Lectures on symplectic geometry:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2001
|
| Series: | Lecture notes in mathematics
1764 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 217 S. graph. Darst. |
| ISBN: | 3540421955 |
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| 245 | 1 | 0 | |a Lectures on symplectic geometry |c Ana Cannas da Silva |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2001 | |
| 300 | |a XII, 217 S. |b graph. Darst. | ||
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| 490 | 1 | |a Lecture notes in mathematics |v 1764 | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Foreword v
Introduction 1
I Symplectic Manifolds 3
1 Symplectic Forms 3
1.1 Skew Symmetric Bilinear Maps 3
1.2 Symplectic Vector Spaces 4
1.3 Symplectic Manifolds 6
1.4 Symplectomorphisms 7
Homework 1: Symplectic Linear Algebra 8
2 Symplectic Form on the Cotangent Bundle 9
2.1 Cotangent Bundle 9
2.2 Tautological and Canonical Forms in Coordinates 9
2.3 Coordinate Free Definitions 10
2.4 Naturality of the Tautological and Canonical Forms 11
Homework 2: Symplectic Volume 13
II Symplectomorphisms 15
3 Lagrangian Submanifolds 15
3.1 Submanifolds 15
3.2 Lagrangian Submanifolds of T*X 16
3.3 Conormal Bundles 18
3.4 Application to Symplectomorphisms 19
Homework 3: Tautological Form and Symplectomorphisms 20
4 Generating Functions 22
4.1 Constructing Symplectomorphisms 22
4.2 Method of Generating Functions 23
4.3 Application to Geodesic Flow 24
Homework 4: Geodesic Flow 27
viii CONTENTS
5 Recurrence 29
5.1 Periodic Points 29
5.2 Billiards 30
5.3 Poincaré Recurrence 32
III Local Forms 35
6 Preparation for the Local Theory 35
6.1 Isotopies and Vector Fields 35
6.2 Tubular Neighborhood Theorem 37
6.3 Homotopy Formula 39
Homework 5: Tubular Neighborhoods in Mn 41
7 Moser Theorems 42
7.1 Notions of Equivalence for Symplectic Structures 42
7.2 Moser Trick 42
7.3 Moser Local Theorem 45
8 Darboux Moser Weinstein Theory 46
8.1 Classical Darboux Theorem 46
8.2 Lagrangian Subspaces 46
8.3 Weinstein Lagrangian Neighborhood Theorem 48
Homework 6: Oriented Surfaces 50
9 Weinstein Tubular Neighborhood Theorem 51
9.1 Observation from Linear Algebra 51
9.2 Tubular Neighborhoods 51
9.3 Application 1:
Tangent Space to the Group of Symplectomorphisms 53
9.4 Application 2:
Fixed Points of Symplectomorphisms 55
IV Contact Manifolds 57
10 Contact Forms 57
10.1 Contact Structures 57
10.2 Examples 58
10.3 First Properties 59
Homework 7: Manifolds of Contact Elements 61
CONTENTS ix
11 Contact Dynamics 63
11.1 Reeb Vector Fields 63
11.2 Symplectization 64
11.3 Conjectures of Seifert and Weinstein 65
V Compatible Almost Complex Structures 67
12 Almost Complex Structures 67
12.1 Three Geometries 67
12.2 Complex Structures on Vector Spaces 68
12.3 Compatible Structures 70
Homework 8: Compatible Linear Structures 72
13 Compatible Triples 74
13.1 Compatibility 74
13.2 Triple of Structures 75
13.3 First Consequences 75
Homework 9: Contractibility 77
14 Dolbeault Theory 78
14.1 Splittings 78
14.2 Forms of Type (£, m) 79
14.3 J Holomorphic Functions 80
14.4 Dolbeault Cohomology 81
Homework 10: Integrability 82
VI Kâhler Manifolds 83
15 Complex Manifolds 83
15.1 Complex Charts 83
15.2 Forms on Complex Manifolds 85
15.3 Differentials 86
Homework 11: Complex Projective Space 89
16 Kâhler Forms 90
16.1 Kâhler Forms 90
16.2 An Application 92
16.3 Recipe to Obtain Kâhler Forms 92
16.4 Local Canonical Form for Kâhler Forms 94
Homework 12: The Fubini Study Structure 96
x CONTENTS
17 Compact Kâhler Manifolds 98
17.1 Hodge Theory 98
17.2 Immediate Topological Consequences 100
17.3 Compact Examples and Counterexamples 101
17.4 Main KÃ hler Manifolds 103
VII Hamiltonian Mechanics 105
18 Hamiltonian Vector Fields 105
18.1 Hamiltonian and Symplectic Vector Fields 105
18.2 Classical Mechanics , • ¦ • 107
18.3 Brackets 108
18.4 Integrable Systems 109
Homework 13: Simple Pendulum 112
19 Variational Principles 113
19.1 Equations of Motion 113
19.2 Principle of Least Action 114
19.3 Variational Problems 114
19.4 Solving the Euler Lagrange Equations 116
19.5 Minimizing Properties 117
Homework 14: Minimizing Geodesies 119
20 Legendre Transform 121
20.1 Strict Convexity 121
20.2 Legendre Transform 121
20.3 Application to Variational Problems 122
Homework 15: Legendre Transform 125
VIII Moment Maps 127
21 Actions 127
21.1 One Parameter Groups of Diffeomorphisms 127
21.2 Lie Groups 128
21.3 Smooth Actions 128
21.4 Symplectic and Hamiltonian Actions 129
21.5 Adjoint and Coadjoint Representations ... : 130
Homework 16: Hermitian Matrices 132
CONTENTS xi
22 Hainiltonian Actions 133
22.1 Moment and Comoment Maps 133
22.2 Orbit Spaces 135
22.3 Preview of Reduction 136
22.4 Classical Examples 137
Homework 17: Coadjoint Orbits 139
IX Symplectic Reduction 141
23 The Marsden Weinstein Meyer Theorem 141
23.1 Statement 141
23.2 Ingredients 142
23.3 Proof of the Marsden Weinstein Meyer Theorem 145
24 Reduction 147
24.1 Noether Principle 147
24.2 Elementary Theory of Reduction 147
24.3 Reduction for Product Groups 149
24.4 Reduction at Other Levels 149
24.5 Orbifolds 150
Homework 18: Spherical Pendulum 152
X Moment Maps Revisited 155
25 Moment Map in Gauge Theory 155
25.1 Connections on a Principal Bundle 155
25.2 Connection and Curvature Forms 156
25.3 Symplectic Structure on the Space of Connections 158
25.4 Action of the Gauge Group 158
25.5 Case of Circle Bundles 159
Homework 19: Examples of Moment Maps 162
26 Existence and Uniqueness of Moment Maps 164
26.1 Lie Algebras of Vector Fields 164
26.2 Lie Algebra Cohomology 165
26.3 Existence of Moment Maps 166
26.4 Uniqueness of Moment Maps 167
Homework 20: Examples of Reduction 169
xii CONTENTS
27 Convexity 170
27.1 Convexity Theorem 170
27.2 Effective Actions 171
27.3 Examples 172
Homework 21: Connectedness 175
XI Symplectic Toric Manifolds 177
28 Classification of Symplectic Toric Manifolds 177
28.1 Delzant Polytopes 177
28.2 Delzant Theorem 179
28.3 Sketch of Delzant Construction 180
29 Delzant Construction 183
29.1 Algebraic Set Up 183
29.2 The Zero Level 183
29.3 Conclusion of the Delzant Construction 185
29.4 Idea Behind the Delzant Construction 186
Homework 22: Delzant Theorem 189
30 Duistermaat Heckman Theorems 191
30.1 Duistermaat Heckman Polynomial 191
30.2 Local Form for Reduced Spaces 192
30.3 Variation of the Symplectic Volume 195
Homework 23: ^ Equivariant Cohomology 197
References 199
Index 207
|
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| indexdate | 2024-07-09T18:51:17Z |
| institution | BVB |
| isbn | 3540421955 |
| language | English |
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| physical | XII, 217 S. graph. Darst. |
| publishDate | 2001 |
| publishDateSearch | 2001 |
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| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spelling | Silva, Ana Cannas da 1968- Verfasser (DE-588)122891333 aut Lectures on symplectic geometry Ana Cannas da Silva Berlin [u.a.] Springer 2001 XII, 217 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1764 Symplektische Geometrie Symplektische Geometrie (DE-588)4194232-2 gnd rswk-swf Symplektische Geometrie (DE-588)4194232-2 s DE-604 Lecture notes in mathematics 1764 (DE-604)BV000676446 1764 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009397332&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Silva, Ana Cannas da 1968- Lectures on symplectic geometry Lecture notes in mathematics Symplektische Geometrie Symplektische Geometrie (DE-588)4194232-2 gnd |
| subject_GND | (DE-588)4194232-2 |
| title | Lectures on symplectic geometry |
| title_auth | Lectures on symplectic geometry |
| title_exact_search | Lectures on symplectic geometry |
| title_full | Lectures on symplectic geometry Ana Cannas da Silva |
| title_fullStr | Lectures on symplectic geometry Ana Cannas da Silva |
| title_full_unstemmed | Lectures on symplectic geometry Ana Cannas da Silva |
| title_short | Lectures on symplectic geometry |
| title_sort | lectures on symplectic geometry |
| topic | Symplektische Geometrie Symplektische Geometrie (DE-588)4194232-2 gnd |
| topic_facet | Symplektische Geometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009397332&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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