Progress in Mathematics Izu Vaisman Symplectic Geometry and Secondary Characteristic Classes Progress in Mathematics Volume 72 Series Editors J. Oesterle A. Weinstein lzu Vaisman Symplectic Geometry and Secondary Characteristic Classes Springer Science+Business Media, LLC 1987 Izu Vaisman Department of Mathematics University of Haifa Mount Carmel, Haifa 31 999 Israel Library of Congress Cataloging-in-Publication Data Vaisman, lzu. Symplectic geometry and secondary characteristic; classes I Izu Vaisman. p. em.-(Progress in mathematics;v. 72) Bibliography: p. Includes index. ISBN 978-1-4757-1962-8 I. Geometry, Differential. 2. Characteristic classes. 3. Maslov index. I. Title. II. Series: Progress in mathematics (Boston, Mass.);vol. 72. QA649. V284 1987 516.3'6-dcl9 87-19855 CIP-Kurztitelaufnahme der Deutschen Bibliothek Vaisman, Izu: Symplectic geometry and secondary characteristic classes I Izu Vaisman. (Progress in mathematics ; Vol. 72) ISBN 978-1-4757-1962-8 NE:GT © Springer Science+Business Media New York 1987 Originally published by Birkhäuser Boston, in 1987 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the copyright owner. ISBN 978-1-4757-1962-8 ISBN 978-1-4757-1960-4 (eBook) DOI 10.1007/978-1-4757-1960-4 Text prepared by the author in camera-ready form. 9 8 7 6 5 4 3 2 l FOREWORD The present work grew out of a study of the Maslov class (e.g. (37]), which is a fundamental invariant in asymptotic analysis of partial differential equations of quantum physics. One of the many in terpretations of this class was given by F. Kamber and Ph. Tondeur (43], and it indicates that the Maslov class is a secondary characteristic class of a complex trivial vector bundle endowed with a real reduction of its structure group. (In the basic paper of V.I. Arnold about the Maslov class (2], it is also pointed out without details that the Maslov class is characteristic in the category of vector bundles mentioned pre viously.) Accordingly, we wanted to study the whole range of secondary characteristic classes involved in this interpretation, and we gave a short description of the results in (83]. It turned out that a complete exposition of this theory was rather lengthy, and, moreover, I felt that many potential readers would have to use a lot of scattered references in order to find the necessary information from either symplectic geometry or the theory of the secondary characteristic classes. On the otherhand, both these subjects are of a much larger interest in differential geome try and topology, and in the applications to physical theories. For all these reasons it seemed to me appropriate to give an exposition of sym plectic geometry and of the general and specific theory of secondary characteristic classes (including the Maslov class) under the form of a monograph, which we bring now before the readers. Our approach to the subject is that of differential geometry, While the motivation for the study of the Maslov class comming from analysis and physics is definitely more important, there is also a geometric moti- vation for this study, and this geometric motivation also asks for the study of appropriate general secondary characteristic classes. Accord ingly, we shall not discuss the asymptotic analysis related to the Maslov class (the interested reader may find it, for instance, in [37)), but we motivate our work by the following geometric problem. Let rr: E ~ M be a symplectic vector bundle, and let L0, L1 be two Lagrangian subbundles of E. We want to study conditions for L and L to be transversal 0 1 subbundles i.e., E = L0 ~ L1, and to measure nontransversality by cor responding obstructions. The theory which we obtain thereby involves a more general case than the one of Kamber-Tondeur [43), which occurs if the bundle E is trivial. Much more important is the fact that the Lagrangian transversality problem is not only of a geometric interest. Indeed, this problem appears in the study of Hamilton-Jacobi equations, and, therefore, in the study of the equations of quantum physics, and this is a basic reason for the study of the transversality of Lagrangian sub bundles. Let us also notice that (as we shall point out at the end) the characteristic classes of the triple (E,L0,L1) can also play an important role in Lagrangian cobordism theory [4], which is an important part of Lagrangian differential topology. Symplectic geometry is the framework for the secondary charac teristic classes of a triple of vector bundles (.E,L0 ,L1) as described previously, Hence, necessarily, we must develop enough symplectic geo metry in our work, But, here again, the importance of the subject trans cends by far such particular problems as transversality of Lagrangian subbundles. Indeed, symplectic geometry is the framework of all the geo metric structures which involve symplectic vector bundles. We discuss general symplectic vector bundles, and natural symplectic vector bundles that appear on symplectic manifolds and their submanifolds, on contact manifolds, etc, In the same setting, we also study various Lagrangian vii subbundles. A last section of this chapter is devoted to local and tubu lar equivalence theorems of symplectic geometry: Darboux-Weinstein, Caratheodory, Lie, etc., and a recent theorem of Marle (56] for submani folds of constant rank of a symplectic manifold. The second half of the book consists of the single Chapter 4 on transversality obstructions of Lagrangian subbundles. Here, we review connections on principal bundles, and then develop a certain version of the theory of the secondary char acteristic classes which is simple and adequate to our purpose, namely the Chern-Simons theory (18] and the connection comparison Bott-Lehmann theory (49]. Among others, we give a simple proof of the derivation formulas of Chern-Simons (18] and Heitsch (39]. Then, we insert a review section of various approaches to the standard Maslov class, and an ori ginal section on general Maslov secondary characteristic classes (in dimensions 4h-3), which are the obstructions to the transversality of two Lagrangian subbundles of a symplectic vector bundle [83]. (Other known definitions of "higher order Maslov classes" (29], (85], (6], (66] are for more particular situations.) This section includes the general definitions, computation methods, and properties of the new Maslov clas ses. The last section, which also has an original character, is devoted to the computation of the general Maslov classes for Lagrangian submani folds of cotangent bundles, endowed with the canonical symplectic struc ture, and for Legendrian submanifolds of contangent unit spheres bundles of riemannian manifolds. It generalizes a result of Morvan about the standard Maslov class (62], and it establishes a general method of com putation showing that the Maslov classes depend on a generalized second fundamental form, and on the curvature of the Lagrangian (Legendrian) submanifold. I should like to point out that we had no intention whatsoever to write an exhaustive book or textbook on symplectic geometry and/or on viii secondary characteristic classes. Our book is just a monograph, directed towards the study of the general Maslov classes seen as transversality obstructions, and the material included is determined by this general purpose. It is also determined, by personal preferences in what concerns the more ample development of symplectic geometry than is needed for Maslov classes. Let us add a few formal explanations. All the objects encoun tered in this work are in the C® category, and this convention holds without further notice. The differentiable manifolds will be denoted by M,N, ••• , and if we write M" ,N" etc. this means that n,h, ••• are the dimensions of the respective manifolds. APM, denotes the space of p- forms on M, and A M is the Grassmann algebra of M. TM is the tan- gent bundle, T*M is the cotangent bundle, etc. Generally, we try to use more or less standard notation. As usual, ffi denotes the real field, [ denotes the complex field, Q the quaternion algebra, ~ the integer n ring, lR" etc. the Euclidean n-space over ffi etc., S the unit sphere in ffi n+l and so on. The end of proofs is marked by the abbreviation Q.e.d. The bibliographical references are far from being exhaustive as well, but we tried to indicate all our sources. We are particularly indepted to such sources as A. Weinstein [87], V. Guillemin and s. Sternberg [37], R. Deheuvels [25], Chern-Simons [18], Bott [15], D. Lehmann [49], J.M. Morvan [62], and to the many other papers quoted in the bibliography at the end of the book. TABLE OF CONTENTS CHAPTER 1. INTRODUCTION AND MOTIVATION , •••• 1 1.1. Equations of the Hamilton-Jacobi type 1 1.2. Some more symplectic geometry , 7 1,3. Some more mathematical physics 14 CHAPTER 2. SYMPLECTIC VECTOR SPACES 23 2.1. Symplectic vector spaces and their automorphisms 23 2.2. Subspaces of symplectic vector spaces 30 2.3. Complex structures in real symplectic spaces 40 CHAPTER 3. SYMPLECTIC GEOMETRY ON MANIFOLDS 53 3.1. Symplectic vector bundles 53 3.2. Symplectic vector bundles of geometric structures 59 3.3. Lagrangian subbundles • • 71 3,4. Local equivalence theorems in symplectic geometry 83 CHAPTER 4. TRANSVERSALITY OBSTRUCTIONS OF LAGRANGIAN SUBBUNDLES (MASLOV CLASSES) , , • 103 4,1. Connections on principal bundles 103 4,2. Secondary characteristic classes 113 4,3. The Maslov class and index 138 4.4. Maslov secondary characteristic classes 151 4.5. Computations in cotangent bundles • • • 177 REFERENCES 207 INDEX 213