Introduction to symplectic geometry

✍ Scribed by Koszul J.-L., Zou Y.M

• Publisher: Springer
• Year: 2019
• Tongue: English
• Leaves: 166
• Category: Library

✦ Table of Contents

Foreword 1. About This Book......Page 5
Jean-Louis Koszul’s Life: The Spirit of Geometry and the Spirit of Finesse of an “Esprit raffiné”......Page 8
The Genesis of this Translation of Koszul’s Book “Introduction to Symplectic Geometry”......Page 14
Koszul’s Book: A Joint Source of Geometric Heat Theory and Information Geometry......Page 18
Affine Representation of Lie Groups and Lie Algebras: Koszul and Souriau’s Pillars Based on Elie Cartan’s Seminal Work......Page 25
Conclusion......Page 30
References......Page 31
The Content and Originality of this Book......Page 34
Some Developments in Symplectic and Poisson Geometry......Page 40
References......Page 42
Preface......Page 44
Notations......Page 46
Contents......Page 48
1.1 Skew-Symmetric Forms......Page 50
1.2 Orthogonality Defined by a Skew-Symmetric 2-Form......Page 52
1.3 Symplectic Vector Spaces, Symplectic Bases......Page 55
1.4 The Canonical Linear Representation of sl(2,k) in the Algebra …......Page 57
1.5 Symplectic Groups......Page 60
1.6 Symplectic Complex Structures......Page 65
2.1 Symplectic Structures on Manifolds......Page 69
2.2 Operators of the Algebra of Differential Forms on a Symplectic Manifold......Page 74
2.3 Symplectic Coordinates......Page 78
2.4 Hamiltonian Vector Fields and Symplectic Vector Fields......Page 83
2.5 Poisson Brackets Under Symplectic Coordinates......Page 92
2.6 Submanifolds of Symplectic Manifolds......Page 96
3.1 Liouville Forms and Canonical Symplectic Structures on Cotangent Bundles......Page 104
3.2 Symplectic Vector Fields on a Cotangent Bundle......Page 108
3.3 Lagrangian Submanifolds of a Cotangent Bundle......Page 115
4 Symplectic G-Spaces......Page 121
4.1 Definitions and Examples......Page 122
4.2 Hamiltonian mathfrakg-Spaces and Moment Maps......Page 125
4.3 Equivariance of Moment Maps......Page 133
5.1.1 The Schouten–Nijenhuis Bracket......Page 137
5.2 The Leaves of a Poisson Manifold......Page 141
5.3 Poisson Structures on the Dual of a Lie Algebra......Page 144
6.1 (0,n)-Dimensional Supermanifolds......Page 154
6.2 (0,n)-Dimensional Symplectic Supermanifolds......Page 159
6.3 The Canonical Symplectic Structure on TastP......Page 160
BookmarkTitle:......Page 162
Index......Page 164