Geometry of differential forms:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English Japanese |
| Published: |
Providence, RI
American Math. Soc.
2001
|
| Series: | Translations of mathematical monographs
201 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Transl. from the Japan. |
| Physical Description: | XXIV, 321 S. graph. Darst. |
| ISBN: | 0821810456 |
Staff View
MARC
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| 100 | 1 | |a Morita, Shigeyuki |e Verfasser |4 aut | |
| 240 | 1 | 0 | |a Bibun keishiki no kikagaku |
| 245 | 1 | 0 | |a Geometry of differential forms |c Shigeyuki Morita |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 2001 | |
| 300 | |a XXIV, 321 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Translations of mathematical monographs |v 201 | |
| 490 | 0 | |a Iwanami series in modern mathematics | |
| 500 | |a Transl. from the Japan. | ||
| 650 | 4 | |a Geometrie - Differentialform | |
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Record in the Search Index
| _version_ | 1804128793172901888 |
|---|---|
| adam_text | Contents
Preface xiii
Preface to the English Edition xvii
Outline and Goal of the Theory xix
Chapter 1 Manifolds 1
1.1 What is a manifold? 2
(a) The n dimensional numerical space Rn 2
(b) Topology of Rn 3
(c) C°° functions and diffeomorphisms 4
(d) Tangent vectors and tangent spaces of Mn 6
(e) Necessity of an abstract definition 10
1.2 Definition and examples of manifolds 11
(a) Local coordinates and topological manifolds 11
(b) Definition of differentiable manifolds 13
(c) 1 and general surfaces in it 16
(d) Submanifolds 19
(e) Projective spaces 21
(f) Lie groups 22
1.3 Tangent vectors and tangent spaces 23
(a) Cx functions and Cx mappings on manifolds 23
(b) Practical construction of Cx functions on a man¬
ifold 25
(c) Partition of unity 27
(d) Tangent vectors 29
(e) The differential of maps 33
(f) Immersions and embeddings 34
1.4 Vector fields 36
(a) Vector fields 36
(b) The bracket of vector fields 38
vii
viii CONTENTS
(c) Integral curves of vector fields and one parameter
group of local transformations 39
(d) Transformations of vector fields by diffeomorphism 44
1.5 Fundamental facts concerning manifolds 44
(a) Manifolds with boundary 44
(b) Orientation of a manifold 46
(c) Group actions 49
(d) Fundamental groups and covering manifolds 51
Summary 54
Exercises 55
Chapter 2 Differential Forms 57
2.1 Definition of differential forms 57
(a) Differential forms on Rn 57
(b) Differential forms on a general manifold 61
(c) The exterior algebra 61
(d) Various definitions of differential forms 66
2.2 Various operations on differential forms 69
(a) Exterior product 69
(b) Exterior differentiation 70
(c) Pullback by a map 72
(d) Interior product and Lie derivative 72
(e) The Cartan formula and properties of Lie deriva¬
tives 73
(f) Lie derivative and one parameter group of local
transformations 77
2.3 Frobenius theorem 80
(a) Frobenius theorem — Representation by vector
fields 80
(b) Commutative vector fields 82
(c) Proof of the Frobenius theorem 83
(d) The Frobenius theorem Representation by dif¬
ferential forms 86
2.4 A few facts 89
(a) Differential forms with values in a vector space 89
(b) The Maurer Cartan form of a Lie group 90
Summary 92
Exercises 93
Chapter 3 The de Rham Theorem 95
3.1 Homology of manifolds 96
CONTENTS ix
(a) Homology of simplicial complexes 96
(b) Singular homology 99
(c) Cx triangulation of C00 manifolds 100
(d) Cx singular chain complexes of Cx manifolds 103
3.2 Integral of differential forms and the Stokes theorem 104
(a) Integral of n forms on n dimensional manifolds 104
(b) The Stokes theorem (in the case of manifolds) 107
(c) Integral of differential forms on chains, and the
Stokes theorem 109
3.3 The de Rham theorem 111
(a) de Rham cohomology 111
(b) The de Rham theorem 113
(c) Poincare lemma 116
3.4 Proof of the de Rham theorem 119
(a) Cech cohomology 119
(b) Comparison of de Rham cohomology and Cech co¬
homology 121
(c) Proof of the de Rham theorem 126
(d) The de Rham theorem and product structure 131
3.5 Applications of the de Rham theorem 133
(a) Hopf invariant 133
(b) The Massey product 136
(c) Cohomology of compact Lie groups 137
(d) Mapping degree 138
(e) Integral expression of the linking number by Gauss 140
Summary 142
Exercises 142
Chapter 4 Laplacian and Harmonic Forms 145
4.1 Differential forms on Riemannian manifolds 145
(a) Riemannian metric 145
(b) Riemannian metric and differentieal forms 148
(c) The * operator of Hodge 150
4.2 Laplacian and harmonic forms 153
4.3 The Hodge theorem 158
(a) The Hodge theorem and the Hodge decomoposi
tion of differential forms 158
(b) The idea for the proof of the Hodoge decomposi¬
tion 160
4.4 Applications of the Hodge theorem 162
x CONTENTS
(a) The Poincare duality theorem 162
(b) Manifolds and Euler number 164
(c) Intersection number 165
Summary 166
Exercises 16
Chapter 5 Vector Bundles and Characteristic Classes 169
5.1 Vector bundles 169
(a) The tangent bundle of a manifold 169
(b) Vector bundles 170
(c) Several constructions of vector bundles 173
5.2 Geodesies and parallel translation of vectors 180
(a) Geodesies 180
(b) Covariant derivative 181
(c) Parallel displacement of vectors and curvature 183
5.3 Connections in vector bundles and 185
(a) Connections 185
(b) Curvature 186
(c) Connection form and curvature form 188
(d) Transformation rules of the local expressions for a
connection and its curvature 190
(e) Differential forms with values in a vector bundle 191
5.4 Pontrjagin classes 193
(a) Invariant polynomials 193
(b) Definition of Pontrjagin classes 197
(c) Levi Civita connection 201
5.5 Chern classes 204
(a) Connection and curvature in a complex vector bun¬
dle 204
(b) Definition of Chern classes 205
(c) Whitney formula 207
(d) Relations between Pontrjagin and Chern classes 208
5.6 Euler classes 211
(a) Orientation of vector bundles 211
(b) The definition of the Euler class 211
(c) Properties of the Euler class 214
5.7 Applications of characteristic classes 216
(a) The Gauss Bonnet theorem 216
(b) Characteristic classes of the complex projective
space 223
CONTENTS xi
(c) Characteristic numbers 225
Summary 228
Exercises 229
Chapter 6 Fiber Bundles and Characteristic Classes 231
6.1 Fiber bundle and principal bundle 231
(a) Fiber bundle 231
(b) Structure group 233
(c) Principal bundle 236
(d) The classification of fiber bundles and character¬
istic classes 238
(e) Examples of fiber bundles 239
6.2 51 bundles and Euler classes 240
(a) S1 bundle 241
(b) Euler class of an S1 bundle 241
(c) The classification of S1 bundles 246
(d) Defining the Euler class for an S bundle by using
differential forms 249
(e) The primary obstruction class and the Euler class
of the sphere bundle 254
(f) Vector fields on a manifold and Hopf index theo¬
rem 255
6.3 Connections 257
(a) Connections in general fiber bundles 257
(b) Connections in principal bundles 260
(c) Differential form representation of a connection in
a principal bundle 262
6.4 Curvature 265
(a) Curvature form 265
(b) Weil algebra 268
(c) Exterior differentiation of the Weil algebra 270
6.5 Characteristic classes 275
(a) Weil homomorphism 275
(b) Invariant polynomials for Lie groups 279
(c) Connections for vector bundles and principal bun¬
dles 282
(d) Characterisric classes 284
6.6 A couple of items 285
(a) Triviality of the cohomology of the Weil algebra 285
(b) Chern Simons forms 287
xii CONTENTS
(c) Flat bundles and holonomy homomorphisms 287
Summary 291
Exercises 292
Perspectives 295
Answers to Exercises 299
Chapter 1 299
Chapter 2 302
Chapter 3 305
Chapter 4 308
Chapter 5 310
Chapter 6 311
References 315
Index 317
|
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| indexdate | 2024-07-09T18:54:53Z |
| institution | BVB |
| isbn | 0821810456 |
| language | English Japanese |
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| physical | XXIV, 321 S. graph. Darst. |
| publishDate | 2001 |
| publishDateSearch | 2001 |
| publishDateSort | 2001 |
| publisher | American Math. Soc. |
| record_format | marc |
| series | Translations of mathematical monographs |
| series2 | Translations of mathematical monographs Iwanami series in modern mathematics |
| spelling | Morita, Shigeyuki Verfasser aut Bibun keishiki no kikagaku Geometry of differential forms Shigeyuki Morita Providence, RI American Math. Soc. 2001 XXIV, 321 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Translations of mathematical monographs 201 Iwanami series in modern mathematics Transl. from the Japan. Geometrie - Differentialform Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 gnd rswk-swf Differentialform (DE-588)4149772-7 gnd rswk-swf Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 s Differentialform (DE-588)4149772-7 s DE-604 Translations of mathematical monographs 201 (DE-604)BV000002394 201 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009543822&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Morita, Shigeyuki Geometry of differential forms Translations of mathematical monographs Geometrie - Differentialform Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 gnd Differentialform (DE-588)4149772-7 gnd |
| subject_GND | (DE-588)4012269-4 (DE-588)4149772-7 |
| title | Geometry of differential forms |
| title_alt | Bibun keishiki no kikagaku |
| title_auth | Geometry of differential forms |
| title_exact_search | Geometry of differential forms |
| title_full | Geometry of differential forms Shigeyuki Morita |
| title_fullStr | Geometry of differential forms Shigeyuki Morita |
| title_full_unstemmed | Geometry of differential forms Shigeyuki Morita |
| title_short | Geometry of differential forms |
| title_sort | geometry of differential forms |
| topic | Geometrie - Differentialform Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 gnd Differentialform (DE-588)4149772-7 gnd |
| topic_facet | Geometrie - Differentialform Differenzierbare Mannigfaltigkeit Differentialform |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009543822&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000002394 |
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