Riemannian Geometry of Contact and Symplectic Manifolds
This monograph deals with the Riemannian geometry of both symplectic and contact manifolds, with particular emphasis on the latter. The text is carefully presented. Topics unfold systematically from Chapter 1, which examines the general theory of symplect
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Series Editors Hyman Bass Joseph Oesterle Alan Weinstein
David E. Blair
Riemannian Geometry of Contact and Symplectic Manifolds
Springer Science+Business Media, LLC
David E. Blair Department of Mathematics Michigan State University East Lansing, MI 48824-1027 U.S.A.
Library of Congress Cataloging-in-Publication Data Blair, Oavid E., 1940 Riemannian geometry of contact and symplectic manifolds 1 Oavid E. Blair. p. cm. - (Progress in mathematics (Boston, Mass.» Includes bibliographical references and index. I. Contact manifolds. 2. Symplectic manifolds. 3. Geometry, Riemannian. I. Title. 11. Series. QA614.3.B53 2001 516.3'73-dc21
2001052707 CIP
AMS Subject Classifications: 53B35, 53C15, 53C25, 53C26, 53C40, 53C42, 53C55, 53C56, 53005, 53010, 53012,53015,53022,53025,53035,58Ell Printed on acid-free paper © 2002 Springer Science+Business Media New York Originally published by Birkhiiuser Boston in 2002. Softcover reprint of the hardcover I st edition 2002 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC, except for brief eKcerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
ISBN 978-1-4757-3606-9 ISBN 978-1-4757-3604-5 (eBook) DOI 10.1007/978-1-4757-3604-5 Reformatted from author's files by TEXniques, Inc., Cambridge, MA
987654321
To Rebecca in appreciation of all her love and support
Contents
1
1 Symplectic Manifolds 1.1 Definitions and examples . . . . . . 1.2 Lagrangian submanifolds . . . . . . 1.3 The Darboux-Weinstein theorems. 1.4 Symplectomorphisms........
1 5 7 9
2 Principal Sl-bundles 2.1 The set of principal Sl-bundles as a group 2.2 Connections on a principal bundle.
11 11 14
3 Contact Manifolds 3.1 Definitions .. 3.2 Examples . . . 3.2.1 ]R2n+1 3.2.2 ]Rn+l x p]Rn . 3.2.3 M2n+1 C ]R2n+2 with TmM2n+1 n {O} 3.2.4 TtM, TIM 3.2.5 T* M x ]R 3.2.6 T3 . . . . . 3.2.7 T5 . . . . . 3.2.8 Overtwisted contact structures. 3.2.9 Contact circles . . . . 3.3 The Boothby-Wang fibration 3.4 The Weinstein conjecture. . .
17 17
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=0
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Contents
4 Associated Metrics 4.1 Almost complex and almost contact structures 4.2 Polarization and associated metrics .. 4.3 Polarization of metrics as a projection 4.3.1 Some linear algebra . . . . . . . 4.3.2 Results on the set A . . . . . . 4.4 Action of symplectic and contact transformations 4.5 Examples of almost contact metric manifolds. 4.5.1 1R2n +l . . . . . . . . . . . . . . . 4.5.2 M2n+l C M2n+2 almost complex. 4.5.3 8
