Lectures on symplectic geometry
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin ; Heidelberg
Springer
2008
|
| Ausgabe: | Corrected 2nd printing |
| Schriftenreihe: | Lecture notes in mathematics
1764 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
MARC
| LEADER | 00000nam a22000008cb4500 | ||
|---|---|---|---|
| 001 | BV035057286 | ||
| 003 | DE-604 | ||
| 005 | 20231124 | ||
| 007 | t| | ||
| 008 | 080917s2008 gw a||| |||| 00||| eng d | ||
| 020 | |a 9783540421955 |c print |9 978-3-540-42195-5 | ||
| 035 | |a (OCoLC)244058705 | ||
| 035 | |a (DE-599)BVBBV035057286 | ||
| 040 | |a DE-604 |b ger |e rda | ||
| 041 | 0 | |a eng | |
| 044 | |a gw |c DE | ||
| 049 | |a DE-824 |a DE-29T |a DE-355 |a DE-91G |a DE-83 |a DE-11 |a DE-188 |a DE-19 |a DE-20 | ||
| 082 | 0 | |a 516.36 |2 22/ger | |
| 084 | |a SI 850 |0 (DE-625)143199: |2 rvk | ||
| 084 | |a SK 370 |0 (DE-625)143234: |2 rvk | ||
| 084 | |a 53Dxx |2 msc/2000 | ||
| 084 | |a MAT 500f |2 stub | ||
| 100 | 1 | |a Silva, Ana Cannas da |d 1968- |0 (DE-588)122891333 |4 aut | |
| 245 | 1 | 0 | |a Lectures on symplectic geometry |c Ana Cannas da Silva |
| 250 | |a Corrected 2nd printing | ||
| 264 | 1 | |a Berlin ; Heidelberg |b Springer |c 2008 | |
| 300 | |a xiv, 247 Seiten |b Illustrationen | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture notes in mathematics |v 1764 | |
| 650 | 7 | |a Symplectische ruimten |2 gtt | |
| 650 | 4 | |a Symplectic geometry | |
| 650 | 0 | 7 | |a Symplektische Geometrie |0 (DE-588)4194232-2 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Symplektische Geometrie |0 (DE-588)4194232-2 |D s |
| 689 | 0 | |5 DE-604 | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-3-540-45330-7 |
| 830 | 0 | |a Lecture notes in mathematics |v 1764 |w (DE-604)BV000676446 |9 1764 | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u 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 |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-016725855 | |
Datensatz im Suchindex
| DE-19_call_number | 1601/SI 850-1764(.008) |
|---|---|
| DE-19_location | 95 |
| DE-BY-TUM_call_number | 0102 MAT 001z 2001 B 999 |
| DE-BY-TUM_katkey | 1694182 |
| DE-BY-TUM_location | 01 |
| DE-BY-TUM_media_number | 040010152159 |
| DE-BY-UBM_katkey | 4757989 |
| DE-BY-UBM_media_number | 41620570670014 |
| _version_ | 1823055230179213312 |
| 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
|
| any_adam_object | 1 |
| author | Silva, Ana Cannas da 1968- |
| author_GND | (DE-588)122891333 |
| author_facet | Silva, Ana Cannas da 1968- |
| author_role | aut |
| author_sort | Silva, Ana Cannas da 1968- |
| author_variant | a c d s acd acds |
| building | Verbundindex |
| bvnumber | BV035057286 |
| classification_rvk | SI 850 SK 370 |
| classification_tum | MAT 500f |
| ctrlnum | (OCoLC)244058705 (DE-599)BVBBV035057286 |
| dewey-full | 516.36 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.36 |
| dewey-search | 516.36 |
| dewey-sort | 3516.36 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Corrected 2nd printing |
| format | Book |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01750nam a22004458cb4500</leader><controlfield tag="001">BV035057286</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20231124 </controlfield><controlfield tag="007">t|</controlfield><controlfield tag="008">080917s2008 gw a||| |||| 00||| eng d</controlfield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9783540421955</subfield><subfield code="c">print</subfield><subfield code="9">978-3-540-42195-5</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)244058705</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV035057286</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">rda</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="044" ind1=" " ind2=" "><subfield code="a">gw</subfield><subfield code="c">DE</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-824</subfield><subfield code="a">DE-29T</subfield><subfield code="a">DE-355</subfield><subfield code="a">DE-91G</subfield><subfield code="a">DE-83</subfield><subfield code="a">DE-11</subfield><subfield code="a">DE-188</subfield><subfield code="a">DE-19</subfield><subfield code="a">DE-20</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">516.36</subfield><subfield code="2">22/ger</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SI 850</subfield><subfield code="0">(DE-625)143199:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 370</subfield><subfield code="0">(DE-625)143234:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">53Dxx</subfield><subfield code="2">msc/2000</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">MAT 500f</subfield><subfield code="2">stub</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Silva, Ana Cannas da</subfield><subfield code="d">1968-</subfield><subfield code="0">(DE-588)122891333</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Lectures on symplectic geometry</subfield><subfield code="c">Ana Cannas da Silva</subfield></datafield><datafield tag="250" ind1=" " ind2=" "><subfield code="a">Corrected 2nd printing</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Berlin ; Heidelberg</subfield><subfield code="b">Springer</subfield><subfield code="c">2008</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">xiv, 247 Seiten</subfield><subfield code="b">Illustrationen</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="1" ind2=" "><subfield code="a">Lecture notes in mathematics</subfield><subfield code="v">1764</subfield></datafield><datafield tag="650" ind1=" " ind2="7"><subfield code="a">Symplectische ruimten</subfield><subfield code="2">gtt</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Symplectic geometry</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Symplektische Geometrie</subfield><subfield code="0">(DE-588)4194232-2</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Symplektische Geometrie</subfield><subfield code="0">(DE-588)4194232-2</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="776" ind1="0" ind2="8"><subfield code="i">Erscheint auch als</subfield><subfield code="n">Online-Ausgabe</subfield><subfield code="z">978-3-540-45330-7</subfield></datafield><datafield tag="830" ind1=" " ind2="0"><subfield code="a">Lecture notes in mathematics</subfield><subfield code="v">1764</subfield><subfield code="w">(DE-604)BV000676446</subfield><subfield code="9">1764</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">Digitalisierung UB Regensburg</subfield><subfield code="q">application/pdf</subfield><subfield code="u">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</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="943" ind1="1" ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-016725855</subfield></datafield></record></collection> |
| id | DE-604.BV035057286 |
| illustrated | Illustrated |
| indexdate | 2025-02-03T17:41:49Z |
| institution | BVB |
| isbn | 9783540421955 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016725855 |
| oclc_num | 244058705 |
| open_access_boolean | |
| owner | DE-824 DE-29T DE-355 DE-BY-UBR DE-91G DE-BY-TUM DE-83 DE-11 DE-188 DE-19 DE-BY-UBM DE-20 |
| owner_facet | DE-824 DE-29T DE-355 DE-BY-UBR DE-91G DE-BY-TUM DE-83 DE-11 DE-188 DE-19 DE-BY-UBM DE-20 |
| physical | xiv, 247 Seiten Illustrationen |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| 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_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 |
| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT silvaanacannasda lecturesonsymplecticgeometry |