Differential geometry: bundles, connections, metrics and curvature
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Oxford [u.a.]
Oxford Univ. Press
2011
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | Oxford graduate texts in mathematics
23 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024386452&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024386452&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XIII, 298 S. graph. Darst. |
| ISBN: | 9780199605880 9780199605873 |
Internformat
MARC
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| 020 | |a 9780199605873 |c pbk. |9 978-0-19-960587-3 | ||
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| 100 | 1 | |a Taubes, Clifford |d 1954- |e Verfasser |0 (DE-588)172877423 |4 aut | |
| 245 | 1 | 0 | |a Differential geometry |b bundles, connections, metrics and curvature |c Clifford Henry Taubes |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Oxford [u.a.] |b Oxford Univ. Press |c 2011 | |
| 300 | |a XIII, 298 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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Datensatz im Suchindex
| DE-BY-TUM_call_number | 0202 MAT 530f 2012 A 5872 |
|---|---|
| DE-BY-TUM_katkey | 1779090 |
| DE-BY-TUM_location | 02 |
| DE-BY-TUM_media_number | 040020795533 |
| _version_ | 1821933560196497409 |
| adam_text | geometry
and theoretical physics. The purpose of this book is to supply a graduate student in
mathematics or theoretical physics with the fundamentals of these objects.
The book introduces as needed many of the basic results about differentiable manifolds,
smooth maps, differential forms, vector fields, Lie groups and Grassmannians. This
introduction includes many of the basic tools used in differential topology. The differential
geometry topics include the basic theorems about geodesies and Jacobi fields, the classifi¬
cation theorem for flat connections, the definition of characteristic classes, and an
introduction to complex and
Kahler
geometry.
The book also contains many of the classical examples and applications for the subjects
it covers. The philosophy is to use these examples to bring abstract ideas to life. The
presentation is essentially self-contained as proofs are given for most all assertions, either
in the text or in chapter appendices.
ALSO AVAILABLE FROM
OXFORD UNIVERSITY PRESS
The Many Facets ofGeometi
A Tribute to Nigel Hitchin
Edited bv Oscar Garcia-
Prada,
lean Pierre
Bourguignon
and Simon
Salamon
.
Donaldson and P. B. Kronheimer
With Applications to Physics
Second Edition
Robert H.
Wasserman
Contents
1
Smooth manifolds
1.1
Smooth manifolds
1
1.2
The inverse function theorem and implicit function theorem
3
1.3
Submanifolds of Rm
4
1.4
Submanifolds of manifolds
7
1.5
More constructions of manifolds
8
1.6
More smooth manifolds: The Grassmannians
9
Appendix
1.1
How to prove the inverse function and implicit
function theorems
11
Appendix
1.2
Partitions of unity
13
Additional reading
13
2
Matrices and Lie groups
14
2.1
The general linear group
14
2.2
Lie groups
15
2.3
Examples of Lie groups
16
2.4
Some complex Lie groups
17
2.5
The groups Sl(n; C), U(n) and SU(n)
19
2.6
Notation with regards to matrices and differentials
21
Appendix
2.1
The transition functions for the Grassmannians
22
Additional reading
24
3
Introduction to vector bundles
25
3.1
The definition
25
3.2
The standard definition
27
3.3
The first examples of vector bundles
28
3.4
The tangent bundle
29
viii Contents
3.5
Tangent
bundle examples
31
3.6
The cotangent bundle
33
3.7
Bundle homomorphisms
34
3.8
Sections of vector bundles
35
3.9
Sections of TM and T*M
36
Additional reading
38
4
Algebra of vector bundles
39
4.1
Subbundles
39
4.2
Quotient bundles
40
4.3
The dual bundle
41
4.4
Bundles of homomorphisms
42
4.5
Tensor product bundles
43
4.6
The direct sum
43
4.7
Tensor powers
44
Additional reading
46
5
Maps and vector bundles
48
5.1
The pull-back construction
48
5.2
Pull-backs and Grassmannians
49
5.3
Pull-back of differential forms and push-forward of
vector fields
50
5.4
Invariant forms and vector fields on Lie groups
52
5.5
The exponential map on a matrix group
53
5.6
The exponential map and right/left
invariance
on Gl(n; C)
and its subgroups
55
5.7
Immersion, submersion and transversality
57
Additional reading
58
6
Vector bundles with Cn as fiber
59
6.1
Definitions
59
6.2
Comparing definitions
60
6.3
Examples: The complexification
62
6.4
Complex bundles over surfaces in
Ш3
63
6.5
The tangent bundle to a surface in R3
64
6.6
Bundles over 4-dimensional submanifolds in
Ш5
64
6.7
Complex bundles over 4-dimensional manifolds
65
6.8
Complex Grassmannians
65
Contents ix
6.9
The exterior product construction
68
6.10
Algebraic operations
69
6.11
Pull-back
70
Additional reading
71
7
Metrics on vector bundles
72
7.1
Metrics and transition functions for real vector bundles
73
7.2
Metrics and transition functions for complex
vector bundles
75
7.3
Metrics, algebra and maps
75
7.4
Metrics on TM
77
Additional reading
77
8
Geodesies
78
8.1
Riemannian metrics and distance
78
8.2
Length minimizing curves
79
8.3
The existence of geodesies
81
8.4
First examples
82
8.5
Geodesies on SO(n)
85
8.6
Geodesies on U(n) and SU(n)
89
8.7
Geodesies and matrix groups
92
Appendix
8.1
The proof of the vector field theorem
93
Additional reading
94
9
Properties of geodesies
96
9.1
The maximal extension of a geodesic
96
9.2
The exponential map
96
9.3
Gaussian coordinates
98
9.4
The proof of the geodesic theorem
100
Additional reading
103
10
Principal bundles
104
10.1
The definition
104
10.2
A cocycle definition
105
10.3
Principal bundles constructed from vector bundles
106
10.4
Quotients of Lie groups by subgroups
108
Contents
10.5
Examples of Lie group quotients
110
10.6
Cocycle construction examples
113
10.7
Pull-backs of principal bundles
116
10.8
Reducible principal bundles
118
10.9
Associated vector bundles
119
Appendix
10.1
Proof of Proposition
10.1 121
Additional reading
124
11
Covariant derivatives and connections
125
11.1
Covariant derivatives
125
11.2
The space of covariant derivatives
126
11.3
Another construction of covariant derivatives
127
11.4
Principal bundles and connections
128
11.5
Connections and covariant derivatives
134
11.6
Horizontal lifts
135
11.7
An application to the classification of principal
G-bundles up to Isomorphism
136
11.8
Connections, covariant derivatives and pull-back bundles
137
Additional reading
138
12
Covariant derivatives, connections and curvature
139
12.1
Exterior derivative
139
12.2
Closed forms, exact forms, diffeomorphisms and
De Rham
cohomology
141
12.3
Lie derivative
143
12.4
Curvature and covariant derivatives
144
12.5
An example
146
12.6
Curvature and commutators
148
12.7
Connections and curvature
148
12.8
The horizontal subbundle revisited
150
Additional reading
151
13
Flat connections and holonomy
152
13.1
Flat connections
152
13.2
Flat connections on bundles over the circle
153
13.3
Foliations
155
13.4
Automorphisms of a principal bundle
156
13.5
The fundamental group
157
Contents xi
13.6
The flat connections on bundles over
M
159
13.7
The universal covering space
159
13.8
Holonomy and curvature
160
13.9
Proof of the classification theorem for flat connections
162
Appendix
13.1
Smoothing maps
164
Appendix
13.2
The proof of the Frobenius theorem
166
Additional reading
169
14
Curvature polynomials and characteristic classes
170
14.1
The BianchMdentity
170
14.2
Characteristic forms
171
14.3
Characteristic classes: Part
1 174
14.4
Characteristic classes: Part
2 175
14.5
Characteristic classes for complex vector bundles
and the Chern classes
177
14.6
Characteristic classes for real vector bundles and
the Pontryagin classes
179
14.7
Examples of bundles with nonzero Chern classes
180
14.8
The degree of the map
g
->
gm from
SU
(2)
to itself
189
14.9
A Chern-Simons form
190
Appendix
14.1
The ad-invariant functions on M(n; C)
190
Appendix
14.2
Integration on manifolds
192
Appendix
14.3
The degree of a map
197
Additional reading
204
15
Covariant derivatives and metrics
205
15.1
Metric compatible covariant derivatives
205
15.2
Torsion free covariant derivatives on T*M
208
15.3
The Levi-Civita connection/covariant derivative
210
15.4
A formula for the Levi-Civita connection
211
15.5
Covariantly constant sections
212
15.6
An example of the Levi-Civita connection
214
15.7
The curvature of the Levi-Civita connection
216
Additional reading
218
16
The Riemann curvature tensor
220
16.1
Spherical metrics, flat metrics and hyperbolic metrics
220
16.2
The Schwarzchild metric
223
xii Contents
16.3
Curvature conditions
224
16.4
Manifolds of dimension
2:
The Gauss-Bonnet formula
227
16.5
Metrics on manifolds of dimension
2 229
16.6
Conformai
changes
230
16.7
Sectional curvatures and universal covering spaces
232
16.8
The Jacobi field equation
233
16.9
Constant sectional curvature and the Jacobi
field equation
236
16.10
Manifolds of dimension
3 238
16.11
The Riemannian curvature of a compact matrix group
239
Additional reading
244
17
Complex manifolds
245
17.1
Some basics concerning holomorphic functions on Cn
246
17.2
The definition of a complex manifold
247
17.3
First examples of complex manifolds
248
17.4
The Newlander-Nirenberg theorem
251
17.5
Metrics and almost complex structures on TM
255
17.6
The almost
Kahler
2-form
255
17.7
Symplectic forms
256
17.8 Kahler
manifolds
257
17.9
Complex manifolds with closed almost
Kähier form 258
17.10
Examples of
Kahler
manifolds
259
Appendix
17.1
Compatible almost complex structures
261
Additional reading
267
18
Holomorphic submanifolds, holomorphic sections
and curvature
268
18.1
Holomorphic submanifolds of a complex manifold
268
18.2
Holomorphic submanifolds of
projective
spaces
269
18.3
Proof of Proposition
18.2,
about holomorphic
submanifolds in OP
271
18.4
The curvature of
a
Kahler
metric
272
18.5
Curvature with no
(0, 2)
part
275
18.6
Holomorphic sections
277
18.7
Example on CF
279
Additional reading
281
Contents xiii
19
The Hodge star
282
19.1
Definition of the Hodge star
282
19.2
Representatives of
De
Rham cohomology
283
19.3
A fairy tale
284
19.4
The Hodge theorem
285
19.5
Self-duality
286
Additional reading
287
List of lemmas, propositions, corollaries and theorems
289
List of symbols
291
Index
295
|
| any_adam_object | 1 |
| author | Taubes, Clifford 1954- |
| author_GND | (DE-588)172877423 |
| author_facet | Taubes, Clifford 1954- |
| author_role | aut |
| author_sort | Taubes, Clifford 1954- |
| author_variant | c t ct |
| building | Verbundindex |
| bvnumber | BV039534238 |
| classification_rvk | SK 370 |
| classification_tum | MAT 530f |
| ctrlnum | (OCoLC)750874291 (DE-599)OBVAC08461656 |
| discipline | Mathematik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV039534238 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T15:56:46Z |
| institution | BVB |
| isbn | 9780199605880 9780199605873 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-024386452 |
| oclc_num | 750874291 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-824 DE-11 DE-19 DE-BY-UBM DE-384 DE-20 DE-188 DE-703 DE-29T |
| owner_facet | DE-91G DE-BY-TUM DE-824 DE-11 DE-19 DE-BY-UBM DE-384 DE-20 DE-188 DE-703 DE-29T |
| physical | XIII, 298 S. graph. Darst. |
| publishDate | 2011 |
| publishDateSearch | 2011 |
| publishDateSort | 2011 |
| publisher | Oxford Univ. Press |
| record_format | marc |
| series | Oxford graduate texts in mathematics |
| series2 | Oxford graduate texts in mathematics |
| spellingShingle | Taubes, Clifford 1954- Differential geometry bundles, connections, metrics and curvature Oxford graduate texts in mathematics Komplexe Mannigfaltigkeit (DE-588)4031996-9 gnd Vektorraumbündel (DE-588)4187470-5 gnd Differentialgeometrie (DE-588)4012248-7 gnd |
| subject_GND | (DE-588)4031996-9 (DE-588)4187470-5 (DE-588)4012248-7 |
| title | Differential geometry bundles, connections, metrics and curvature |
| title_auth | Differential geometry bundles, connections, metrics and curvature |
| title_exact_search | Differential geometry bundles, connections, metrics and curvature |
| title_full | Differential geometry bundles, connections, metrics and curvature Clifford Henry Taubes |
| title_fullStr | Differential geometry bundles, connections, metrics and curvature Clifford Henry Taubes |
| title_full_unstemmed | Differential geometry bundles, connections, metrics and curvature Clifford Henry Taubes |
| title_short | Differential geometry |
| title_sort | differential geometry bundles connections metrics and curvature |
| title_sub | bundles, connections, metrics and curvature |
| topic | Komplexe Mannigfaltigkeit (DE-588)4031996-9 gnd Vektorraumbündel (DE-588)4187470-5 gnd Differentialgeometrie (DE-588)4012248-7 gnd |
| topic_facet | Komplexe Mannigfaltigkeit Vektorraumbündel Differentialgeometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024386452&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024386452&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV011416591 |
| work_keys_str_mv | AT taubesclifford differentialgeometrybundlesconnectionsmetricsandcurvature |
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