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The Geometry of the Group of Symplectic Diffeomorphism : Lectures in Mathematics Eth Zurich - Leonid Polterovich

The Geometry of the Group of Symplectic Diffeomorphism

Lectures in Mathematics Eth Zurich

Paperback

Published: 1st March 2001
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The group of Hamiltonian diffeomorphisms Ham(M, 0) of a symplectic mani- fold (M, 0) plays a fundamental role both in geometry and classical mechanics. For a geometer, at least under some assumptions on the manifold M, this is just the connected component of the identity in the group of all symplectic diffeomorphisms. From the viewpoint of mechanics, Ham(M, O) is the group of all admissible motions. What is the minimal amount of energy required in order to generate a given Hamiltonian diffeomorphism I? An attempt to formalize and answer this natural question has led H. Hofer [HI] (1990) to a remarkable discovery. It turns out that the solution of this variational problem can be interpreted as a geometric quantity, namely as the distance between I and the identity transformation. Moreover this distance is associated to a canonical biinvariant metric on Ham(M, 0). Since Hofer's work this new ge- ometry has been intensively studied in the framework of modern symplectic topology. In the present book I will describe some of these developments. Hofer's geometry enables us to study various notions and problems which come from the familiar finite dimensional geometry in the context of the group of Hamiltonian diffeomorphisms. They turn out to be very different from the usual circle of problems considered in symplectic topology and thus extend significantly our vision of the symplectic world.

Preface
Introducing the Group
Introducing he Geometry
Lagrangian Submanifolds
The [actual symbol not reproducible]-Equation
Linearization
Lagrangian Intersections
Diameter
Growth and Dynamics
Length Spectrum
Deformations of Symplectic Forms
Ergodic Theory
Geodesics
Floer Homology
Non-Hamiltonian Diffeomorphisms
Bibliography
Index
List of Symbols
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9783764364328
ISBN-10: 3764364327
Series: Lectures in Mathematics Eth Zurich
Audience: General
Format: Paperback
Language: English
Number Of Pages: 136
Published: 1st March 2001
Publisher: Birkhauser Verlag AG
Country of Publication: CH
Dimensions (cm): 24.41 x 16.99  x 0.84
Weight (kg): 0.25