Symplectic geometry of integrable Hamiltonian systems:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel [u.a.]
Birkhäuser
2003
|
| Series: | Advanced courses in mathematics, CRM Barcelona
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 225 S. graph. Darst. |
| ISBN: | 3764321679 |
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| 245 | 1 | 0 | |a Symplectic geometry of integrable Hamiltonian systems |c Michèle Audin ; Ana Cannas da Silva ; Eugene Lerman |
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 2003 | |
| 300 | |a X, 225 S. |b graph. Darst. | ||
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| 490 | 0 | |a Advanced courses in mathematics, CRM Barcelona | |
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| 650 | 7 | |a Sistemas hamiltonianos |2 larpcal | |
| 650 | 4 | |a Systèmes hamiltoniens | |
| 650 | 7 | |a Variedades simpléticas |2 larpcal | |
| 650 | 4 | |a Variétés symplectiques | |
| 650 | 4 | |a Hamiltonian systems | |
| 650 | 4 | |a Symplectic manifolds | |
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Record in the Search Index
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| adam_text | Contents
Preface ix
A Lagrangian Submanifolds
Michele Audin 1
Introduction 3
I Lagrangian and special Lagrangian immersions in C 5
1.1 Symplectic form on C . symplectic vector spaces 5
I.I.a Symplectic vector spaces 5
I.l.b Symplectic bases 6
I.l.c The symplectic form as a differential form 7
I.l.d The symplectic group 8
I.l.e Orthogonality, isotropy 9
1.2 Lagrangian subspaces 9
1.2.a Definition of Lagrangian subspaces 9
I.2.b The symplectic reduction 10
1.3 The Lagrangian Grassmannian 11
1.3.a The Grassmannian An as a homogeneous space f 1
1.3.b The manifold A,, 11
I.3.C The tautological vector bundle 13
I.3.d The tangent bundle to A,, 14
I.3.e The case of oriented Lagrangian subspaces 15
I.3.f The determinant and the Maslov class 15
1.4 Lagrangian submanifolds in C 16
1.4.a Lagrangian submanifolds described by functions 16
I.4.b Wave fronts 20
I.4.C Other examples 25
I.4.d The Gauss map 28
vi Contents
1.5 Special Lagrangian submanifolds in C 29
1.5.a Special Lagrangian subspaces 29
I.5.b Special Lagrangian submanifolds 32
I.5.C Graphs of forms 33
I.5.d Normal bundles of surfaces 35
I.5.e From integrable systems 36
1.5.f Special Lagrangian submanifolds invariant under SO(n) . . 38
1.6 Appendices 39
1.6.a The topology of the symplectic group 39
I.6.b Complex structures 41
I.6.c Hamiltonian vector fields, integrable systems 41
Exercises 44
II Lagrangian and special Lagrangian submanifolds in symplectic and
Calabi Yau manifolds 49
II. 1 Symplectic manifolds 49
11.2 Lagrangian submanifolds and immersions 51
II.2.a In cotangent bundles 51
11.3 Tubular neighborhoods of Lagrangian submanifolds 53
II.3.a Moser s method 53
II.3.b Tubular neighborhoods 57
II.3.c Moduli space of Lagrangian submanifolds 59
11.4 Calabi Yau manifolds 60
II.4.a Definition of the Calabi Yau manifolds 61
II.4.b Yau s theorem 62
II.4.C Examples of Calabi Yau manifolds 63
II.4.d Special Lagrangian submanifolds 66
11.5 Special Lagrangians in real Calabi Yau manifolds 66
II.5.a Real manifolds 66
II.5.b Real Calabi Yau manifolds 67
II.5.c The example of elliptic curves 68
II.5.d Special Lagrangians in real Calabi Yau manifolds 68
11.6 Moduli space of special Lagrangian submanifolds 70
11.7 Towards mirror symmetry? 73
II.7.a Fibrations in special Lagrangian submanifolds 74
II.7.b Mirror symmetry 75
Exercises 76
Bibliography 81
Contents vii
B Symplectic Toric Manifolds
Ana Cannas da Silva 85
Foreword 87
I Symplectic Viewpoint 89
1.1 Symplectic Toric Manifolds 89
1.1.1 Symplectic Manifolds 89
1.1.2 Hamiltonian Vector Fields 90
1.1.3 Integrable Systems 91
1.1.4 Hamiltonian Actions 94
1.1.5 Hamiltonian Torus Actions 96
1.1.6 Symplectic Toric Manifolds 98
1.2 Classification 101
1.2.1 Delzant s Theorem 101
1.2.2 Orbit Spaces 103
1.2.3 Symplectic Reduction 104
1.2.4 Extensions of Symplectic Reduction 106
1.2.5 Delzant s Construction 109
1.2.6 Idea Behind Delzant s Construction 113
1.3 Moment Polytopes 118
1.3.1 Equivariant Darboux Theorem 118
1.3.2 Morse Theory 119
1.3.3 Homology of Symplectic Toric Manifolds 120
1.3.4 Symplectic Blow Up 122
1.3.5 Blow Up of Toric Manifolds 124
1.3.6 Symplectic Cutting 126
II Algebraic Viewpoint 129
11.1 Toric Varieties 129
11.1.1 Affine Varieties 129
11.1.2 Rational Maps on Affine Varieties 131
11.1.3 Projective Varieties 132
II. 1.4 Rational Maps on Projective Varieties 135
11.1.5 Quasiprojective Varieties 136
11.1.6 Toric Varieties 137
11.2 Classification 140
11.2.1 Spectra 140
11.2.2 Toric Varieties Associated to Semigroups 142
11.2.3 Classification of Affine Toric Varieties 143
11.2.4 Fans 144
11.2.5 Toric Varieties Associated to Fans 148
11.2.6 Classification of Normal Toric Varieties 153
viii Contents
II.3 Moment Polytopes 154
11.3.1 Equivariantly Projective Toric Varieties 154
11.3.2 Weight Polytopes 155
11.3.3 Orbit Decomposition 156
11.3.4 Fans from Polytopes 158
11.3.5 Classes of Toric Varieties 160
11.3.6 Symplectic vs. Algebraic 161
Bibliography 165
Index 169
C Geodesic Flows and Contact Toric Manifolds
Eugene Lerman 175
Foreword 177
I From toric integrable geodesic flows to contact toric manifolds 179
1.1 Introduction 179
1.2 Symplectic cones and contact manifolds 184
II Contact group actions and contact moment maps 193
III Proof of Theorem 1.38 197
111.1 Homogeneous vector bundles and slices 198
111.2 The 3 dimensional case 202
111.3 Uniqueness of symplectic toric manifolds 205
III.3.1 Cech cohomology 211
111.4 Proof of Theorem 1.38, part three 213
III.4.1 Morse theory on orbifolds 216
Appendix Hypersurfaces of contact type 221
Bibliography 223
|
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| spelling | Symplectic geometry of integrable Hamiltonian systems Michèle Audin ; Ana Cannas da Silva ; Eugene Lerman Basel [u.a.] Birkhäuser 2003 X, 225 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Advanced courses in mathematics, CRM Barcelona Hamiltonianen gtt Sistemas hamiltonianos larpcal Systèmes hamiltoniens Variedades simpléticas larpcal Variétés symplectiques Hamiltonian systems Symplectic manifolds Hamiltonsches System (DE-588)4139943-2 gnd rswk-swf Symplektische Geometrie (DE-588)4194232-2 gnd rswk-swf Integrables System (DE-588)4114032-1 gnd rswk-swf (DE-588)1071861417 Konferenzschrift 2001 Barcelona gnd-content Hamiltonsches System (DE-588)4139943-2 s Integrables System (DE-588)4114032-1 s Symplektische Geometrie (DE-588)4194232-2 s DE-604 Audin, Michèle 1954- Sonstige (DE-588)14151552X oth Silva, Ana Cannas da 1968- Verfasser (DE-588)122891333 aut Lerman, Eugene Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010441739&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Silva, Ana Cannas da 1968- Lerman, Eugene Symplectic geometry of integrable Hamiltonian systems Hamiltonianen gtt Sistemas hamiltonianos larpcal Systèmes hamiltoniens Variedades simpléticas larpcal Variétés symplectiques Hamiltonian systems Symplectic manifolds Hamiltonsches System (DE-588)4139943-2 gnd Symplektische Geometrie (DE-588)4194232-2 gnd Integrables System (DE-588)4114032-1 gnd |
| subject_GND | (DE-588)4139943-2 (DE-588)4194232-2 (DE-588)4114032-1 (DE-588)1071861417 |
| title | Symplectic geometry of integrable Hamiltonian systems |
| title_auth | Symplectic geometry of integrable Hamiltonian systems |
| title_exact_search | Symplectic geometry of integrable Hamiltonian systems |
| title_full | Symplectic geometry of integrable Hamiltonian systems Michèle Audin ; Ana Cannas da Silva ; Eugene Lerman |
| title_fullStr | Symplectic geometry of integrable Hamiltonian systems Michèle Audin ; Ana Cannas da Silva ; Eugene Lerman |
| title_full_unstemmed | Symplectic geometry of integrable Hamiltonian systems Michèle Audin ; Ana Cannas da Silva ; Eugene Lerman |
| title_short | Symplectic geometry of integrable Hamiltonian systems |
| title_sort | symplectic geometry of integrable hamiltonian systems |
| topic | Hamiltonianen gtt Sistemas hamiltonianos larpcal Systèmes hamiltoniens Variedades simpléticas larpcal Variétés symplectiques Hamiltonian systems Symplectic manifolds Hamiltonsches System (DE-588)4139943-2 gnd Symplektische Geometrie (DE-588)4194232-2 gnd Integrables System (DE-588)4114032-1 gnd |
| topic_facet | Hamiltonianen Sistemas hamiltonianos Systèmes hamiltoniens Variedades simpléticas Variétés symplectiques Hamiltonian systems Symplectic manifolds Hamiltonsches System Symplektische Geometrie Integrables System Konferenzschrift 2001 Barcelona |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010441739&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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