**Graduate Studies in Mathematics**

Volume: 26;
2001;
195 pp;
Hardcover

MSC: Primary 53; 20; 81;

**Print ISBN: 978-0-8218-2056-8
Product Code: GSM/26**

List Price: $48.00

AMS Member Price: $38.40

MAA Member Price: $43.20

**Electronic ISBN: 978-1-4704-2081-9
Product Code: GSM/26.E**

List Price: $45.00

AMS Member Price: $36.00

MAA Member Price: $40.50

# An Introduction to Symplectic Geometry

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*Rolf Berndt*

Symplectic geometry is a central topic of
current research in mathematics. Indeed, symplectic methods are key
ingredients in the study of dynamical systems, differential equations,
algebraic geometry, topology, mathematical physics and representations
of Lie groups.

This book is a true introduction to symplectic geometry, assuming only a
general background in analysis and familiarity with linear algebra. It starts
with the basics of the geometry of symplectic vector spaces. Then, symplectic
manifolds are defined and explored. In addition to the essential classic
results, such as Darboux's theorem, more recent results and ideas are also
included here, such as symplectic capacity and pseudoholomorphic curves. These
ideas have revolutionized the subject. The main examples of symplectic
manifolds are given, including the cotangent bundle, Kähler manifolds, and
coadjoint orbits. Further principal ideas are carefully examined, such as
Hamiltonian vector fields, the Poisson bracket, and connections with contact
manifolds.

Berndt describes some of the close connections between symplectic
geometry and mathematical physics in the last two chapters of the
book. In particular, the moment map is defined and explored, both
mathematically and in its relation to physics. He also introduces
symplectic reduction, which is an important tool for reducing the
number of variables in a physical system and for constructing new
symplectic manifolds from old. The final chapter is on quantization,
which uses symplectic methods to take classical mechanics to quantum
mechanics. This section includes a discussion of the Heisenberg group
and the Weil (or metaplectic) representation of the symplectic
group.

Several appendices provide background material on vector bundles,
on cohomology, and on Lie groups and Lie algebras and their
representations.

Berndt's presentation of symplectic geometry is a clear and concise
introduction to the major methods and applications of the subject, and requires
only a minimum of prerequisites. This book would be an excellent text for a
graduate course or as a source for anyone who wishes to learn about symplectic
geometry.

#### Readership

Graduate students and research mathematicians interested in differential geometry.

#### Table of Contents

# Table of Contents

## An Introduction to Symplectic Geometry

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Preface xi12 free
- Chapter 0. Some Aspects of Theoretical Mechanics 118 free
- Chapter 1. Symplectic Algebra 926
- Chapter 2. Symplectic Manifolds 3552
- Chapter 3. Hamiltonian Vector Fields and the Poisson Bracket 7188
- Chapter 4. The Moment Map 93110
- Chapter 5. Quantization 111128
- Appendix A. Differentiable Manifolds and Vector Bundles 135152
- Appendix B. Lie Groups and Lie Algebras 163180
- Appendix C. A Little Cohomology Theory 171188
- Appendix D. Representations of Groups 177194
- Bibliography 185202
- Index 189206
- Symbols 193210
- Back Cover Back Cover1213