文档介绍：Lectures on Symplectic Geometry
Ana Cannas da Silva1
August 17, 2000
1E-mail: ******@ or ******@
Foreword
The goal of these notes is to provide a fast introduction to symplectic geometry
– the geometry of manifolds equipped with a closed nondegenerate 2-form.
Two centuries ago, symplectic geometry1 provided a language for classical
mechanics. Through its recent huge development, it conquered an independent
and rich territory. Symplectic geometry is significantly stimulated by impor-
tant interactions with global analysis, mathematical physics, low-dimensional
topology, dynamical systems, algebraic geometry, integrable systems, microlo-
cal analysis, partial differential equations, representation theory, quantization,
equivariant cohomology, binatorics, etc.
Parts I-III explain classical topics, including cotangent bundles, symplecto-
morphisms, lagrangian submanifolds and local forms. Parts IV-VI concentrate
on important related areas, such as contact geometry and K¨ahler geometry.
Classical hamiltonian theory enters only in Parts VII-VIII, starting the second
half of this book, which is devoted to a selection from hamiltonian dynamical
systems and symmetry. Parts IX-XI discuss the moment map whose preponder-
ance has been growing steadily for the past twenty years. There are scattered
short exercises throughout the text. At the end of most lectures, some longer
guided problems, called homework, were designed plement the exposition
or extend the reader’s understanding.
These notes approximately transcribe a 15-week course on symplectic geom-
etry I taught at UC Berkeley in the Fall of 1997, with 2 hour-and-a-half lectures
per week. The course targeted second-year graduate students in mathemat-
ics, though the audience was more diverse, including advanced undergraduates,
post-docs and graduate students from other departments. The present text
should hence still be appropriate for a second-year graduate