**CBMS Regional Conference Series in Mathematics**

Volume: 29;
1977;
48 pp;
Softcover

MSC: Primary 58;
Secondary 47; 53; 70

Print ISBN: 978-0-8218-1679-0

Product Code: CBMS/29

List Price: $20.00

Individual Price: $16.00

**Electronic ISBN: 978-1-4704-2389-6
Product Code: CBMS/29.E**

List Price: $20.00

Individual Price: $16.00

# Lectures on Symplectic Manifolds

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*Alan Weinstein*

A co-publication of the AMS and CBMS

The first six sections of these notes contain a description of some of the basic constructions and results on symplectic manifolds and lagrangian submanifolds. Section 7, on intersections of largrangian submanifolds, is still mostly internal to symplectic geometry, but it contains some applications to machanics and dynamical systems. Sections 8, 9, and 10 are devoted to various aspects of the quantization problem. In Section 10 there is a feedback of ideas from quantization theory into symplectic geometry itslef.

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#### Reviews & Endorsements

This volume of lecture notes is devoted to some problems in differential topology arising from the study of Hamiltonian systems and geometrical quantization … After the necessary definitions are given, some topological facts and problems concerning symplectic and Lagrangian manifolds are presented and their origin in abstract Hamiltonian mechanics is outlined. The author treats the classification problem for symplectic manifolds and discusses the relevant theorems of Darboux and Moser, and comments on the embedding problem and intersection theory for Lagrangian submanifolds. The last three chapters are devoted to geometric quantization in connection with representation theory of Lie groups, the quasi-classical approximation in quantum mechanics, and the theory of Fourier integral operators. The author states several open problems and supplies a detailed bibliography.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Lectures on Symplectic Manifolds

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Introduction 18 free
- Lecture 1. Symplectic manifolds and lagrangian submanifolds, examples 310 free
- Lecture 2. Lagrangian splittings, real and complex polarizations, Kahler manifolds 714
- Lecture 3. Reduction, the calculus of canonical relations, intermediate polarizations 1118
- Lecture 4. Hamiltonian systems and group actions on symplectic manifolds 1522
- Lecture 5. Normal forms 2229
- Lecture 6. Lagrangian submanifolds and families of functions 2532
- Lecture 7. Intersection Theory of lagrangian submanifolds 2936
- Lecture 8. Quantization on cotangent bundles 3138
- Lecture 9. Quantization and polarizations 3542
- Lecture 10. Quantizing lagrangian submanifolds and subspaces, construction of the Maslov bundle 3946
- References 4552
- Back Cover Back Cover157