Modern differential geometry for physicists:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New Jersey [u.a.]
World Scientific
2005
|
| Ausgabe: | 2. ed., repr. |
| Schriftenreihe: | World Scientific lecture notes in physics
61 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIII, 290 S. graph. Darst. |
| ISBN: | 9810235623 9810235550 |
Internformat
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| 245 | 1 | 0 | |a Modern differential geometry for physicists |c Chris J. Isham |
| 250 | |a 2. ed., repr. | ||
| 264 | 1 | |a New Jersey [u.a.] |b World Scientific |c 2005 | |
| 300 | |a XIII, 290 S. |b graph. Darst. | ||
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Datensatz im Suchindex
| _version_ | 1804136662188425216 |
|---|---|
| adam_text | Contents
1 An
Introduction
to Topology
1
1.1
Preliminary Remarks
................... 1
1.1.1
Remarks on differential geometry
........ 1
1.1.2
Remarks on topology
............... 2
1.2
Metric Spaces
. ...................... 3
1.2.1
The simple idea of convergence
......... 3
1.2.2
The idea of a metric space
............ 5
1.2.3
Examples of metric spaces
............ 8
1.2.4
Operations on metrics
.............. 10
1.2.5
Some topological concepts in metric spaces
... 11
1.3
Partially Ordered Sets and Lattices
........... 14
1.3.1
Partially ordered sets
............... 14
1.3.2
Lattices
...................... 18
1.4
General Topology
..................... 23
1.4.1
An example of non-metric convergence
..... 23
1.4.2
The idea of a neighbourhood space
....... 25
1.4.3
Topological spaces
................ 32
1.4.4
Some examples of topologies on a finite set
... 37
1.4.5
A topology as a lattice
.............. 40
1.4.6
The lattice of topologies
т(Х)
on a set X
.... 42
χ
CONTENTS
1.4.7
Some properties of convergence in a general topo-
logical space
.................... 45
1.4.8
The idea of a compact space
........... 46
1.4.9
Maps between topological spaces
........ 48
1.4.10
The idea of a homeomorphism
.......... 51
1.4.11
Separation axioms
................ 52
1.4.12
Frames and locales
. ............... 54
2
Differentiable Manifolds
59
2.1
Preliminary Remarks
................... 59
2.2
The Main Definitions
................... 60
2.2.1
Coordinate charts
................. 60
2.2.2
Some examples of differentiable manifolds
... 64
2.2.3
Differentiable maps
................ 68
2.3
Tangent Spaces
...................... 70
2.3.1
The intuitive idea
. . ............... 70
2.3.2
A tangent vector as an equivalence class of curves
72
2.3.3
The vector space structure on TpAi
....... 76
2.3.4
The push-forward of an equivalence class of curves.
77
2.3.5
Tangent vectors as derivations
.......... 79
2.3.6
The tangent space TVV of a vector space V
... 90
2.3.7
A simple example of the push-forward operation
91
2.3.8
The tangent space of a product manifold
.... 92
3
Vector Fields and n-Forms
97
3.1
Vector Fields
....................... 97
3.1.1
The main definition
................ 97
3.1.2
The vector field commutator
........... 102
3.1.3
h-related vector fields
............... 104
CONTENTS xi
3.2 Integral
Curves
and Flows
................107
3.2.1
Complete vector fields
..............107
3.2.2
One-parameter groups of diffeomorphisms
. . .111
3.2.3
Local flows
.....................115
3.2.4
Some concrete examples of integral curves and
flows
........................117
3.3
Cotangent Vectors
....................121
3.3.1
The algebraic dual of a vector space
.......121
3.3.2
The main definitions
...............123
3.3.3
The pull-back of a one-form
........... 126
3.3.4
A simple example of the pull-back operation
. . 129
3.3.5
The Lie derivative
................130
3.4
General Tensors and
η
-Forms..............
132
3.4.1
The tensor product operation
..........132
3.4.2
The idea of an n-form
..............135
3.4.3
The definition of the exterior derivative
.....137
3.4.4
The local nature of the exterior derivative
. . .138
3.5
DeRham Cohomology
...................140
4
Lie Groups
149
4.1
The Basic Ideas
......................149
4.1.1
The first definitions
................149
4.1.2
The orthogonal group
...............155
4.2
The Lie Algebra of a Lie Group
.............157
4.2.1
Left-invariant vector fields
............ 157
4.2.2
The completeness of a left-invariant vector field
162
4.2.3
The exponential map
.............. . 165
4.2.4
The Lie algebra of
GX(n,
IR)
........... 169
4.3
Left-Invariant Forms
................... 170
xii CONTENTS
4.3.1
The basic definitions
...............170
4.3.2
The Cartan-Maurer form
.............172
4.4
Transformation Groups
..................175
4.4.1
The basic definitions
...............175
4.4.2
Different types of group action
..........179
4.4.3
The main theorem for transitive group actions
. 183
4.4.4
Some important transitive actions
........ 185
4.5
Infinitesimal IVansformations
..............190
4.5.1
The induced vector field
.............190
4.5.2
The main result
..................195
5
Fibre Bundles
199
5.1
Bundles in General
....................199
5.1.1
Introduction
....................199
5.1.2
The definition of a bundle
............ 201
5.1.3
The idea of a cross-section
............207
5.1.4
Covering spaces and sheaves
...........210
5.1.5
The definition of a sub-bundle
. .........213
5.1.6
Maps between bundles
..............214
5.1.7
The pull-back operation
.............216
5.1.8
Universal bundles
.................218
5.2
Principal Fibre Bundles
.................220
5.2.1
The main definition
................220
5.2.2
Principal bundle maps
..............224
5.2.3
Cross-sections of a principal bundle
.......230
5.3
Associated Bundles
....................232
5.3.1
The main definition
................232
5.3.2
Associated bundle maps
.............236
CONTENTS xiii
5.3.3
Restricting and extending the structure group
. 240
5.3.4
Riemannian metrics as reductions of
Ђ(М)
. . 243
5.3.5
Cross-sections as functions on the principle bun¬
dle
.........................246
5.4
Vector Bundles
......................248
5.4.1
The main definitions
...............248
5.4.2
Vector bundles as associated bundles
.......249
6
Connections in a Bundle
253
6.1
Connections in a Principal Bundle
............253
6.1.1
The definition of a connection
..........253
6.1.2
Local representatives of a connection
......256
6.1.3
Local gauge transformations
...........258
6.1.4
Connections in the frame bundle
........261
6.2
Parallel Transport
.....................262
6.2.1
Parallel transport in a principal bundle
.....262
6.2.2
Parallel transport in an associated bundle
. . . 267
6.2.3
Covariant differentiation
.............269
6.2.4
The curvature two-form
.............271
BIBLIOGRAPHY
277
INDEX
281
|
| adam_txt |
Contents
1 An
Introduction
to Topology
1
1.1
Preliminary Remarks
. 1
1.1.1
Remarks on differential geometry
. 1
1.1.2
Remarks on topology
. 2
1.2
Metric Spaces
. . 3
1.2.1
The simple idea of convergence
. 3
1.2.2
The idea of a metric space
. 5
1.2.3
Examples of metric spaces
. 8
1.2.4
Operations on metrics
. 10
1.2.5
Some topological concepts in metric spaces
. 11
1.3
Partially Ordered Sets and Lattices
. 14
1.3.1
Partially ordered sets
. 14
1.3.2
Lattices
. 18
1.4
General Topology
. 23
1.4.1
An example of non-metric convergence
. 23
1.4.2
The idea of a neighbourhood space
. 25
1.4.3
Topological spaces
. 32
1.4.4
Some examples of topologies on a finite set
. 37
1.4.5
A topology as a lattice
. 40
1.4.6
The lattice of topologies
т(Х)
on a set X
. 42
χ
CONTENTS
1.4.7
Some properties of convergence in a general topo-
logical space
. 45
1.4.8
The idea of a compact space
. 46
1.4.9
Maps between topological spaces
. 48
1.4.10
The idea of a homeomorphism
. 51
1.4.11
Separation axioms
. 52
1.4.12
Frames and locales
. . 54
2
Differentiable Manifolds
59
2.1
Preliminary Remarks
. 59
2.2
The Main Definitions
. 60
2.2.1
Coordinate charts
. 60
2.2.2
Some examples of differentiable manifolds
. 64
2.2.3
Differentiable maps
. 68
2.3
Tangent Spaces
. 70
2.3.1
The intuitive idea
. . . 70
2.3.2
A tangent vector as an equivalence class of curves
72
2.3.3
The vector space structure on TpAi
. 76
2.3.4
The push-forward of an equivalence class of curves.
77
2.3.5
Tangent vectors as derivations
. 79
2.3.6
The tangent space TVV of a vector space V
. 90
2.3.7
A simple example of the push-forward operation
91
2.3.8
The tangent space of a product manifold
. 92
3
Vector Fields and n-Forms
97
3.1
Vector Fields
. 97
3.1.1
The main definition
. 97
3.1.2
The vector field commutator
. 102
3.1.3
h-related vector fields
. 104
CONTENTS xi
3.2 Integral
Curves
and Flows
.107
3.2.1
Complete vector fields
.107
3.2.2
One-parameter groups of diffeomorphisms
. . .111
3.2.3
Local flows
.115
3.2.4
Some concrete examples of integral curves and
flows
.117
3.3
Cotangent Vectors
.121
3.3.1
The algebraic dual of a vector space
.121
3.3.2
The main definitions
.123
3.3.3
The pull-back of a one-form
. 126
3.3.4
A simple example of the pull-back operation
. . 129
3.3.5
The Lie derivative
.130
3.4
General Tensors and
η
-Forms.
132
3.4.1
The tensor product operation
.132
3.4.2
The idea of an n-form
.135
3.4.3
The definition of the exterior derivative
.137
3.4.4
The local nature of the exterior derivative
. . .138
3.5
DeRham Cohomology
.140
4
Lie Groups
149
4.1
The Basic Ideas
.149
4.1.1
The first definitions
.149
4.1.2
The orthogonal group
.155
4.2
The Lie Algebra of a Lie Group
.157
4.2.1
Left-invariant vector fields
. 157
4.2.2
The completeness of a left-invariant vector field
162
4.2.3
The exponential map
. . 165
4.2.4
The Lie algebra of
GX(n,
IR)
. 169
4.3
Left-Invariant Forms
. 170
xii CONTENTS
4.3.1
The basic definitions
.170
4.3.2
The Cartan-Maurer form
.172
4.4
Transformation Groups
.175
4.4.1
The basic definitions
.175
4.4.2
Different types of group action
.179
4.4.3
The main theorem for transitive group actions
. 183
4.4.4
Some important transitive actions
. 185
4.5
Infinitesimal IVansformations
.190
4.5.1
The induced vector field
.190
4.5.2
The main result
.195
5
Fibre Bundles
199
5.1
Bundles in General
.199
5.1.1
Introduction
.199
5.1.2
The definition of a bundle
. 201
5.1.3
The idea of a cross-section
.207
5.1.4
Covering spaces and sheaves
.210
5.1.5
The definition of a sub-bundle
. .213
5.1.6
Maps between bundles
.214
5.1.7
The pull-back operation
.216
5.1.8
Universal bundles
.218
5.2
Principal Fibre Bundles
.220
5.2.1
The main definition
.220
5.2.2
Principal bundle maps
.224
5.2.3
Cross-sections of a principal bundle
.230
5.3
Associated Bundles
.232
5.3.1
The main definition
.232
5.3.2
Associated bundle maps
.236
CONTENTS xiii
5.3.3
Restricting and extending the structure group
. 240
5.3.4
Riemannian metrics as reductions of
Ђ(М)
. . 243
5.3.5
Cross-sections as functions on the principle bun¬
dle
.246
5.4
Vector Bundles
.248
5.4.1
The main definitions
.248
5.4.2
Vector bundles as associated bundles
.249
6
Connections in a Bundle
253
6.1
Connections in a Principal Bundle
.253
6.1.1
The definition of a connection
.253
6.1.2
Local representatives of a connection
.256
6.1.3
Local gauge transformations
.258
6.1.4
Connections in the frame bundle
.261
6.2
Parallel Transport
.262
6.2.1
Parallel transport in a principal bundle
.262
6.2.2
Parallel transport in an associated bundle
. . . 267
6.2.3
Covariant differentiation
.269
6.2.4
The curvature two-form
.271
BIBLIOGRAPHY
277
INDEX
281 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Isham, Chris J. |
| author_facet | Isham, Chris J. |
| author_role | aut |
| author_sort | Isham, Chris J. |
| author_variant | c j i cj cji |
| building | Verbundindex |
| bvnumber | BV022546460 |
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| callnumber-label | QA641 |
| callnumber-raw | QA641 |
| callnumber-search | QA641 |
| callnumber-sort | QA 3641 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 370 |
| ctrlnum | (OCoLC)254048759 (DE-599)BSZ261665960 |
| dewey-full | 516.3/6 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3/6 |
| dewey-search | 516.3/6 |
| dewey-sort | 3516.3 16 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 2. ed., repr. |
| format | Book |
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| id | DE-604.BV022546460 |
| illustrated | Illustrated |
| index_date | 2024-07-02T18:12:03Z |
| indexdate | 2024-07-09T20:59:57Z |
| institution | BVB |
| isbn | 9810235623 9810235550 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015752824 |
| oclc_num | 254048759 |
| open_access_boolean | |
| owner | DE-706 DE-384 |
| owner_facet | DE-706 DE-384 |
| physical | XIII, 290 S. graph. Darst. |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | World Scientific |
| record_format | marc |
| series | World Scientific lecture notes in physics |
| series2 | World Scientific lecture notes in physics |
| spelling | Isham, Chris J. Verfasser aut Modern differential geometry for physicists Chris J. Isham 2. ed., repr. New Jersey [u.a.] World Scientific 2005 XIII, 290 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier World Scientific lecture notes in physics 61 Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Physik (DE-588)4045956-1 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 s Physik (DE-588)4045956-1 s DE-604 Mathematische Physik (DE-588)4037952-8 s 1\p DE-604 World Scientific lecture notes in physics 61 (DE-604)BV000819327 61 Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015752824&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Isham, Chris J. Modern differential geometry for physicists World Scientific lecture notes in physics Mathematische Physik (DE-588)4037952-8 gnd Differentialgeometrie (DE-588)4012248-7 gnd Physik (DE-588)4045956-1 gnd |
| subject_GND | (DE-588)4037952-8 (DE-588)4012248-7 (DE-588)4045956-1 |
| title | Modern differential geometry for physicists |
| title_auth | Modern differential geometry for physicists |
| title_exact_search | Modern differential geometry for physicists |
| title_exact_search_txtP | Modern differential geometry for physicists |
| title_full | Modern differential geometry for physicists Chris J. Isham |
| title_fullStr | Modern differential geometry for physicists Chris J. Isham |
| title_full_unstemmed | Modern differential geometry for physicists Chris J. Isham |
| title_short | Modern differential geometry for physicists |
| title_sort | modern differential geometry for physicists |
| topic | Mathematische Physik (DE-588)4037952-8 gnd Differentialgeometrie (DE-588)4012248-7 gnd Physik (DE-588)4045956-1 gnd |
| topic_facet | Mathematische Physik Differentialgeometrie Physik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015752824&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000819327 |
| work_keys_str_mv | AT ishamchrisj moderndifferentialgeometryforphysicists |