Holdings: Lectures on symplectic geometry

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Main Author:	Silva, Ana Cannas da 1968- (Author)
Format:	Book
Language:	English
Published:	Berlin ; Heidelberg Springer 2008
Edition:	Corrected 2nd printing
Series:	Lecture notes in mathematics 1764
Subjects:	Symplectische ruimten Symplectic geometry Symplektische Geometrie
Online Access:	Inhaltsverzeichnis
Physical Description:	xiv, 247 Seiten Illustrationen
ISBN:	9783540421955

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adam_text	Contents Foreword .......................................................... v Introduction ....................................................... xiii Part I Symplectic Manifolds 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 ............... 10 2.3 Coordinate-Free Definitions .................................. 10 2.4 Naturality of the Tautological and Canonical Forms .............. 11 Homework 2: Symplectic Volume ................................ 14 Part II Symplectomorphisms 3 Lagrangian Submanifolds ...................................: · ■ · 17 3.1 Submanifolds .............................................. 17 3.2 Lagrangian Submanifolds of T*X ............................. 18 3.3 Conormal Bundles .......................................... 19 3.4 Application to Symplectomorphisms .......................... 20 Homework 3: Tautological Form and Symplectomorphisms ......... 22 viii Contents 4 Generating Functions ........................................... 25 4.1 Constructing Symplectomorphisms ............................ 25 4.2 Method of Generating Functions .............................. 26 4.3 Application to Geodesic Flow ................................ 28 Homework 4: Geodesic Flow .................................... 30 5 Recurrence .................................................... 33 5.1 Periodic Points ............................................. 33 5.2 Billiards .................................................. 35 5.3 Poincaré Recurrence ........................................ 36 Partili Local Forms 6 Preparation for the Local Theory ................................ 41 6.1 Isotopies and Vector Fields ................................... 41 6.2 Tubular Neighborhood Theorem .............................. 43 6.3 Homotopy Formula ......................................... 45 Homework 5: Tubular Neighborhoods in Ш ....................... 47 7 Moser Theorems ............................................... 49 7.1 Notions of Equivalence for Symplectic Structures ............... 49 7.2 Moser Trick ............................................... 50 7.3 Moser Relative Theorem .................................... 52 8 Darboux-Moser- Weinstein Theory ............................... 55 8.1 Darboux Theorem .......................................... 55 8.2 Lagrangian Subspaces ....................................... 56 8.3 Weinstein Lagrangian Neighborhood Theorem .................. 57 Homework 6: Oriented Surfaces ................................. 60 9 Weinstein Tubular Neighborhood Theorem ....................... 61 9.1 Observation from Linear Algebra ............................. 61 9.2 Tubular Neighborhoods ..................................... 61 9.3 Application 1 : Tangent Space to the Group of Symplectomorphisms ............................... 63 9.4 Application 2: Fixed Points of Symplectomorphisms ............. 65 Part IV Contact Manifolds 10 Contact Forms ................................................. 69 10.1 Contact Structures .......................................... 69 10.2 Examples ................................................. 70 10.3 First Properties ............................................ 71 Homework 7: Manifolds of Contact Elements ..................... 73 Contents ix 11 Contact Dynamics.............................................. 75 11.1 Reeb Vector Fields ......................................... 75 11.2 Symplectization ............................................ 76 11.3 Conjectures of Seifert and Weinstein .......................... 77 Part V Compatible Almost Complex Structures 12 Almost Complex Structures ..................................... 83 12.1 Three Geometries .......................................... 83 12.2 Complex Structures on Vector Spaces .......................... 84 12.3 Compatible Structures ....................................... 86 Homework 8: Compatible Linear Structures ...................... 88 13 Compatible Triples ............................................. 89 13.1 Compatibility .............................................. 89 13.2 Triple of Structures ......................................... 90 13.3 First Consequences ......................................... 91 Homework 9: Contractibility .................................... 92 14 Dolbeault Theory .............................................. 93 14.1 Splittings ................................................. 93 14.2 Forms of Type (£,m) ........................................ 94 14.3 J-Holomorphic Functions .................................... 95 14.4 Dolbeault Cohomology ...................................... 96 Homework 10: Integrability ..................................... 98 Part VI Kahler Manifolds 15 Complex Manifolds ............................................101 15.1 Complex Charts ............................................101 15.2 Forms on Complex Manifolds ................................103 15.3 Differentials ...............................................104 Homework 11: Complex Projective Space .........................107 16 Kahler Forms ..................................................109 16.1 Kahler Forms ..............................................109 16.2 An Application ............................................111 16.3 Recipe to Obtain Kahler Forms ...............................112 16.4 Local Canonical Form for Kahler Forms .......................113 Homework 12: The Fubini-Study Structure .......................115 x Contents 17 Compact Kahler Manifolds .....................................117 17.1 Hodge Theory .............................................117 17.2 Immediate Topological Consequences .........................119 17.3 Compact Examples and Counterexamples ......................120 17.4 Main Kahler Manifolds ......................................122 Part VII Hamiltonian Mechanics 18 Hamiltonian Vector Fields ......................................127 18.1 Hamiltonian and Symplectic Vector Fields ......................127 18.2 Classical Mechanics ........................................129 18.3 Brackets ..................................................130 18.4 Integrable Systems .........................................131 Homework 13: Simple Pendulum ................................134 19 Variational Principles ...........................................135 19.1 Equations of Motion ........................................135 19.2 Principle of Least Action ....................................136 19.3 Variational Problems ........................................137 19.4 vSolving the Euler-Lagrange Equations .........................138 19.5 Minimizing Properties ......................................140 Homework 14: Minimizing Geodesies ............................141 20 Legendre Transform ............................................143 20.1 Strict Convexity ............................................143 20.2 Legendre Transform ........................................144 20.3 Application to Variational Problems ...........................145 Homework 15: Legendre Transform ..............................147 Part VIII Moment Maps 21 Actions .......................................................151 21.1 One-Parameter Groups of Diffeomorphisms ....................151 21.2 Lie Groups ................................................152 21.3 Smooth Actions ............................................152 21.4 Symplectic and Hamiltonian Actions ..........................153 21.5 Adjoint and Coadjoint Representations .........................154 Homework 16: Hermitian Matrices ..............................156 22 Hamiltonian Actions ...........................................157 22.1 Moment and Comoment Maps ................................157 22.2 Orbit Spaces ...............................................159 Contents xi 22.3 Preview of Reduction....................................... 160 22.4 Classical Examples .........................................161 Homework 17: Coadjoint Orbits .................................163 Part IX Symplectic Reduction 23 The Marsden- Weinstein-Meyer Theorem .........................167 23.1 Statement .................................................167 23.2 Ingredients ................................................168 23.3 Proof of the Marsden- Weinstein-Meyer Theorem ................171 24 Reduction .....................................................173 24.1 Noether Principle ...........................................173 24.2 Elementary Theory of Reduction ..............................173 24.3 Reduction for Product Groups ................................175 24.4 Reduction at Other Levels ...................................176 24.5 Orbifolds .................................................176 Homework 18: Spherical Pendulum ..............................178 Part X Moment Maps Revisited 25 Moment Map in Gauge Theory ..................................183 25.1 Connections on a Principal Bundle ............................183 25.2 Connection and Curvature Forms .............................184 25.3 Symplectic Structure on the Space of Connections ...............186 25.4 Action of the Gauge Group ..................................187 25.5 Case of Circle Bundles ......................................187 Homework 19: Examples of Moment Maps .......................191 26 Existence and Uniqueness of Moment Maps .......................193 26.1 Lie Algebras of Vector Fields ................................193 26.2 Lie Algebra Cohomology ....................................194 26.3 Existence of Moment Maps ..................................195 26.4 Uniqueness of Moment Maps ................................ Ì 96 Homework 20: Examples of Reduction ...........................198 27 Convexity .....................................................199 27.1 Convexity Theorem .........................................199 27.2 Effective Actions ...........................................201 27.3 Examples .................................................202 Homework 21: Connectedness ...................................204 xii Contents Part XI Symplectic Toric Manifolds 28 Classification of Symplectic Toric Manifolds .......................209 28.1 Delzant Polytopes ..........................................209 28.2 Delzant Theorem ...........................................211 28.3 Sketch of Delzant Construction ...............................212 29 Delzant Construction ...........................................215 29.1 Algebraic Set-Up ...........................................215 29.2 The Zero-Level ............................................216 29.3 Conclusion of the Delzant Construction ........................218 29.4 Idea Behind the Delzant Construction ..........................219 Homework 22: Delzant Theorem .................................221 30 Duistermaat-Heckman Theorems ................................223 30.1 Duistermaat-Heckman Polynomial ............................223 30.2 Local Form for Reduced Spaces ..............................225 30.3 Variation of the Symplectic Volume ...........................227 Homework 23: S -Equivariant Cohomology .......................229 References .........................................................233 Index ...................................... ........239
adam_txt	Contents Foreword . v Introduction . xiii Part I Symplectic Manifolds 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 . 10 2.3 Coordinate-Free Definitions . 10 2.4 Naturality of the Tautological and Canonical Forms . 11 Homework 2: Symplectic Volume . 14 Part II Symplectomorphisms 3 Lagrangian Submanifolds .: · ■ · 17 3.1 Submanifolds . 17 3.2 Lagrangian Submanifolds of T*X . 18 3.3 Conormal Bundles . 19 3.4 Application to Symplectomorphisms . 20 Homework 3: Tautological Form and Symplectomorphisms . 22 viii Contents 4 Generating Functions . 25 4.1 Constructing Symplectomorphisms . 25 4.2 Method of Generating Functions . 26 4.3 Application to Geodesic Flow . 28 Homework 4: Geodesic Flow . 30 5 Recurrence . 33 5.1 Periodic Points . 33 5.2 Billiards . 35 5.3 Poincaré Recurrence . 36 Partili Local Forms 6 Preparation for the Local Theory . 41 6.1 Isotopies and Vector Fields . 41 6.2 Tubular Neighborhood Theorem . 43 6.3 Homotopy Formula . 45 Homework 5: Tubular Neighborhoods in Ш" . 47 7 Moser Theorems . 49 7.1 Notions of Equivalence for Symplectic Structures . 49 7.2 Moser Trick . 50 7.3 Moser Relative Theorem . 52 8 Darboux-Moser- Weinstein Theory . 55 8.1 Darboux Theorem . 55 8.2 Lagrangian Subspaces . 56 8.3 Weinstein Lagrangian Neighborhood Theorem . 57 Homework 6: Oriented Surfaces . 60 9 Weinstein Tubular Neighborhood Theorem . 61 9.1 Observation from Linear Algebra . 61 9.2 Tubular Neighborhoods . 61 9.3 Application 1 : Tangent Space to the Group of Symplectomorphisms . 63 9.4 Application 2: Fixed Points of Symplectomorphisms . 65 Part IV Contact Manifolds 10 Contact Forms . 69 10.1 Contact Structures . 69 10.2 Examples . 70 10.3 First Properties . 71 Homework 7: Manifolds of Contact Elements . 73 Contents ix 11 Contact Dynamics. 75 11.1 Reeb Vector Fields . 75 11.2 Symplectization . 76 11.3 Conjectures of Seifert and Weinstein . 77 Part V Compatible Almost Complex Structures 12 Almost Complex Structures . 83 12.1 Three Geometries . 83 12.2 Complex Structures on Vector Spaces . 84 12.3 Compatible Structures . 86 Homework 8: Compatible Linear Structures . 88 13 Compatible Triples . 89 13.1 Compatibility . 89 13.2 Triple of Structures . 90 13.3 First Consequences . 91 Homework 9: Contractibility . 92 14 Dolbeault Theory . 93 14.1 Splittings . 93 14.2 Forms of Type (£,m) . 94 14.3 J-Holomorphic Functions . 95 14.4 Dolbeault Cohomology . 96 Homework 10: Integrability . 98 Part VI Kahler Manifolds 15 Complex Manifolds .101 15.1 Complex Charts .101 15.2 Forms on Complex Manifolds .103 15.3 Differentials .104 Homework 11: Complex Projective Space .107 16 Kahler Forms .109 16.1 Kahler Forms .109 16.2 An Application .111 16.3 Recipe to Obtain Kahler Forms .112 16.4 Local Canonical Form for Kahler Forms .113 Homework 12: The Fubini-Study Structure .115 x Contents 17 Compact Kahler Manifolds .117 17.1 Hodge Theory .117 17.2 Immediate Topological Consequences .119 17.3 Compact Examples and Counterexamples .120 17.4 Main Kahler Manifolds .122 Part VII Hamiltonian Mechanics 18 Hamiltonian Vector Fields .127 18.1 Hamiltonian and Symplectic Vector Fields .127 18.2 Classical Mechanics .129 18.3 Brackets .130 18.4 Integrable Systems .131 Homework 13: Simple Pendulum .134 19 Variational Principles .135 19.1 Equations of Motion .135 19.2 Principle of Least Action .136 19.3 Variational Problems .137 19.4 vSolving the Euler-Lagrange Equations .138 19.5 Minimizing Properties .140 Homework 14: Minimizing Geodesies .141 20 Legendre Transform .143 20.1 Strict Convexity .143 20.2 Legendre Transform .144 20.3 Application to Variational Problems .145 Homework 15: Legendre Transform .147 Part VIII Moment Maps 21 Actions .151 21.1 One-Parameter Groups of Diffeomorphisms .151 21.2 Lie Groups .152 21.3 Smooth Actions .152 21.4 Symplectic and Hamiltonian Actions .153 21.5 Adjoint and Coadjoint Representations .154 Homework 16: Hermitian Matrices .156 22 Hamiltonian Actions .157 22.1 Moment and Comoment Maps .157 22.2 Orbit Spaces .159 Contents xi 22.3 Preview of Reduction. 160 22.4 Classical Examples .161 Homework 17: Coadjoint Orbits .163 Part IX Symplectic Reduction 23 The Marsden- Weinstein-Meyer Theorem .167 23.1 Statement .167 23.2 Ingredients .168 23.3 Proof of the Marsden- Weinstein-Meyer Theorem .171 24 Reduction .173 24.1 Noether Principle .173 24.2 Elementary Theory of Reduction .173 24.3 Reduction for Product Groups .175 24.4 Reduction at Other Levels .176 24.5 Orbifolds .176 Homework 18: Spherical Pendulum .178 Part X Moment Maps Revisited 25 Moment Map in Gauge Theory .183 25.1 Connections on a Principal Bundle .183 25.2 Connection and Curvature Forms .184 25.3 Symplectic Structure on the Space of Connections .186 25.4 Action of the Gauge Group .187 25.5 Case of Circle Bundles .187 Homework 19: Examples of Moment Maps .191 26 Existence and Uniqueness of Moment Maps .193 26.1 Lie Algebras of Vector Fields .193 26.2 Lie Algebra Cohomology .194 26.3 Existence of Moment Maps .195 26.4 Uniqueness of Moment Maps . Ì 96 Homework 20: Examples of Reduction .198 27 Convexity .199 27.1 Convexity Theorem .199 27.2 Effective Actions .201 27.3 Examples .202 Homework 21: Connectedness .204 xii Contents Part XI Symplectic Toric Manifolds 28 Classification of Symplectic Toric Manifolds .209 28.1 Delzant Polytopes .209 28.2 Delzant Theorem .211 28.3 Sketch of Delzant Construction .212 29 Delzant Construction .215 29.1 Algebraic Set-Up .215 29.2 The Zero-Level .216 29.3 Conclusion of the Delzant Construction .218 29.4 Idea Behind the Delzant Construction .219 Homework 22: Delzant Theorem .221 30 Duistermaat-Heckman Theorems .223 30.1 Duistermaat-Heckman Polynomial .223 30.2 Local Form for Reduced Spaces .225 30.3 Variation of the Symplectic Volume .227 Homework 23: S'-Equivariant Cohomology .229 References .233 Index . .239
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spelling	Silva, Ana Cannas da 1968- (DE-588)122891333 aut Lectures on symplectic geometry Ana Cannas da Silva Corrected 2nd printing Berlin ; Heidelberg Springer 2008 xiv, 247 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1764 Symplectische ruimten gtt Symplectic geometry Symplektische Geometrie (DE-588)4194232-2 gnd rswk-swf Symplektische Geometrie (DE-588)4194232-2 s DE-604 Erscheint auch als Online-Ausgabe 978-3-540-45330-7 Lecture notes in mathematics 1764 (DE-604)BV000676446 1764 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016725855&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 Symplectische ruimten gtt Symplectic geometry 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_exact_search_txtP	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	Symplectische ruimten gtt Symplectic geometry Symplektische Geometrie (DE-588)4194232-2 gnd
topic_facet	Symplectische ruimten Symplectic geometry Symplektische Geometrie
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016725855&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
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work_keys_str_mv	AT silvaanacannasda lecturesonsymplecticgeometry

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