Lectures on symplectic geometry /
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| Author / Creator: | Silva, Ana Cannas da |
|---|---|
| Edition: | Corr. 2. printing. |
| Imprint: | Berlin ; New York : Springer, 2008, c2001. |
| Description: | xiv, 247 p. : ill. ; 24 cm |
| Language: | English |
| Series: | Lecture notes in mathematics ; 1764 Lecture notes in mathematics (Springer-Verlag) ; 1764. |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/7249419 |
Table of Contents:
- Foreword
- Introduction
- I. Symplectic Manifolds
- 1. Symplectic Forms
- 1.1. Skew-Symmetric Bilinear Maps
- 1.2. Symplectic Vector Spaces
- 1.3. Symplectic Manifolds
- 1.4. Symplectomorphisms
- Homework 1. Symplectic Linear Algebra
- 2. Symplectic Form on the Cotangent Bundle
- 2.1. Cotangent Bundle
- 2.2. Tautological and Canonical Forms in Coordinates
- 2.3. Coordinate-Free Definitions
- 2.4. Naturality of the Tautological and Canonical Forms
- Homework 2. Symplectic Volume
- II. Symplectomorphisms
- 3. Lagrangian Submanifolds
- 3.1. Submanifolds
- 3.2. Lagrangian Submanifolds of T*X
- 3.3. Conormal Bundles
- 3.4. Application to Symplectomorphisms
- Homework 3. Tautological Form and Symplectomorphisms
- 4. Generating Functions
- 4.1. Constructing Symplectomorphisms
- 4.2. Method of Generating Functions
- 4.3. Application to Geodesic Flow
- Homework 4. Geodesic Flow
- 5. Recurrence
- 5.1. Periodic Points
- 5.2. Billiards
- 5.3. Poincaré Recurrence
- III. Local Forms
- 6. Preparation for the Local Theory
- 6.1. Isotopies and Vector Fields
- 6.2. Tubular Neighborhood Theorem
- 6.3. Homotopy Formula
- Homework 5. Tubular Neighborhoods in R n
- 7. Moser Theorems
- 7.1. Notions of Equivalence for Symplectic Structures
- 7.2. Moser Trick
- 7.3. Moser Local Theorem
- 8. Darboux-Moser-Weinstein Theory
- 8.1. Classical Darboux Theorem
- 8.2. Lagrangian Subspaces
- 8.3. Weinstein Lagrangian Neighborhood Theorem
- Homework 6. Oriented Surfaces
- 9. Weinstein Tubular Neighborhood Theorem
- 9.1. Observation from Linear Algebra
- 9.2. Tubular Neighborhoods
- 9.3. Application 1: Tangent Space to the Group of Symplectomorphisms
- 9.4. Application 2: Fixed Points of Symplectomorphisms
- IV. Contact Manifolds
- 10. Contact Forms
- 10.1. Contact Structures
- 10.2. Examples
- 10.3. First Properties
- Homework 7. Manifolds of Contact Elements
- 11. Contact Dynamics
- 11.1. Reeb Vector Fields
- 11.2. Symplectization
- 11.3. Conjectures of Seifert and Weinstein
- V. Compatible Almost Complex Structures
- 12. Almost Complex Structures
- 12.1. Three Geometries
- 12.2. Complex Structures on Vector Spaces
- 12.3. Compatible Structures
- Homework 8. Compatible Linear Structures
- 13. Compatible Triples
- 13.1. Compatibility
- 13.2. Triple of Structures
- 13.3. First Consequences
- Homework 9. Contractibility
- 14. Dolbeault Theory
- 14.1. Splittings
- 14.2. Forms of Type (l, m)
- 14.3. J-Holomorphic Functions
- 14.4. Dolbeault Cohomology
- Homework 10. Integrability
- VI. Kähler Manifolds
- 15. Complex Manifolds
- 15.1. Complex Charts
- 15.2. Forms on Complex Manifolds
- 15.3. Differentials
- Homework 11. Complex Projective Space
- 16. Kähler Forms
- 16.1. Kähler Forms
- 16.2. An Application
- 16.3. Recipe to Obtain Kähler Forms
- 16.4. Local Canonical Form for Kähler Forms
- Homework 12. The Fubini-Study Structure
- 17. Compact Kähler Manifolds
- 17.1. Hodge Theory
- 17.2. Immediate Topological Consequences
- 17.3. Compact Examples and Counterexamples
- 17.4. Main Kähler Manifolds
- VII. Hamiltonian Mechanics
- 18. Hamiltonian Vector Fields
- 18.1. Hamiltonian and Symplectic Vector Fields
- 18.2. Classical Mechanics
- 18.3. Brackets
- 18.4. Integrable Systems
- Homework 13. Simple Pendulum
- 19. Variational Principles
- 19.1. Equations of Motion
- 19.2. Principle of Least Action
- 19.3. Variational Problems
- 19.4. Solving the Euler-Lagrange Equations
- 19.5. Minimizing Properties
- Homework 14. Minimizing Geodesies
- 20. Legendre Transform
- 20.1. Strict Convexity
- 20.2. Legendre Transform
- 20.3. Application to Variational Problems
- Homework 15. Legendre Transform
- VIII. Moment Maps
- 21. Actions
- 21.1. One-Parameter Groups of Diffeomorphisms
- 21.2. Lie Groups
- 21.3. Smooth Actions
- 21.4. Symplectic and Hamiltonian Actions
- 21.5. Adjoint and Coadjoint Representations
- Homework 16. Hermitian Matrices
- 22. Hamiltonian Actions
- 22.1. Moment and Comoment Maps
- 22.2. Orbit Spaces
- 22.3. Preview of Reduction
- 22.4. Classical Examples
- Homework 17. Coadjoint Orbits
- IX. Symplectic Reduction
- 23. The Marsden-Weinstein-Meyer Theorem
- 23.1. Statement
- 23.2. Ingredients
- 23.3. Proof of the Marsden-Weinstein-Meyer Theorem
- 24. Reduction
- 24.1. Noether Principle
- 24.2. Elementary Theory of Reduction
- 24.3. Reduction for Product Groups
- 24.4. Reduction at Other Levels
- 24.5. Orbifolds
- Homework 18. Spherical Pendulum
- X. Moment Maps Revisited
- 25. Moment Map in Gauge Theory
- 25.1. Connections on a Principal Bundle
- 25.2. Connection and Curvature Forms
- 25.3. Symplectic Structure on the Space of Connections
- 25.4. Action of the Gauge Group
- 25.5. Case of Circle Bundles
- Homework 19. Examples of Moment Maps
- 26. Existence and Uniqueness of Moment Maps
- 26.1. Lie Algebras of Vector Fields
- 26.2. Lie Algebra Cohomology
- 26.3. Existence of Moment Maps
- 26.4. Uniqueness of Moment Maps
- Homework 20. Examples of Reduction
- 27. Convexity
- 27.1. Convexity Theorem
- 27.2. Effective Actions
- 27.3. Examples
- Homework 21. Connectedness
- XI. Symplectic Toric Manifolds
- 28. Classification of Symplectic Toric Manifolds
- 28.1. Delzant Polytopes
- 28.2. Delzant Theorem
- 28.3. Sketch of Delzant Construction
- 29. Delzant Construction
- 29.1. Algebraic Set-Up
- 29.2. The Zero-Level
- 29.3. Conclusion of the Delzant Construction
- 29.4. Idea Behind the Delzant Construction
- Homework 22. Delzant Theorem
- 30. Duistermaat-Heckman Theorems
- 30.1. Duistermaat-Heckman Polynomial
- 30.2. Local Form for Reduced Spaces
- 30.3. Variation of the Symplectic Volume
- Homework 23. S 1 -Equivariant Cohomology
- References
- Index