Symplectic Geometry and Secondary Characteristic Classes
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
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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.
98765432 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 previously.)
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 geometry and topology, and in the applications to physical theories.
For all
these reasons it seemed to me appropriate to give an exposition of symplectic 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
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