Riemannian Foliations
Foliation theory has its origins in the global analysis of solutions of ordinary differential equations: on an n-dimensional manifold M, an [autonomous] differential equation is defined by a vector field X ; if this vector field has no singularities, then
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Series Editors 1. Oesterle A. Weinstein
Pierre Molino
Riemannian Foliations Translated by Grant Cairns With Appendices by G. Cairns Y. Carriere E. Ghys E. Salem V. Sergiescu
1988
Birkhauser Boston . Basel
Pierre Molino Universite des Sciences et Techniques du Languedoc Institut de Mathematiques 34060 Montpellier Cedex France
Grant Cairns (Translator) Department of Pure Mathematics University of Waterloo Waterloo, Ontario, N2L 3Gl Canada
Library of Congress Cataloging-in-Publication Data Molino, Pierre, 1935Riemannian foliations / Pierre Molino; translated by Grant Caims ; with appendices by Grant Caims . . . lct a1.1. p. cm - (Progress in mathematics;v.73) Translation Bibliography: p. Includes index. ISBN 978-1-4684-8672-8 ISBN 978-1-4684-8670-4 (eBook) DOI 10.1007/978-1-4684-8670-4 I. Foliations (Mathematics) 2. Riemannian manifolds. I. Title. 11. Series: Progress in mathematics (Boston, Mass.);vo!. 73. QA6I3.62.M65 1988 514'.72-dcl9 87-29963
CIP-Kurztitelaufnahme der Deutschen Bibliothek Molino, Pierre: Riemannian Foliations / Pierre Molino. Trans!. by Grant Cairns. With app. by Grant Cairns ... - Boston; Basel: Birkhäuser, 1988. (Progress in mathematics ; Vo!. 73) NE:GT © Birkhäuscr Boston, 1988 Softcover reprint ofthe hardcover 1st edition 1988 All rights reserved. No part of this publication may bc reproduced. stored in a retrieval system. or transmitted, in any form or by any means, electronic, mcchanical. photocopying, recording or otherwise. without prior permission of the copyright owner.
Text prepared by the translator using an APOLLO word processor.
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Table of Contents
1
Elements of Foliation theory
1
1.1. Foliated atlases; foliations
1
1.2. Distributions and foliations
4
1.3. The leaves of a foliation
9
1.4. Particular cases and elementary examples
14
1.5. The space of leaves and the saturated topology
18
1.6. Transverse submanifolds ; proper leaves
20
and closed leaves 1.7. Leaf holonomy
22
1.8. Exercises
29
2
33
Transverse Geometry
2.1. Basic functions
34
2.2. Foliate vector fields and transverse fields
35
2.3. Basic forms
38
2.4. The transverse frame bundle
41
2.5. Transverse connections and G-structures
48
2.6. Folklted bundles and projectable connections
53
2.7. Transverse equivalence of foliations
61
2.8. Exercises
65
- vi -
3
Basic Properties of Riemannian Foliations
69
3.l. Elements of Riemannian geometry
69
3.2. Riemannian foliations: bundle-like metrics
76
3.3. The Transverse Levi-Civita connection and
80
the associated transverse parallelism
3.4. Properties of geodesics for bundle-like metrics
86
3.5. The case of compact manifolds : the universal
87
covering of the leaves
3.6. Riemannian foliations with compact leaves
88
and Satake manifolds
3.7. Riemannian foliations defined by suspension
96
3.8. Exercises
99
4
Transversally Parallelizable Foliations
103
4.1. The basic fibration
104
4.2. Complete Lie foliations
110
4.3. The structure of transversally parallelizable foliations
117
4.4. The comm
