Table Of ContentDedicated to my parents
Vladimir Y. Rovenskii
Foliations on Riemannian
Manifolds and Submanifolds
Birkhauser
Boston • Basel • Berlin
Vladimir Y. Rovenskii
Mathematics Depanment/Geometry Chair
Pedagogical Institute
Krasnoyarsk 49
660049 Russia
Library of Congress Cataloging In-Publication Data
Rovenskii. Vladimir. 1953-
Foliations on Riemannian manifolds and submanifolds 1 Vladimir
Rovenskii.
p. cm.
Includes bibliographical references and indel(.
ISBN-13: 978-1-4612-8717-9
(hardcover acid-free)
I. Riemannian manifolds. 2. Foliations (Mathematics) I. Tille
QA613.62.R68 1997 96-5737
516.3·73--dc20 CIP
AMS Classification: 52, 51, 53, 57
Printed on acid-free paper
~
" 1998 Birkhallser Boston Birkhiiuser
Softcover reprint oftre hardcover 1st edition 1998
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ISBN-13: 978-1-4612-8717-9 e·ISBN·13: 978·1-4612-4270·3
DOl: 10.1007/978-1·4612-4270·3
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CONTENTS
Preface vii
I. Foliations on Manifolds 1
1.1 Definitions and examples of foliations 1
1.2 Holonomy . . . . . . 13
1.3 Ehresmann foliations 19
1.4 Foliations and curvature 25
II. Local Riemannian Geometry of Foliations 31
2.1 The main tensors and their invariants 31
2.2 A Riemannian almost-product structure 36
2.3 Constructions of geodesic and umbilic foliations 40
2.4 Curvature identities 48
2.5 Riemannian foliations 55
III. T -Parallel Fields and Mixed Curvature 64
3.1 Jacobi and Riccati equations . . 64
3.2 T -parallel vector fields and the Jacobi equation 73
3.3 L-parallel vector fields and variations of curves 77
3.4 Positive mixed curvature 79
IV. Rigidity and Splitting of Foliations 95
4.1 Foliations on space forms . 96
4.2 Area and volume of a T -parallel vector field 98
4.3 Riccati and Raychaudhuri equations 116
v.
Submanifolds with Generators 129
5.1 Submanifolds with generators in Riemannian spaces 129
5.2 Submanifolds with generators in space forms . . . 137
5.3 Submanifolds with nonpositive extrinsic q-Ricci curvature 151
5.4 Ruled submanifolds with conditions on mean curvature 164
5.5 Submanifolds with spherical generators ...... 170
vi Contents
VI. Decomposition of Ruled Submanifolds . . . . . . . . 175
6.1 Cylindricity of submanifolds in a Riemannian space
of nonnegative curvature . . . . . . . . . . . . 176
6.2 Ruled submanifolds in CROSS and the Segre embedding 183
6.3 Ruled submanifolds in a Riemannian space of positive
curvature and Segre type embeddings 192
VII. Decomposition of Parabolic Submanifolds 201
7.1 Parabolic submanifolds in CROSS 201
7.2 Parabolic submanifolds in a Riemannian space
of positive curvature ........... 211
7.3 Remarks on pseudo-Riemannian isometric immersions 215
Appendix A Great Sphere Foliations and Manifolds
with Curvature Bounded Above 218
Al Great circle foliations 218
A2 Extremal theorem for manifolds
with curvature bounded above 223
Appendix B. Submersions of Riemannian Manifolds
with Compact Leaves .... 235
Appendix C. Foliations by Closed Geodesics
with Positive Mixed Sectional Curvature 247
References 255
Index 283
PREFACE
This monograph is based on the author's results on the Riemannian ge
ometry of foliations with nonnegative mixed curvature and on the geometry of
sub manifolds with generators (rulings) in a Riemannian space of nonnegative
curvature. The main idea is that such foliated (sub) manifolds can be decom
posed when the dimension of the leaves (generators) is large. The methods of
investigation are mostly synthetic.
The work is divided into two parts, consisting of seven chapters and three
appendices. Appendix A was written jointly with V. Toponogov.
Part 1 is devoted to the Riemannian geometry of foliations. In the first few
sections of Chapter I we give a survey of the basic results on foliated smooth
manifolds (Sections 1.1-1.3), and finish in Section 1.4 with a discussion of the
key problem of this work: the role of Riemannian curvature in the study of
foliations on manifolds and submanifolds.
Chapter II contains local notions and results not only for a foliated Rie
mannian manifold, but also for the slightly more general situation of a pair
of complementary orthogonal distributions. The latter is sometimes called an
"almost-product structure." Beginning with the well-known "co-nullity opera
tor" of D. Ferus, P. Dombrowski and others, we introduce in Chapter II a pair of
structural tensors, which 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 II concludes with an overview of 1) integral formulas
containing the mixed curvature along a compact total manifold or a compact
leaf, 2) recent studies of Riemmanian foliations.
The next two chapters are devoted to the case which is local on transversal
directions, i.e., a foliation is given on a neighborhood of one complete leaf.
Chapter III starts off with Ferus's result (1970) on the optimum (largest)
dimension of a totally geodesic foliation with constant positive mixed sectional
curvature on a given manifold. It contains a surprisingly deep relationship
betweeen the Riemannian geometry of foliations and the topological notion of
the number of vector fields on an n-sphere. In Section 3.4 we discuss Ferus's
scheme: if the dimension of a (totally geodesic) foliation is "large," then there
viii Preface
exists an L -parallel (Jacobi) vector field (i.e., induced by a foliation along some
leaf geodesic) with a parallel initial value and first derivative. This is impossible
in the case of constant curvature. From this point of view Chapter III is devoted
to results on vector fields that are induced by a foliation or by a distribution
given on a Riemannian manifold.
For non-constant positive mixed curvature, the dimension of a compact
foliation is easily estimated by half of the totally geodesic dimension of the
total manifold. The idea (by T. Frankel) is that two compact submanifolds (for
instance, great spheres in a round sphere) in a Riemannian space of positive
curvature must intersect if the sum of their dimensions is not less than the
dimension of the total space. In Section 3.4 we extend this result to the case of
positive partial mixed Ricci curvature.
The problem by V. Toponogov is to obtain a Ferus type estimate for the
dimension of a foliation on a (compact) Riemannian manifold with positive
mixed curvature. Chapter IV contains the author's results on the above prob
lem. In the beginning of Section 4.1 a brief survey of foliations on space forms
is given. In Section 4.2 we introduce a new variation procedure based on the
volume 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
nonnegative mixed curvature, and in particular, to generalize Ferus's result. In
Section 4.3 the Riccati equation procedure is extended to foliations. Combining
some ideas, the integral inequality with mixed scalar curvature along a complete
leaf is obtained. The case of when a structural tensor of a foliation is almost
symmetric or skew-symmetric (i.e., the norm of the second fundamental form
of the leaves or the norm of an integrability tensor for horizontal distribution is
bounded above by some term with mixed Ricci curvature) is also studied.
In Part 2 we consider a popular class of submanifolds equipped with the
additional structure of a foliation whose leaves (generators) are totally geodesic,
umbilic etc., in an ambient Riemannian space. The ruled, canal and tubular
submanifolds (well-known in space forms) are intrinsically geodesic or umbilic
foliations.
Systematic investigations of the local and global structure of submanifolds
in (pseudo -) Riemannian spaces include a study of the relationships between
intrinsic and extrinsic properties, the tests for totally geodesic, cylindrical sub
manifolds, estimates of codimension, etc.
Attention given to foliated submanifolds has increased due to studies of
Preface ix
some special embeddings with degenerate second fundamental form: (strongly)
parabolic, k-saddle, having non positive extrinsic curvature, small codimension,
and others.
During the years 1960--1970 it was popular to examine submanifolds Min
a Riemannian space with positive relative null-index J.1(M). Different names
for these sub manifolds with constant nullity, strongly parabolic, with constant
rank, tangential degenerate, and k-developable were introduced by geometers.
In the case of a curvature invariant sub manifold (for example, when the am
bient manifold is a space form) the regularity domain on M, which consists of the
points with minimal value of relative null-index, has a ruled developable struc
ture (with constant Kmix in the case of a space form), whose J.1(M)-dimensional
rulings are tangent to the relative nullity distribution.
The k-saddle submanifolds and the submanifolds with nonpositive extrinsic
curvature and small codimension are important examples of strongly parabolic
submanifolds.
More general parabolic submanifolds, which were first introduced in space
forms by A. Borisenko in 1972, have a large number of rulings under a certain
condition on the curvature tensor of an ambient Riemannian space.
Part 2 begins with the bound (from above) on the dimension of a complete
ruling on a ruled submanifold in a round sphere as half of the dimension of the
ambient space; the best (Ferus's type) estimate holds in the case of a developable
ruled submanifold. Then we continue the investigation of Toponogov's problem
in the case of foliated submanifolds.
In Chapter V a detailed survey of all studies of ruled submanifolds in space
forms and in arbitrary Riemannian spaces is given as well as the concepts and
facts necessary for the subsequent chapters. We introduce the new (syntheti
cally defined) class of uniquely projectable submanifolds along generators in
a Riemannian space, which is deeply related to the class of ruled submanifolds
with nonnegative (positive) mixed curvature. Also, in Section 5.3, we extend
some recent results on submanifolds and foliations with nonpositive extrinsic
sectional curvature for the case of extrinsic partial Ricci curvature.
The central result of Chapter VI is that a ruled submanifold in a sphere or
in a complex projective space with a ruling of "large" dimension and a "small"
norm of its second fundamental form is congruent to the Segre embedding;
the latter plays the role of a "cylinder" in a space form of positive curvature.
Combined with our method for the volume of an L -parallel vector field (from
Chapter IV) we obtain the test for the Segre type decomposition of a ruled
x Preface
submanifold in a Riemannian space of positive curvature.
In Chapter VII, which is based on ideas from Chapter VI, we obtain for
the first time the Decomposition Theorems for parabolic submanifolds (Le.,
with degenerate second quadratic forms). The central result is that a complete
parabolic submanifold in a sphere or in a complex projective space with "small"
rank and a "small" norm of its second fundamental form is congruent to the Segre
embedding. Combined with our method for a volume of an L -parallel vector
field, as in Chapter VI, we obtain the test of Segre type decomposition of a
parabolic submanifold in a Riemannian space of positive curvature.
Appendix A is devoted to Toponogov's conjecture: if a complete simply
:s
connected Riemannian manifold with sectional curvature 4 and injectivity
I I'
radius::: has extremal diameter then it is isometric to CROSS (Le., com
pact rank one symmetric spaces). In Section A.l the relationship between this
problem and geodesic foliations of a round sphere (in particular, skew-Hopf
foliations) is considered, but the proof of the conjecture along these lines is in
complete. In Section A.2 the proof is given, based on recent results, topological
methods, and estimates of volumes for Blaschke manifolds.
Appendix B contains the author's results on submersions where some con
cepts of submanifolds with relative nullity are used.
In Appendix C we construct an even-dimensional Riemannian manifold
with fibration on closed geodesics and (0 :::::: 1)-pinched mixed sectional cur
vature. This example is important in view of results in Chapters IV, VI, VII.
Acknowledgments. Victor Toponogov (Novosibirsk), for scientific and hu
man help for many years, Professor Pawel Walczak and other colleagues from
the Polish Academy of Sciences for discussions on the results from the Work
shop "Foliations: Geometry and Dynamics" (1995, Warsaw), Robert Wolak
(Krakow) for a useful discussion and improvements concerning the entire text,
colleagues and friends from Novosibirsk, Kazan, Bamaul, Kharkov and Kras
noyarsk. Finally, I would like to warmly thank Ann Kostant for human support
and help in the publishing process. My apologies for any remaining errors due
to language and thanks to all the copy editors who tried to make the book as
readable as possible.
Vladimir Rovenskii
1997
