Introduction to Riemannian manifolds:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York
Springer
[2018]
|
| Edition: | Second edition |
| Series: | Graduate texts in mathematics
176 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | xiii, 437 Seiten Illustrationen |
| ISBN: | 9783319917542 9783030801069 |
Staff View
MARC
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| 100 | 1 | |a Lee, John M. |d 1950- |e Verfasser |0 (DE-588)122260880 |4 aut | |
| 240 | 1 | 0 | |a Riemannian manifolds (an introduction to curvature) |
| 245 | 1 | 0 | |a Introduction to Riemannian manifolds |c John M. Lee |
| 250 | |a Second edition | ||
| 264 | 1 | |a New York |b Springer |c [2018] | |
| 300 | |a xiii, 437 Seiten |b Illustrationen | ||
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| 490 | 1 | |a Graduate texts in mathematics |v 176 | |
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Record in the Search Index
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| adam_text | Contents Preface............................................................................................................. v 1 What Is Curvature?................................................................................... The Euclidean Plane..................................................................................... Surfaces in Space......................................................................................... Curvature in Higher Dimensions................................................................ 1 1 4 7 2 Riemannian Metrics................................................................................... Definitions..................................................................................................... Methods for Constructing Riemannian Metrics........................................ Basic Constructions on Riemannian Manifolds........................................ Lengths and Distances ................................................................................ Pseudo-Riemannian Metrics......................................................................... Other Generalizations of Riemannian Metrics........................................... Problems........................................................................................................ 9 9 15 25 33 40 46 47 3 Model Riemannian Manifolds.................................................................. Symmetries of Riemannian Manifolds...................................................... Euclidean
Spaces......................................................................................... Spheres.......................................................................................................... Hyperbolic Spaces....................................................................................... Invariant Metrics on Lie Groups................................................................ Other Homogeneous Riemannian Manifolds............................................. Model Pseudo-Riemannian Manifolds...................................................... Problems........................................................................................................ 55 55 57 58 62 67 72 79 80 4 Connections................................................................................................... The Problem of Differentiating Vector Fields.......................................... Connections................................................................................................... Covariant Derivatives of Tensor Fields .................................................... Vector and Tensor Fields Along Curves.................................................... 85 85 88 95 100 xi
xii Contents Geodesics........................................................................................................ Parallel Transport.......................................................................................... Pullback Connections................................................................................... Problems........................................................................................................ 103 105 110 Ill 5 The Levi-Civita Connection.................................................................... The Tangential Connection Revisited......................................................... Connections on Abstract Riemannian Manifolds...................................... The Exponential Map................................................................................... Normal Neighborhoods and Normal Coordinates................................... Tubular Neighborhoods and Fermi Coordinates...................................... Geodesics of the Model Spaces.................................................................. Euclidean and Non-Euclidean Geometries......................... Problems................................................................................ 1..................... 115 115 117 126 131 133 136 142 145 6 Geodesics and Distance.............................................................................. Geodesics and Minimizing Curves............................................................. Uniformly Normal
Neighborhoods............................................................. Completeness................................................................................................. Distance Functions....................................................................................... Semigeodesic Coordinates........................................................................... Problems........................................................................................................ 151 151 163 166 174 181 185 7 Curvature..................................................................................................... Local Invariants............................................................................................ The Curvature Tensor.................................................................................. Flat Manifolds.............................................................................................. Symmetries of the Curvature Tensor........................................................ The Ricci Identities....................................................................................... Ricci and Scalar Curvatures......................................................................... The Weyl Tensor.......................................................................................... Curvatures of Conformally Related Metrics............................................. Problems........................................................................................................ 193 193 196 199 202 205 207
212 216 222 8 Riemannian Submanifolds......................................................................... The Second Fundamental Form.................................................................. Hypersurfaces................................................................................................. Hypersurfaces in Euclidean Space............................................................. Sectional Curvatures..................................................................................... Problems........................................................................................................ 225 225 234 244 250 255 9 The Gauss-Bonnet Theorem.................................................................... 263 Some Plane Geometry................................................................................... 263 The Gauss-Bonnet Formula....................................................................... 271
Contents xiii The Gauss-Bonnet Theorem....................................................................... 276 Problems........................................................................................................ 281 10 Jacobi Fields................................................................................................ The Jacobi Equation..................................................................................... Basic Computations with Jacobi Fields.................................................... Conjugate Points............................................................................................ The Second Variation Formula.................................................................. Cut Points............................................................................. Problems........................................................................................................ 283 284 287 297 300 307 313 11 Comparison Theory..................................... Jacobi Fields, Hessians, and Riccati Equations........................................ Comparisons Based on Sectional Curvature............................................. Comparisons Based on Ricci Curvature.................................................... Problems........................................................................................................ 319 320 327 336 342 12 Curvature and Topology........................................................................... Manifolds of Constant
Curvature................................................................ Manifolds of Nonpositive Curvature......................................................... Manifolds of Positive Curvature................................................................ Problems........................................................................................................ 345 345 352 361 368 Appendix A: Review of Smooth Manifolds.................................................... Topological Preliminaries............................................................................ Smooth Manifolds and Smooth Maps........................................................ Tangent Vectors........................................................................................... Submanifolds.................................................................................................. Vector Bundles.............................................................................................. The Tangent Bundle and Vector Fields...................................................... Smooth Covering Maps................................................................................ 371 371 374 376 378 382 384 388 Appendix B: Review of Tensors....................................................................... Tensors on a Vector Space.......................................................................... Tensor Bundles and Tensor Fields............................................................ Differential Forms and
Integration.............................................................. Densities......................................................................................................... 391 391 396 400 405 Appendix C: Review of Lie Groups................................................................. Definitions and Properties............................................................................ The Lie Algebra of a Lie Group................................................................. Group Actions on Manifolds....................................................................... 407 407 408 411 References............................................................................................................. 415 Notation Index....................................................................................................... 419 Subject Index......................................................................................................... 423
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| any_adam_object | 1 |
| author | Lee, John M. 1950- |
| author_GND | (DE-588)122260880 |
| author_facet | Lee, John M. 1950- |
| author_role | aut |
| author_sort | Lee, John M. 1950- |
| author_variant | j m l jm jml |
| building | Verbundindex |
| bvnumber | BV045403804 |
| classification_rvk | SK 370 |
| classification_tum | MAT 537f |
| ctrlnum | (OCoLC)1085413534 (DE-599)BVBBV045403804 |
| 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 |
| edition | Second edition |
| format | Book |
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| id | DE-604.BV045403804 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T08:17:16Z |
| institution | BVB |
| isbn | 9783319917542 9783030801069 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-030789837 |
| oclc_num | 1085413534 |
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| physical | xiii, 437 Seiten Illustrationen |
| publishDate | 2018 |
| publishDateSearch | 2018 |
| publishDateSort | 2018 |
| publisher | Springer |
| record_format | marc |
| 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) Introduction to Riemannian manifolds John M. Lee Second edition New York Springer [2018] xiii, 437 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 176 Hier auch später erschienene, unveränderte Nachdrucke Geometría de Riemann Krümmung (DE-588)4128765-4 gnd rswk-swf Riemannsche Geometrie (DE-588)4128462-8 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 Erscheint auch als Online-Ausgabe 978-3-319-91755-9 Graduate texts in mathematics 176 (DE-604)BV000000067 176 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030789837&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lee, John M. 1950- Introduction to Riemannian manifolds Graduate texts in mathematics Geometría de Riemann Krümmung (DE-588)4128765-4 gnd Riemannsche Geometrie (DE-588)4128462-8 gnd Riemannscher Raum (DE-588)4128295-4 gnd |
| subject_GND | (DE-588)4128765-4 (DE-588)4128462-8 (DE-588)4128295-4 |
| title | Introduction to Riemannian manifolds |
| title_alt | Riemannian manifolds (an introduction to curvature) |
| title_auth | Introduction to Riemannian manifolds |
| title_exact_search | Introduction to Riemannian manifolds |
| title_full | Introduction to Riemannian manifolds John M. Lee |
| title_fullStr | Introduction to Riemannian manifolds John M. Lee |
| title_full_unstemmed | Introduction to Riemannian manifolds John M. Lee |
| title_short | Introduction to Riemannian manifolds |
| title_sort | introduction to riemannian manifolds |
| topic | Geometría de Riemann Krümmung (DE-588)4128765-4 gnd Riemannsche Geometrie (DE-588)4128462-8 gnd Riemannscher Raum (DE-588)4128295-4 gnd |
| topic_facet | Geometría de Riemann Krümmung Riemannsche Geometrie Riemannscher Raum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030789837&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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