Editorial Board
James Humphreys (Chair)
David Saltman David Sattinger Ronald Stern
2000 Mathematics Subject Classification. Primary 53C15, 53Dxx, 20G20, 81S10.
Originally published in the German language by Friedr. Vieweg & Sohn Verlagsge-
sellschaft mbH, D-65189 Wiesbaden, Germany, as "Rolf Berndt: Einfiihrung in die Sym-
plektische Geometrie. 1. Auflage (1st edition)" © by Friedr. Vieweg & Sohn Verlagsge-
sellschaft mbH, Braunschweig/Wiesbaden, 1998.
Translated from the German by Michael Klucznik
ABSTRACT. The notions of symplectic form, symplectic manifold and symplectic group appear
in many different contexts in analysis, geometry, function theory and dynamical systems. This
book assembles tools from different mathematical regions necessary to define these notions and to
introduce their application. Among the topics treated here are
symplectic and Kahler vector spaces,
the symplectic group and Siegel's half space,
symplectic and contact manifolds, the theorem of Darboux,
methods of constructing symplectic manifolds: Kahler manifolds, coadjoint orbits and sym-
plectic reduction,
Hamiltonian systems,
the moment map,
and a glimpse into geometric quantization (in particular the theorem of Groenewold and
van Hove) leading to some rudiments of the representation theory of the Heisenberg and
the Jacobi group.
The goal of the book is to provide an entrance into a fascinating area linking several mathematical
disciplines and parts of theoretical physics.
Library of Congress Cataloging-in-Publication Data
Berndt, Rolf.
[Einfiihrung in die symplektische Geometrie. English]
An introduction to sympletic geometry / Rolf Berndt ; translated by Michael Klucznik.
p. cm. (Graduate studies in mathematics, ISSN 1065-7339 ; v. 26)
Includes bibliographical references and index.
ISBN 0-8218-2056-7 (alk. paper)
1. Symplectic manifolds. 2. Geometry, Differential I. Title. II. Series.
QA649 .B47 2000
516.3'6—dc21 00-033139
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is permitted only under license from the American Mathematical Society. Requests for such
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© 2001 by the American Mathematical Society. All rights reserved.
The American Mathematical Society retains all rights
except those granted to the United States Government.
Printed in the United States of America.
@ The paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
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