Riemannian manifolds: an introduction to curvature
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
1997
|
| Series: | Graduate texts in mathematics
176 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 224 S. graph. Darst. |
| ISBN: | 9780387983226 038798271X 0387983228 |
Staff View
MARC
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| 245 | 1 | 0 | |a Riemannian manifolds |b an introduction to curvature |c John M. Lee |
| 264 | 1 | |a New York [u.a.] |b Springer |c 1997 | |
| 300 | |a XV, 224 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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Record in the Search Index
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| adam_text | Contents
Preface vii
1 What Is Curvature? 1
The Euclidean Plane 2
Surfaces in Space 4
Curvature in Higher Dimensions 8
2 Review of Tensors, Manifolds, and Vector Bundles 11
Tensors on a Vector Space 11
Manifolds 14
Vector Bundles 16
Tensor Bundles and Tensor Fields 19
3 Definitions and Examples of Riemannian Metrics 23
Riemannian Metrics 23
Elementary Constructions Associated with Riemannian Metrics . 27
Generalizations of Riemannian Metrics 30
The Model Spaces of Riemannian Geometry 33
Problems 43
4 Connections 47
The Problem of Differentiating Vector Fields 48
Connections 49
Vector Fields Along Curves 55
xiv Contents
Geodesies 58
Problems 63
5 Riemannian Geodesies 65
The Riemannian Connection 65
The Exponential Map 72
Normal Neighborhoods and Normal Coordinates 76
Geodesies of the Model Spaces 81
Problems 87
6 Geodesies and Distance 91
Lengths and Distances on Riemannian Manifolds 91
Geodesies and Minimizing Curves 96
Completeness 108
Problems 112
7 Curvature 115
Local Invariants 115
Flat Manifolds . 119
Symmetries of the Curvature Tensor 121
Ricci and Scalar Curvatures 124
Problems 128
8 Riemannian Submanifolds 131
Riemannian Submanifolds and the Second Fundamental Form . . 132
Hypersurfaces in Euclidean Space 139
Geometric Interpretation of Curvature in Higher Dimensions . . 145
Problems 150
9 The Gauss Bonnet Theorem 155
Some Plane Geometry 156
The Gauss Bonnet Formula 162
The Gauss Bonnet Theorem 166
Problems 171
10 Jacobi Fields 173
The Jacobi Equation 174
Computations of Jacobi Fields 178
Conjugate Points 181
The Second Variation Formula 185
Geodesies Do Not Minimize Past Conjugate Points 187
Problems 191
11 Curvature and Topology 193
Some Comparison Theorems 194
Manifolds of Negative Curvature 196
Contents xv
Manifolds of Positive Curvature 199
Manifolds of Constant Curvature 204
Problems 208
References 209
Index 213
|
| any_adam_object | 1 |
| author | Lee, John M. 1950- |
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| dewey-full | 516.373 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.373 |
| dewey-search | 516.373 |
| dewey-sort | 3516.373 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T18:13:18Z |
| institution | BVB |
| isbn | 9780387983226 038798271X 0387983228 |
| language | English |
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| physical | XV, 224 S. graph. Darst. |
| publishDate | 1997 |
| publishDateSearch | 1997 |
| publishDateSort | 1997 |
| publisher | Springer |
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| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Lee, John M. 1950- Verfasser (DE-588)122260880 aut Riemannian manifolds an introduction to curvature John M. Lee New York [u.a.] Springer 1997 XV, 224 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 176 Geometría de Riemann Riemannsche Geometrie (DE-588)4128462-8 gnd rswk-swf Krümmung (DE-588)4128765-4 gnd rswk-swf Riemannscher Raum (DE-588)4128295-4 gnd rswk-swf Riemannscher Raum (DE-588)4128295-4 s DE-604 Riemannsche Geometrie (DE-588)4128462-8 s Krümmung (DE-588)4128765-4 s Graduate texts in mathematics 176 (DE-604)BV000000067 176 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007847942&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lee, John M. 1950- Riemannian manifolds an introduction to curvature Graduate texts in mathematics Geometría de Riemann Riemannsche Geometrie (DE-588)4128462-8 gnd Krümmung (DE-588)4128765-4 gnd Riemannscher Raum (DE-588)4128295-4 gnd |
| subject_GND | (DE-588)4128462-8 (DE-588)4128765-4 (DE-588)4128295-4 |
| title | Riemannian manifolds an introduction to curvature |
| title_auth | Riemannian manifolds an introduction to curvature |
| title_exact_search | Riemannian manifolds an introduction to curvature |
| title_full | Riemannian manifolds an introduction to curvature John M. Lee |
| title_fullStr | Riemannian manifolds an introduction to curvature John M. Lee |
| title_full_unstemmed | Riemannian manifolds an introduction to curvature John M. Lee |
| title_short | Riemannian manifolds |
| title_sort | riemannian manifolds an introduction to curvature |
| title_sub | an introduction to curvature |
| topic | Geometría de Riemann Riemannsche Geometrie (DE-588)4128462-8 gnd Krümmung (DE-588)4128765-4 gnd Riemannscher Raum (DE-588)4128295-4 gnd |
| topic_facet | Geometría de Riemann Riemannsche Geometrie Krümmung Riemannscher Raum |
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