler) submanifold in a Riemannian ( Khhler) space ~I with rulings {L~}, and let conditions (4.19), (4.20), and (4.21) with k > 0 hold. IJ:~ > fi(n) ( o r , > 2 for the Kiihlerian

case), then M ~+~ is locally isometric to the product L ~ x M~ and codlin M > v n. Moreover, if 1~I = S N or

C P N and codlin M = ~ n, then h i is congruent to the domain of the Segre submanifold.

C o r o l l a r y 4.10. A ruled submanifold M ~+n c ~I satisfying properties (4.19) and (4.20) with K~x > 0 along a ruling L ~ is uniquely projectable. In particular, we have ~ < p(n) (or ~ = 2 for the Kghlerian ease).

Condition (4.21) can be replaced (without affecting Theorem 4.28) by the integral inequality for the second quadratic form as follows:

along any leaf geodesic ~/: [0, ~-/(2v/k)] ~ M, for any V• unit vector field ~(t) E T~M • we have

f tlA~(t) Ildt <_

I d e a o f t h e P r o o f of T h e o r e m 4.28. For simplicity, we assume that k = 1. For the totally geodesic

foliation {L} (on the ruled submanifold M) , we define the structural tensor B: TL • x TL • --+ TL. In view

of the given inequality for the dimension, for every point of M, there exist unit vectors x E TL, y E T L • and real A < 0 satisfying the property B(x, y) = Ay. The L-parallel Jacobi field y(t), y(O) = y, along the geodesic ~/(t), ~/(0) = x, can be written as y(t) -- (cos(t) + A sin(t))~ + (sin(t))~, where ~ and '~ are obtained

by using the V-parallel transport of the vectors y and v = h(x, y) # 0 along ~/. Note that the normal vector

v under the V• transport along the path 7(t), 0 < t < to, where to -- co t - l ( -A) _< 7~/2, turns in

/~I through the angle 7r/2, although the velocity of rotation is less than 1. In view of the extremal situation,

we find that the normal vector v turns uniformly in some plane along 7(t), 0 < t < 7r/2, through the angle

lr/2 and A = 0 holds. Hence we deduce that we can "chip off' the vector field ZI: L -+ TL • containing the

unit vector II A~x: V~Z~ = R(Z~, u)u = O, w 1 Z~ ~ B(u, w) _1_ Z~. We will repeat this process n times and

obtain the splitting of M along T L and TL • Using Codazzi equations, we find the metric decomposition of M with codim M _> v n.

Next we give similar decomposition and rigidity results for parabolic submanifolds in CROSS. The cur- vature tensor of a submanifold M in SN(k) or a K~i~hler submanifold M in CPN(k) has the following simple form:

= o, x, y e TM, z • (x A y), (4.23) [~(x, y)z I I Jz , x, y e TM, z I (x A y A Jx A Jy) for the K~hlerian case,

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R(y, x )x = kx2y, (x, y E T M for the Kiihlerian case y_l_ (x A Jx)) . (4.24)

We will use the following inequality (cf. (4.21)):

A~x + A~u <_k (x, u E TM, x L u , [x[ = ]al = I~] = l). (4.25)

Note that from (4.25) and (4.24) in a real case, by virtue of the Gauss equation (2.3 a), we obtain KM _> 0. Moreover, (4.25) follows from the inequality (cf. (4.22))

]]A~I] 2 _< 2k, I~] = 1. (4.26)

De f in i t i on 4.15. For a submanifold M C ~I and any s-dimensional subspace U~ C TqM • let h(U~) = Ph:

TqM x TqM -+ Us be a bilineax form, where h is the second fundamental form of M, and let P: TqM • -+ U~

be an orthoprojector. The s-nuU index of M, where s _< codimM, is defined by the formula #~(M) = min{dimN(Us): U, C Tq.~'I • q E 1~1}, where N(Us) is the null space of the bilineax form h(Us).

Note that this definition is different from that given in [54]. The following inequalities hold, where p = codim M:

#(M) = p~(M) _<... _< #2(M) _~ #I(M) = d i m M - r(~I) . (4.27)

L e m m a 4.9. The Segre submanifold M '~+~ = f ( S ~ x S n) C S g satisfies the conditions r(M) = 2 rain{y, n} and # I (M) = #2(M) + 1.

Thus, the classical Segre embedding is a parabolic submanifold when u ~ n.

T h e o r e m 4.29. Let M n C l~f n+p be a complete submanifold satisfying properties (4.23 a), (4.24), 0 < r( M) < n - ~(n), and one of the following inequalities:

1 (a) (4.25), (b) KM_>0, (c) ~RiCM:>n- -2 ,

where k > 0 and #I(M) <_ #.z(M) + 1 for (b) and (c). Then M is locally isometric to the product (a), (b)

• (c) s • M] with r (M) = 2, respectively, and p > r(M)(n - r ( M ) / 2 ) . Moreover,

if 1~I = Sin(k) and p = �89 - r (M)/2) , then the submanifold AI ~ is congruent to the Segre embedding.

The p r o o f of Theorem 4.29 is based on the relationship between KM > 0 and the synthetic concept of a t-uniquely projectable submanifoId. The only difference with the proof of Theorem 4.28 is that together with every vector field Zi, i < r (M) /2 , we also split off the field ~ in which the mixed sectional curw~ture is k.

The decomposition of K~hler parabolic submanifolds can be deduced from weaker assumptions on the rank r (M) and on the curvature.

T h e o r e m 4.30. Let M n C ~[~+P be a complete Kbhlerian submanifold satisfying properties (4.23b), (4.24),

0 < r (M) < n, and one of the following inequalities: (a) -~Rici > n, (b) Ih(x,y)I/( ix[ �9 lY[) -< v~ , (c) the

(bi)sectional curvature of M is nonnegative. Then M is locally isometric to the product (a) CP'~ -~ x 2 ~

with r (M) = 4, (b), (c) CP~ -~(M)/4 • M2 (M)/2, respectively; and p > �89 - r (M) /2) . If, in addition,

~I = CPm(k) and p = �89 - r ( M ) / 2 ) , then M '~ is congruent to the complex Segre submanifold.

C o r o l l a r y 4.11. Let M ~ C ~[ be a complete submanifold satisfying the properties KM > 0, (4.23), and (4.24), and let u(M) >_ y(n) (or ~ > 0 for the gdhlerian case). Then M is totally geodesic.

4.4.2. R u l e d a n d pa rabo l i c s u b m a n i f o l d s in a R i e m a n n i a n space of p o s i t i v e cu rva tu r e . For a ruled submanifold M in a Riemannian space ~I with nonconstant mixed sectional curvature along the rulings, we can assume that

k s_> /< (x ,y )_>k~>0 , x E TL, y E T L • (4.28)

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We will also use the following condition that generalizes (4.21):

(A~x) 2 + (A~u) 2 <_ k{1 - 5(1 - kl/k2)(a(Lo)2/k + 1)} 2

(x, u TL, x_L u, Ixl = lul = 141 = 1). (4.29)

Combining the area-of-L-parallel-fields method for foliations (see Theorem 3.6) with the synthetic method of Theorem 4.28 for ruled submanifolds in space forms, we obtain the Segre-type decomposition of submanifolds in a Riemannian space with positive sectional curvature.

T h e o r e m 4.31. Let M '~+v C Y/l be a ruled submanifold with a ruling {LV}, and let conditions (4.19), (4.28),

and (4.29) with k = (ke + kl ) /2 hold. If v > p(n) (v > 2 for the Kdhlerian case), then k2 = k~ and M '~+~ is locally isometric to the product L ~ x M~, and codlin M _> v n.

C o r o l l a r y 4.12. Let M '~+~" C i~I be a ruled submanifold with a ruling {L~}, and let there exist a point m E M

such that along any geodesic 7: [0, ~r/ v~'] -+ Lo, 7(0) = m, the following conditions hold: (4.19), (4.28), and

Ih(7',y)l/lYl < v {1 - 5 ( 1 - kl/k ) (a(Lo)21k + 1)}, y �9 T~L • (4.30)

with k = (k., + kl)12. Then v < p(n) or v = 2 and n is divisible by 4 for the Kghlerian case.

Combining the method of 3-dimensional volume of L-parallel fields (see Theorem 3.8) with the synthetic method of Theorem 4.28, we can also obtain a Segre-type decomposition of ruled submanifolds in a Riemannian space similar to Theorem 4.31. However, at present, we cannot obtain a more precise cm'vature pinching.

Next we give similar decomposition and rigidity results for parabolic submanifolds in a Riemannian space of positive sectional curvature. For the nonconstant sectional curvature of/ITI along a submanifold M

k~>_f((x,y)>_]c~ >0, x, y E T M , (4.31)

below we will use the following condition generalizing (4.25) or (4.26):

]]Ar _< v ~ { 1 - 5(1 - kl/k2)(a(5o)2/k + 1)}, ]5] = 1, (4.32)

or the slightly weaker inequality

Ih(x,y)l /( lxl " lyl) -< x/k{1 - 5(1 - k~/k2)(a(~o)2/k + 1)}, x, y �9 TM. (4.33)

The following real a(~) is related to the turbulence from Sec. 3.5.

D e f i n i t i o n 4.16. Let M C 5:I be a complete submanifold with property (4.23), and let k > 0 be a constant.

For the unit normal vector ~ �9 T, ,M • with the condition r(~) = r(M) < dim M, we consider F(m, ~), the

sheaf of geodesics 7: [0, 7r/v~] --+ M (7(0) = m) with the initial directions 7'(0) �9 kerA o and define a real

I(V~,d<(,,)yl + k

a(~) = sup iA~(,)zl ,

where y,z_L kerA~(~), y . L z , [y] = Izl = 1, ~(0) = 4, V~,~(t) = 0, and 7 �9 F(m,~).

We can ~e that a(~) is an analog of the turbulence a(L) for some foliation by generators on a donmin

of M. In view of Corollary 4.2, a(L) is well defined. Combining the method of variations of Theorem 3.6 for foliations with the synthetic method of Theorems

4.29 and 4.30 for parabolic submanifolds in CROSS, we arrive at the following theorem.

T h e o r e m 4.32. Let M "~ C 1~i "~+p be a complete submanifold satisfying conditions (4.23), (4.31), and (4.32) or

(4.33) with #~(M) < ]~2(M) + 1 (in the Khhterian case without inequalities for p~(M)), where k = (k~ + k2)/2

and ~ is any 'unit normal vector having r(~) = r (M) . I f

O < r(M) < ! n - fi(n), - ( n - 1 for the KKhlerian case,

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then k2 = kl, M is locally isometric to the product L n-r(M)/2 x J]/I2 (M)/2 and p > �89 - r ( M ) / 2 ) .

If (4.33) in Theorem 4.32 is given with strict inequality, then we obtain tests of totally geodesic subman- i folds.

�9 C o r o l l a r y 4.13. Let M ~ C t ~ be a complete submanifold satisfying conditions (4.23), (4.31), and (4.33) with

strict inequality (k and ~ are as in Theorem 4.32), and let

r(M) < { n - u(n), - n - 1 for the KShlerian case.

Then M is totally geodesic.

4.4.3. P s e u d o - R i e m a n n i a n i s o m e t r i c i m m e r s i o n s . The universal covering of a complete pseudo- Rieinannian manifold M~ (of dimension n and with a metric of siglaature (p, n - p)) of constant curvature k

can be represented by (a hypersurface in the first and second cases)

p+t q 2 2 ~-~x i ~ v ~ i n ~ n + t i f p # l , k > O , -- Xp+l+j = p+l

i=I j=l p q+l

E x i - E . + j .+1 2 x 2 = i n l r " + l i f q # l , k < 0 ,

i=1 j=l R~ +1 a pseudo-Euclidean space if k = 0.

Hence the case k > 0 is dual to the case k < 0. In [4], tile relative nullity foliation of an indefinite Riemannian manifold isometrically immersed into an

indefinite space form is studied. This foliation is totally geodesic and gives rise to a Riccati-type matr ix ODE along a geodesic in a leaf. This differential equation can be applied in several cases for es t imat ing the index of relative nullity for geodesically complete Lorentz submanifolds in tile Lorentz sphere. In [51, pseudo- Riemannian Kghlerian strongly parabolic embeddings are studied. Iu [140], products of complex submanifolds of indefinite complex space forms are studied, and the criteria for a submanifold corresponding to the SegTe embedding m'e given.

The decomposition results (similar to Corollary 4.11 and Theorein 4.29) are valid for subinanifolds with rulings of a pseudo-Riemannian manifold, which generalize the results given in [4]. We will give t hem in their simplest forms (see [246]). Let M ~ (S~(1)) be a pseudo-Riemannian manifold (a sphere with radius 1) of

dimension m with metric of signature (p, m - p).

T h e o r e m 4.33. Let /}ip "+~ C Sq(1) be a ruled submanifold with geodesically complete rulings and the

following condition on its second quadratic forms:

(A~x) 2 >_ O, (A~x) 2 + (Acu) 2 < 1 (x, u E TL, ~2 = x 2 = ,u 2 = 1, (x, u) = 0).

Y I f v >_ p + fi(n), then Mp +~ is locally isometric to the product Sp x B~, and m >_ n + v + vp, q > n p holds;

if, in addition, m = n + v + vp and q = n p , then the submanifold M~ +~ is congruent to a domain of the

pseudo-Riemannian Segre embedding Sp x S'~ C ,q"+~+~ --rip

T h e o r e m 4.34. Let M~ C S ~+P(1) be a geodesically complete submanifold satisfying one of the follovzing

conditions: (a) (A~x) 2 >0 , (A~x) 2 +(A~y) 2 <1 ( ~ 2 = x 2 = y 2 = 1 , ( x , y ) = 0 ) ,

(b) 0 < K ( x , y) < 1, for (b) (c): #~(M) < #2(M) + 1, (c) K ( x , y) _< 1, Ric (x) _> n - - 2,

where x 2 = y2 = 1, (x ,y) = O, and the restriction of the metric on the plane kerA{ (~2 = 1, r(~) = r ( M ) )

has signature (p, n - r (M) - p). I f r (M) < n - p - f ( n - p), then M~ is locally isometric to the product

(a), (b) S p -r(M)/2 x Bo (M)/2, (c) Sp -1 x B t, r ( M ) ----- 2

1 7 6 6

and p k �89 - r(M)/2), q k �89 Pr(M). If, moreover, p : �89 - r(M)/2) and q : �89 then the submanifold M is congruent to the pseudo-Riemannian Segre embedding.

The existence of a ruled structure on a parabolic submanifold M n with a definite metric in a pseudo- Riemannian space form is proved in [35]. Note that a pseudo-Riemannian manifold whose sectional curvature

preserves its sign actually has a constant curvature (see [35]). The indefinite isometric immersions of constant curvature were studied by many other authors.

T h e o r e m 4.35 [35]. (a) A pseudo-Riemannian manifold Mp of constant negative sectional curvature cannot

be locally isometrically embedded into R27 -2, where n - p = 2 n - 2 - p ' . (b) A complete pseudo-Riemannian

manifold M~ of constant negative sectional curvature cannot be isometrically embedded into R~? -1, where

n - p = 2 n - 1 - p ' # 0, 1,3,7.

E x a m p l e 4.3 [35]. The local isometric immersion of Mp of constant negative curvature into Rp2, ~-~, where

n - p = 2n - 2 - p ' , is given by formulas

{ aiexp(un) cos(~) if 1 < i < p - 1, x2i-1 = ai exp(un)cosh(-~) if p < i < n - 1,

: t l i

aiexp(un) s in(~) if 1 < i < p - 1, x a i

x2i = a iexp(un)s inh(~) if p < i < n - 1,

ltr~

X2i-1 = f V/1 - exp(2t) dt, 0

2 1. where xi are coordinates in [r and E i ai =

The p r o o f of Theorem 4.35 is based on the Otsuki lemma (see Sec. 4.2.2). Obviously, Theorem 4.35 is also valid for n - p = 0, 1, 3, 7 [135]; this is connected with the fact that the parallelizable spheres have dimensions 1, 3, and 7.

4.5. R u l e d S u b m a n i f o l d s w i t h C o n d i t i o n s o n t h e M e a n C u r v a t u r e

Some authors studied ruled submanifolds in space forms with additional conditions on its mean-curvature

vector. The well-known theorem by Catalan states that the only ruled nfiifimal surfaces in ~{3 are the plane

and the helicoid (see [170] for the case of S 3 and [55] for hyperbolic space Ha). These classical results can be generalized in the following three ways.

(1). The cl~sification of minimal ruled submanifolds M ~+" with n = 1 in space forms was obtained

independently in [175], [282], and [16] (for hypersurfaces in R N see [27]). It is shown in these works that the minimal ruled submanifolds are generalized helicoids. This extends the Catalan theorem in one direction.

Def in i t ion 4.17 [16]. Let c = c(t) be a curve in a space form AJ/(k) of constant curvature and with the Fren4t

frame e l , . . . , e,~. We set v = [~] and define the' map f: R "+1 -+ l~I(k) by f(s, h , - - . , t , ) ---- e x p ~~jv__ 1 tje2j(s),

which is called (in any parametrization) the helicoid associated with the curve c.

L e m m a 4.10 [16]. The helicoid f associated with a curve c: R --+ FI(k) describes a minimal immersion,

wherever it is regular. The helicoid f admits a one-parameter subgroup A(s) of rigid motions of IVI(k) such that

d ( s ) f ( t , h , . . . , t ~ ) = f ( s + t , h , . . . , t ~ ) , s , t , h , . . . , t ~ E R .

However, a ruled minimal immersion invariant for a one-parameter subgroup A(s) of rigid motions of

~i(k) is not necessarily a helicoid.

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Defin i t ion 4.18 [16] Let A(s) be a one-parameter subgroup of rigid motions of ~I(k), and let S be a u-

dimensional complete totally geodesic submanifold of ~I(k) orthogonal to the orbits of A(s). Then the map

f: R x S --+ IVI(k) defined by f (s ,p) = A(s)p describes a ruled immersion, whenever it is regular. These maps

(minimal whenever they are regular) are called generalized helicoids.

Note that a generalized helicoid in a sphere or a hyperbolic space is a restriction of a certain submanifold

in Euclidean space. The classification of generalized helicoids in IIr N+~ up to the motion of the ambient space is as follows [16]:

' r /~--T"

f ( s , = t,e (s) + + i=1 i = l

where the vectors VI , . . . , V~v+l constitute some special orthonormal basis of R N+', the fields ei(s) are given

by ei(s) = cos ais V2i-1 + sin ais ~ i (1 < i < u), and a l , . . . , a~, b; s, t l , . . . , t~ are real numbers and variables.

A similar result is valid for H N+I.

T h e o r e m 4.36 [16]. Let M ~+~ be a minimal ruled submanifold of ~IU (k). Then there exist a generalized

helicoid f: R x .~-l~'(k) --+ kiN(k) and an open set U C R x ~I'~(k) such that f restricted to U parametrizes M.

Consider a generalization of helicoids in which, instead of "screw lines," "nicely curved submanifolds" in space forms are used.

The notion of a (locally) symmetric submanifold M of the Euclidean space R '~ is given in [88] as a

submanifold (locally) mapped into itself for each x E M under the reflection of ~ with respect to the (affine) normal space to M at x. Locally symmetric submanifolds of an arbitrary Riemanniau space are studied in [277]

Def in i t ion 4.19. A submanifold M C .([(c) is (locally) symmetric if, for every x E M, M is (locally) mapped

into itself by the geodesic reflection of ~-l(c) with respect to the subspace F~ = exp~(T~M• This F~ is called the submanifold of symmetry.

In [163], the k-symmetric submanifolds of R '~ (k > 2) are defined by using suitable isometries of ~ . The authors of [163] observed that a 2-symmetric submanifold M is invariant under the reflections of 1r with respect to subspaces of the normal spaces of M; therefore, 2-symmetric submanifolds appear as a generalization of symmetric submanifolds.

Def in i t ion 4.20. A submanifold M C l~I(c) is 2-symmetric if, for every x E M, there exists an involutive

isometry a~ of .([(c) such that, locally, M is mapped by a~ into itself and x is a fixed point of cr~ isolated on M.

It is easy to verify that a totally geodesic submanifold of a symmetric submanifold is a 2-symmetric submanifold, but it is not a symmetric submanifold in general.

1 0 The vector space N~ M generated by values of the second fundamental form h =h at a fixed point x E M

is called the first normal space of M at x. Then the kth normal space N~ M at x E M is the vector subspace k- - I

of T~fJ generated by values of the (k - 1)th fundamental form h of M:

k--1 k - 1 1 k - I h : ~ M x ~ M ~ ( ~ M ~ M @ . . . ~ M ) •

k k Note that for k _> l, we have h = 0; for k _< l, we have N , M ~ 0, l is the maximal value of k such

k k that N~ M ~ 0 and the vector spaces N~ M are mutually orthogonal. For the submanifold M, we set

2i+ i lz~ q = ~i--o N~ M, where q = [/-~-].

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T h e o r e m 4.37. If M is a 2-symmetric submanifold in kI(c), then

k ~ ' h = 0 for each k. (4.34)

k D e f i n i t i o n 4.21. A submanifold M C ~I(c) is nicely curved if dim Nx M does not depend on x E M for all k E N .

Recall that a submanifold M of the space form 1~I(c) is said be essential if it is not contained in any

proper totally geodesic submanifold of/17I(c).

L e m m a 4.11. If M is a connected nicely curved submanifold of a real space form 1~I(c), then M is an i

essential submanifold of a totally geodesic submanifold of ~l(c) whose dimension is equal to dim(O~=0 N~ M). i

In particular, the normal space T~M J- of M at x is given by TxM -L --- @~=1 N~ M.

D e f i n i t i o n 4.22. A submanifold M of/17I(e), which is a tubular neighborhood of a nicely curved M satisfying

condition (4.34) in the set (exp~ v}, x E M, v E V~, is foliated by totally geodesic submanifolds of 57/(c) and is called the multihelicoid.

If dim M = 1, then condition (4,34) implies that M is just a curve of constant curvature, and ~I becomes a helicoid associated with a curve in the sense of Definition 4.17.

z

T h e o r e m 4.38 [182]. Any multihelieoid f/l in ~l(c) over a nicely curved submanifold M satisfying condition (4.34) is minimal.

For a nicely curved submanifold M in 2~I(c), we can consider the map L,: M -+ G(p, rh - p) into the 2i-t-1

Grassmannian manifold, where p = dim(~q=0 N~ M), given by the formula

L/(X) = q ( . (4.35) ~i=0 N~ M x E M, q =

Recall that in any Grassmannian manifold G(p, q), there exist flat totally geodesic submanifolds in each dimension m < min(p, q - p). If H is a flat totally geodesic submanifold of G(p, q), then the map exp,:

T~,H --~ H, a E H, is a totally geodesic (local) isometry (see [184]). If we identify ~m with T~H, then exp~ yields a totally geodesic map of ]~m into G(p, q).

Recall that the map f : M --+ N between two manifolds is totally geodesic if the covariant derivative ~7(df) of the differential df is zero (see [78] and Ex~nple 2.1).

T h e o r e m 4.39 [184]. Let M be a complete connected nicely curved submanifold of dimension m in R "-~, and let ~: M --+ G(p, rh - p ) be a totally geodesic map defined by (4.35). Then

(i) i f dim ker dp = 0, then v( M) is a complete totally geodesic submanifold of G(p, rh - p ) , (ii) if dim ker dv ~ O, then ker d~ defines a foliation on M whose leaves are Euclidean spaces of dimension

r; moreover, the leaf space B is a complete totally geodesic submanifold of G(p, Vn - p), the map L, factors into a Riemannian submersion with totally geodesic leaves, ~: M --+ B, followed by a totally geodesic immersion j : B ~ G(p, r'n - p) ( B is a totally geodesic submanifold of G(p, f i t - p) and its connection coincides with the one induced by j), and the fiber space M --+ B has a fiat connection with totally geodesic horizontal fibers,

(iii) M has nonnegative curvature and is locally symmetric.

For a submanifold M C M with a ruling L C M, we can consider a geodesic reflection of ~I with respect to L that maps a neighborhood of L in M back into M.

D e f i n i t i o n 4.23 [182, 183]. A ruled submanifold M ~+'~ is symmetric if the geodesic reflection of M with respect to every ruling L maps some neighborhood of L in M back into M and maps the rulings into rulings.

1769

Since a geodesic reflection of a space form ~l(c) with respect to a total ly geodesic submanifold is an

isometry (rigid motion), a sufficient condition for a ruled submanifold M C tffI(c) to be symmetric is tha t M be locally mapped into itself by a geodesic reflection with respect to each of its rulings.

Any multihelicoid M associated with a nicely curved 2-symmetric submanifold in 1~[(c) is also a symmetric

ruled submanifold in tile sense of Definition 4.23 [183]. The following theorem partially answers the question of how symmetric ruled submanifolds in space forms can be classified.

T h e o r e m 4.40 [183]. A symmetric ruled submanifold M ~+'~ in a space form l~I(c) is a minimal submanifold.

(2) . On the other hand, minimal submanifolds of a Euclidean space can be regarded as a special case of

submanifolds of finite type, which were introduced by B.-Y.Chen in 1984 (for a recent survey, see [62]).

D e f i n i t i o n 4.24 [62]. A submanifold M C ]~N is of finite type if every coordinate of its position vector field f can be writ ten as a finite sum of eigenfunctions of the Laplacian A of M, i.e., f --- f0 + fl + - . . + fro, where f0 is a constant vector and A f i = Aifi, 1 _< i _< m. If, in particular, all eigenvalues {AI, A2, . . . , Am} are pairwise different, then M is of m-type.

Note that every minimal submanifold of a Euclidean space is of 1-type (with A1 = 0), since A f = 0.

Also, we note that a plane curve of finite type is a part of a circle or a straight line. In [71], the Catalan

theorem was extended to ruled submanifolds M ~+~ of a finite type with n = 1 in 1r N.

T h e o r e m 4.41 [71]. A ruled submanifold M ~+1 in R N is of finite type if and only i f M ~+1 is a part of a

cylinder on a curve of finite type or a part of a generalized helicoid. In particular, a ruled submanifold M ~+l

in R~+2 is of finite type if and only i f M ~+~ is a part of a hyperpIane, a circular cylinder, a helicoid F 2 in ~3,

a cone K 3 with vertex p on a minimal ruled surface in a sphere S 3 centered at p, or a cylinder over F 2 or t ( 3 .

Finite-type ruled submanifolds modeled on the basis of spherical submanifolds are studied in [89]. The

test for a finite-type Gauss map for ruled submanifolds is given in [12].

T h e o r e m 4.42 [12]. The only ruled submanifolds M ~+1 in the Euclidean space ~+P with finite type Gauss

map are cylinders over curves of finite type and (~ + 1)-dimensional Euclidean spaces.

(3) . The only ruled submanifolds M "+1 of Ir g of constant nonzero mean curvature are generalized

cylinders, i.e., products of the form R ~ x c(t) C R ~ • R g-~, where c(t) C R g-~ is a curve of constant first

curvature [16]. For k < 0, there are no ruled submanifolds M "+1 of l~I(k) of constant nonzero mean curvature

[16]. If k > 0, the situation is quite different.

T h e o r e m 4.43 [15]. (a) I f M ~+1 is a ruled submanifold in f iN(k) , ]c > 0, of constant nonzero mean

curvature, then N >_ 2v+ 1 and M is locally isometric to the product R x 2~?I'(k). (b) I f f: M T M -+ $2~+1(k)

is a ruled submanifold of a mean curvature of nonzero constant length, then ~ + 1 is even. Given a > O, for any complete smooth curve c in the symmetric space O(2p)/U(p) , there exists an isometric 7nLled immersion

f~,~: R x S 2p-1 --+ S 4p-1 of a mean curvature of constant length a which is well defined up to a rigid motion

o r s ap'l. Moreover, if c and cl coincide in a rigid motion of O(2p)/U(p), then fc,~ and f~,,a are the same up

to a rigid motion of S 4p-1.

Thus, we have a complete classification of ruled submanifolds M ~+1 of codimension v in space forms with mean curvature of constant length. If, in Theorem 4.43, the mean-curvature vector is parallel or the normal

bundle is trivial, then ~ = 1 and M '2 is the product of two circles in S 3 C S ~.

4.6. Subman i fo l d s w i t h S p h e r i c a l G e n e r a t o r s

Let us consider subinanifolds with generators that are totally umbilic in an ambient space.

D e f i n i t i o n 4.25 [223]. A 1-form A on T M • is called the principal curvature of a submanifold M C M if

Tin(A) = {x e TraM: A~x = A(~)x, ~ E T,~M • is at least a 1-dimensional subspace for all m E M.

1770

Let G(A) C M be an open regularity domain, where dimTm(A) takes its minimal value #(A, M). The principal-curvature function A is smooth on G(A).

T h e o r e m 4.44 [223]. Assume that the principal curvature function A of a (complete) submanifold M C l~f(k) with #(A, M) > 1 is given. Then the subspaces TIn(A) form an integrable distribution on G(A) whose leaves

{L(A)} are (complete) totally umbiIic submanifolds (spherical generators) in FI , and, therefore, in M.

There exist isometric immersions without principal curvature, for instance, the immersion of the Veronese surface into S 4. The principal curvat~u'es for hypersurfaces in a conformally flat Riemannian space are studied in [139]. For recent studies of completeness of curvature surfaces of submanifolds in complex space forms and pseudo-Riemannian spaces, see [32].

Coro l l a ry 4.14 [223, 203]. Let M(k) C M ( k ) be an isometric immersion of space forms under the conditions

k > k and dim M < dim M _< 2 dim M - 2. Then, at every point m E M, there exists exactly one principal curvature Am with dim T(Am) > 2; moreover, dim T(Am) > 2 dim M - d i m ATI+l > 3 (for the case dim M + 2 =

dim/~/, see [132]). All statements of Theorem 4.44 can be applied to this principal-curvature function.

Conformally fiat submanifolds of small codimension possess a similar property. Conformally fiat hy- persurfaces are naturally connected with other classes (defined extrinsically) such as special quasiumbilic hypersurfaces, loci of n-spheres, and canal hypersurfaces [60]. The local structure of conformally flat hyper- surfaces was discovered in [57]: they are generically foliated by codimension-1 spheres.

Def in i t ion 4.26 [192]. A submanifold M "~ of ]~m+p p < m, is generically foliated by spheres if there exists a dense open set U C M such that U -- U~=0 Ui, where Ui (for every i) is an open set of h i that is foliated by (m - p +/ ) -d imensional umbilic submanifolds of ~m+p. We denote this class of submanifolds by 9vS~. The points of U are generic points. A submanifold M is strongly generically foliated by spheres if M E .T'S p and

the restriction of T M • to each leaf S of U is parallel in T S • We denote this class of submanifolds by S.TS~.

This definition implies, in particular, tha t the leaves in Ui are open sets of s tandard spheres Sm+P+t Let C9~-~ be the class of conformally fiat submanifolds of R m+p. In view of Theorem 4.44, we have the following theorem.

T h e o r e m 4.45 (local) [192]. /f 1 <_ p <_ rn - 3, then C ~ C S .~S p.

T h e o r e m 4.46 (global) [192]. Let M ~n be an orientable conformally fiat submanifold of ~ m+p (p < m - 3). If M is k-regular (i.e., the leaves b, ave dimension k everywhere and form a regular foliation on M, see Theorem 4.45), then M is a sphere bundle (the fibers are spheres) over an (m - k)-dimensional manifold.

For the classification of hypersurfaces in Theorem 4.46, see [53].

E x a m p l e 4.4 [192]. Consider a standard immersion of the sphere Sin-2(1) into R m-~ and any isometric

immersion of the compact surface H2(-1) of curvature - 1 into ]~z (by the Nash theorem). The product Sin-2(1) • ~2( -1) is then isometrically immersed into R m+~6. If m _> 19, we obtain the situation of Theo- rem 4.46.

The following theorem generalizes the classical investigations by Riemann [231] (a minimal surface in R :3

that is foliated by circles in parallel planes must be either a piece of a catenoid or the example now called the "Riemann staircase") and Enneper [79] ( i f a minimal surface in ~3 is foliated by circles or by their arcs, then the planes containing these curves must be parallel).

T h e o r e m 4.47 [143]. I f M T M is a complete and nonplanar minimal hypersurface in ~+2 (v >_ 2) and an open subset of M is foliated by pieces of u-dimensional spheres, then M is a higher-dimensional catenoid ( a

hypersurface of revolution).

1771

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