An introduction to differentiable manifolds and Riemannian geometry:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Orlando [u.a.]
Acad. Press
1986
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Pure and applied mathematics
120 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVI, 430 S. |
| ISBN: | 0121160521 012116053X |
Internformat
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| 100 | 1 | |a Boothby, William M. |d 1918- |e Verfasser |0 (DE-588)171984412 |4 aut | |
| 245 | 1 | 0 | |a An introduction to differentiable manifolds and Riemannian geometry |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Orlando [u.a.] |b Acad. Press |c 1986 | |
| 300 | |a XVI, 430 S. | ||
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| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Pure and applied mathematics |v 120 | |
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| 650 | 4 | |a Riemann, Variétés de | |
| 650 | 7 | |a Riemann-vlakken |2 gtt | |
| 650 | 4 | |a Variétés différentiables | |
| 650 | 4 | |a Differentiable manifolds | |
| 650 | 4 | |a Riemannian manifolds | |
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Datensatz im Suchindex
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| adam_text | Contents
Preface to the Second Edition xi
Preface to the First Edition xiii
I. Introduction to Manifolds
1. Preliminary Comments on R 1
2. R and Euclidean Space 4
3. Topological Manifolds 6
4. Further Examples of Manifolds. Cutting and Pasting 11
5. Abstract Manifolds. Some Examples 14
Notes 18
II. Functions of Several Variables and Mappings
1. Differentiability for Functions of Several Variables 20
2. Differentiability of Mappings and Jacobians 25
3. The Space of Tangent Vectors at a Point of R 29
4. Another Definition of T.(R ) 32
5. Vector Fields on Open Subsets of R 37
6. The Inverse Function Theorem 41
7. The Rank of a Mapping 46
Notes 50
III. Differentiable Manifolds and Submanifolds
1. The Definition of a Differentiable Manifold 52
2. Further Examples 60
3. Differentiable Functions and Mappings 65
4. Rank of a Mapping. Immersions 69
5. Submanifolds 75
6. Lie Groups 81
7. The Action of a Lie Group on a Manifold. Transformation Groups 89
8. The Action of a Discrete Group on a Manifold 96
9. Covering Manifolds 101
Notes 104
vii
viii CONTENTS
IV. Vector Fields on a Manifold
1. The Tangent Space at a Point of a Manifold 107
2. Vector Fields 116
3. One Parameter and Local One Parameter Groups Acting on a Manifold 123
4. The Existence Theorem for Ordinary Differential Equations 131
5. Some Examples of One Parameter Groups Acting on a Manifold 139
6. One Parameter Subgroups of Lie Groups 147
7. The Lie Algebra of Vector Fields on a Manifold 151
8. Frobenius Theorem 158
9. Homogeneous Spaces 165
Appendix Partial Proof of Theorem 4.1 172
Notes 174
V. Tensors and Tensor Fields on Manifolds
1. Tangent Covectors 177
Covectors on Manifolds 178
Covector Fields and Mappings 180
2. Bilinear Forms. The Riemannian Metric 183
3. Riemannian Manifolds as Metric Spaces 187
4. Partitions of Unity 193
Some Applications of the Partition of Unity 195
5. Tensor Fields 199
Tensors on a Vector Space 199
Tensor Fields 201
Mappings and Covariant Tensors 202
The Symmetrizing and Alternating Transformations 203
6. Multiplication of Tensors 206
Multiplication of Tensors on a Vector Space 206
Multiplication of Tensor Fields 208
Exterior Multiplication of Alternating Tensors 209
The Exterior Algebra on Manifolds 213
7. Orientation of Manifolds and the Volume Element 215
8. Exterior Differentiation 219
An Application to Frobenius Theorem 223
Notes 227
VI. Integration on Manifolds
1 Integration in /? . Domains of Integration 230
Basic Properties of the Riemann Integral 231
2. A Generalization to Manifolds 236
Integration on Riemannian Manifolds 240
3. Integration on Lie Groups 244
4. Manifolds with Boundary 251
5. Stokes s Theorem for Manifolds with Boundary 259
6. Homotopy of Mappings. The Fundamental Group 266
Homotopy of Paths and Loops. The Fundamental Group 268
7. Some Applications of Differential Forms. The de Rham Groups 274
The Homotopy Operator 277
CONTENTS
8. Some Further Applications of de Rham Groups 281
The de Rham Groups of Lie Groups 285
9. Covering Spaces and the Fundamental Group 289
Notes 296
VII. Differentiation on Riemannian Manifolds
1. Differentiation of Vector Fields along Curves in R 298
The Geometry of Space Curves 301
Curvature of Plane Curves 305
2. Differentiation of Vector Fields on Submanifolds of /? 307
Formulas for Covariant Derivatives 312
V^, Y and Differentiation of Vector Fields 314
3. Differentiation on Riemannian Manifolds 317
Constant Vector Fields and Parallel Displacement 323
4. Addenda to the Theory of Differentiation on a Manifold 325
The Curvature Tensor 325
The Riemannian Connection and Exterior Differential Forms 328
5. Geodesic Curves on Riemannian Manifolds 330
6. The Tangent Bundle and Exponential Mapping. Normal Coordinates 335
7. Some Further Properties of Geodesies 342
8. Symmetric Riemannian Manifolds 351
9. Some Examples 357
Notes 364
VIII. Curvature
1. The Geometry of Surfaces in £3 366
The Principal Curvatures at a Point of a Surface 370
2. The Gaussian and Mean Curvatures of a Surface 374
The Theorema Egregium of Gauss 377
3. Basic Properties of the Riemann Curvature Tensor 382
4. The Curvature Forms and the Equations of Structure 389
5. Differentiation of Covariant Tensor Fields 396
6. Manifolds of Constant Curvature 403
Spaces of Positive Curvature 406
Spaces of Zero Curvature 408
Spaces of Constant Negative Curvature 409
Notes 414
References 417
Index 423
|
| any_adam_object | 1 |
| author | Boothby, William M. 1918- |
| author_GND | (DE-588)171984412 |
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| ctrlnum | (OCoLC)12135618 (DE-599)BVBBV000569490 |
| dewey-full | 510 516.3/6 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 510 - Mathematics 516 - Geometry |
| dewey-raw | 510 516.3/6 |
| dewey-search | 510 516.3/6 |
| dewey-sort | 3510 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV000569490 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:15:50Z |
| institution | BVB |
| isbn | 0121160521 012116053X |
| language | English |
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| physical | XVI, 430 S. |
| publishDate | 1986 |
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| publisher | Acad. Press |
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| series | Pure and applied mathematics |
| series2 | Pure and applied mathematics |
| spelling | Boothby, William M. 1918- Verfasser (DE-588)171984412 aut An introduction to differentiable manifolds and Riemannian geometry 2. ed. Orlando [u.a.] Acad. Press 1986 XVI, 430 S. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics 120 Differentieerbaarheid gtt Manifolds gtt Riemann, Variétés de Riemann-vlakken gtt Variétés différentiables Differentiable manifolds Riemannian manifolds Mannigfaltigkeit (DE-588)4037379-4 gnd rswk-swf Differentiation Mathematik (DE-588)4149787-9 gnd rswk-swf Riemannscher Raum (DE-588)4128295-4 gnd rswk-swf Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 gnd rswk-swf Riemannsche Geometrie (DE-588)4128462-8 gnd rswk-swf Mannigfaltigkeit (DE-588)4037379-4 s DE-604 Riemannsche Geometrie (DE-588)4128462-8 s Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 s Riemannscher Raum (DE-588)4128295-4 s Differentiation Mathematik (DE-588)4149787-9 s Pure and applied mathematics 120 (DE-604)BV010177228 120 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000351094&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Boothby, William M. 1918- An introduction to differentiable manifolds and Riemannian geometry Pure and applied mathematics Differentieerbaarheid gtt Manifolds gtt Riemann, Variétés de Riemann-vlakken gtt Variétés différentiables Differentiable manifolds Riemannian manifolds Mannigfaltigkeit (DE-588)4037379-4 gnd Differentiation Mathematik (DE-588)4149787-9 gnd Riemannscher Raum (DE-588)4128295-4 gnd Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 gnd Riemannsche Geometrie (DE-588)4128462-8 gnd |
| subject_GND | (DE-588)4037379-4 (DE-588)4149787-9 (DE-588)4128295-4 (DE-588)4012269-4 (DE-588)4128462-8 |
| title | An introduction to differentiable manifolds and Riemannian geometry |
| title_auth | An introduction to differentiable manifolds and Riemannian geometry |
| title_exact_search | An introduction to differentiable manifolds and Riemannian geometry |
| title_full | An introduction to differentiable manifolds and Riemannian geometry |
| title_fullStr | An introduction to differentiable manifolds and Riemannian geometry |
| title_full_unstemmed | An introduction to differentiable manifolds and Riemannian geometry |
| title_short | An introduction to differentiable manifolds and Riemannian geometry |
| title_sort | an introduction to differentiable manifolds and riemannian geometry |
| topic | Differentieerbaarheid gtt Manifolds gtt Riemann, Variétés de Riemann-vlakken gtt Variétés différentiables Differentiable manifolds Riemannian manifolds Mannigfaltigkeit (DE-588)4037379-4 gnd Differentiation Mathematik (DE-588)4149787-9 gnd Riemannscher Raum (DE-588)4128295-4 gnd Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 gnd Riemannsche Geometrie (DE-588)4128462-8 gnd |
| topic_facet | Differentieerbaarheid Manifolds Riemann, Variétés de Riemann-vlakken Variétés différentiables Differentiable manifolds Riemannian manifolds Mannigfaltigkeit Differentiation Mathematik Riemannscher Raum Differenzierbare Mannigfaltigkeit Riemannsche Geometrie |
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