Lectures on Symplectic Geometry

Ana Cannas da Silva (auth.)3540421955, 9783540421955

The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups.

This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader’s understanding.

There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster.

For this reprint numerous corrections and clarifications have been made, and the layout has been improved.

Table of contents :
Front Matter….Pages i-xiv
Front Matter….Pages 2-2
Symplectic Forms….Pages 3-8
Symplectic Form on the Cotangent Bundle….Pages 9-14
Front Matter….Pages 16-16
Lagrangian Submanifolds….Pages 17-23
Generating Functions….Pages 25-31
Recurrence….Pages 33-37
Front Matter….Pages 40-40
Preparation for the Local Theory….Pages 41-47
Moser Theorems….Pages 49-53
Darboux-Moser-Weinstein Theory….Pages 55-60
Weinstein Tubular Neighborhood Theorem….Pages 61-66
Front Matter….Pages 68-68
Contact Forms….Pages 69-74
Contact Dynamics….Pages 75-79
Front Matter….Pages 82-82
Almost Complex Structures….Pages 83-88
Compatible Triples….Pages 89-92
Dolbeault Theory….Pages 93-98
Front Matter….Pages 100-100
Complex Manifolds….Pages 101-107
Kähler Forms….Pages 109-116
Compact Kähler Manifolds….Pages 117-123
Front Matter….Pages 126-126
Hamiltonian Vector Fields….Pages 127-134
Variational Principles….Pages 135-142
Legendre Transform….Pages 143-148
Front Matter….Pages 150-150
Actions….Pages 151-156
Hamiltonian Actions….Pages 157-163
Front Matter….Pages 166-166
The Marsden-Weinstein-Meyer Theorem….Pages 167-171
Reduction….Pages 173-179
Front Matter….Pages 182-182
Moment Map in Gauge Theory….Pages 183-192
Existence and Uniqueness of Moment Maps….Pages 193-198
Convexity….Pages 199-205
Front Matter….Pages 208-208
Classification of Symplectic Toric Manifolds….Pages 209-214
Delzant Construction….Pages 215-222
Duistermaat-Heckman Theorems….Pages 223-231
Back Matter….Pages 233-247