Differential and Riemannian manifolds:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York u.a.
Springer
1995
|
| Edition: | 3. ed. |
| Series: | Graduate texts in mathematics
160 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Orig.-Ausg. u.d.T.: Lang, Serge: Differential manifolds |
| Physical Description: | XIII, 364 S. graph. Darst. |
| ISBN: | 3540943382 0387943382 |
Staff View
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Record in the Search Index
| _version_ | 1804124352906526720 |
|---|---|
| adam_text | Contents
Preface v
CHAPTER I
Differential Calculus 1
§1. Categories 2
§2. Topological Vector Spaces 3
§3. Derivatives and Composition of Maps 6
§4. Integration and Taylor s Formula 10
§5. The Inverse Mapping Theorem 13
CHAPTER II
Manifolds 20
§1. Atlases, Charts, Morphisms 20
§2. Submanifolds, Immersions, Submersions 23
§3. Partitions of Unity 31
§4. Manifolds with Boundary 36
CHAPTER III
Vector Bundles 4°
§1. Definition, Pull Backs 4°
§2. The Tangent Bundle 48
§3. Exact Sequences of Bundles 49
§4. Operations on Vector Bundles 55
§5. Splitting of Vector Bundles 6°
CHAPTER iv
Vector Fields and Differential Equations 64
§1. Existence Theorem for Differential Equations 65
§2. Vector Fields, Curves, and Flows 86
Xli CONTENTS
§3. Sprays 94
§4. The Flow of a Spray and the Exponential Map 103
§5. Existence of Tubular Neighborhoods 108
§6. Uniqueness of Tubular Neighborhoods 110
CHAPTER V
Operations on Vector Fields and Differential Forms 114
§1. Vector Fields, Differential Operators, Brackets 114
§2. Lie Derivative 120
§3. Exterior Derivative 122
§4. The Poincare Lemma 135
§5. Contractions and Lie Derivative 137
§6. Vector Fields and 1 Forms Under Self Duality 141
§7. The Canonical 2 Form 146
§8. Darboux s Theorem 148
CHAPTER VI
The Theorem of Frobenius 153
§1. Statement of the Theorem 153
§2. Differential Equations Depending on a Parameter 158
§3. Proof of the Theorem 159
§4. The Global Formulation 160
§5. Lie Groups and Subgroups 163
CHAPTER VII
Metrics 169
§1. Definition and Functoriality 169
§2. The Hilbert Group 173
§3. Reduction to the Hilbert Group 176
§4. Hilbertian Tubular Neighborhoods 179
§5. The Morse Palais Lemma 82
§6. The Riemannian Distance 184
§7. The Canonical Spray 1 °
CHAPTER VIII
Covariant Derivatives and Geodesies 191
§1. Basic Properties I91
§2. Sprays and Covariant Derivatives 1
§3. Derivative Along a Curve and Parallelism 1
§4. The Metric Derivative 203
§5. More Local Results on the Exponential Map 209
§6. Riemannian Geodesic Length and Completeness 216
CHAPTER IX
Curvature 225
§1. The Riemann Tensor 22^
§2. Jacobi Lifts 233
CONTENTS Xiii
§3. Application of Jacobi Lifts to de px 240
§4. The Index Form, Variations, and the Second Variation Formula .... 249
§5. Taylor Expansions 257
CHAPTER X
Volume Forms 261
§1. The Riemannian Volume Form 261
§2. Covariant Derivatives 264
§3. The Jacobian Determinant of the Exponential Map 268
§4. The Hodge Star on Forms 273
§5. Hodge Decomposition of Differential Forms 279
CHAPTER XI
Integration of Differential Forms 284
§1. Sets of Measure 0 284
§2. Change of Variables Formula 288
§3. Orientation 297
§4. The Measure Associated with a Differential Form 299
CHAPTER XII
Stokes Theorem 307
§1. Stokes Theorem for a Rectangular Simplex 307
§2. Stokes Theorem on a Manifold 310
§3. Stokes Theorem with Singularities 314
CHAPTER XIII
Applications of Stokes Theorem 321
§1. The Maximal de Rham Cohomology 321
§2. Moser s Theorem 328
§3. The Divergence Theorem 329
§4. The Adjoint of d for Higher Degree Forms 333
§5. Cauchy s Theorem 335
§6. The Residue Theorem 339
APPENDIX
The Spectral Theorem 343
§1. Hilbert Space 343
§2. Functionals and Operators 344
§3. Hermitian Operators 3
Bibliography 355
Index 361
|
| any_adam_object | 1 |
| author | Lang, Serge 1927-2005 |
| author_GND | (DE-588)119305119 |
| author_facet | Lang, Serge 1927-2005 |
| author_role | aut |
| author_sort | Lang, Serge 1927-2005 |
| author_variant | s l sl |
| building | Verbundindex |
| bvnumber | BV009978998 |
| classification_rvk | SK 370 |
| classification_tum | MAT 583f MAT 530f |
| ctrlnum | (OCoLC)758026118 (DE-599)BVBBV009978998 |
| dewey-full | 516.36 516.07 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.36 516.07 |
| dewey-search | 516.36 516.07 |
| dewey-sort | 3516.36 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 3. ed. |
| format | Book |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T17:44:18Z |
| institution | BVB |
| isbn | 3540943382 0387943382 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006613644 |
| oclc_num | 758026118 |
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| physical | XIII, 364 S. graph. Darst. |
| publishDate | 1995 |
| publishDateSearch | 1995 |
| publishDateSort | 1995 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Lang, Serge 1927-2005 Verfasser (DE-588)119305119 aut Differential and Riemannian manifolds Serge Lang 3. ed. New York u.a. Springer 1995 XIII, 364 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 160 Orig.-Ausg. u.d.T.: Lang, Serge: Differential manifolds Riemann, Variétés de ram Variétés différentiables ram calcul forme inriac champ vecteur inriac espace Banach inriac espace Hilbert inriac géométrie différentielle inriac théorème Cartan-Hadamard inriac théorème Stokes inriac topologie différentielle inriac variété différentielle inriac variété riemannienne inriac équation différentielle inriac Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 gnd rswk-swf Differentialtopologie (DE-588)4012255-4 gnd rswk-swf Mannigfaltigkeit (DE-588)4037379-4 gnd rswk-swf Riemannscher Raum (DE-588)4128295-4 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Riemannscher Raum (DE-588)4128295-4 s DE-604 Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 s Differentialtopologie (DE-588)4012255-4 s 1\p DE-604 Differentialgeometrie (DE-588)4012248-7 s 2\p DE-604 Mannigfaltigkeit (DE-588)4037379-4 s 3\p DE-604 Graduate texts in mathematics 160 (DE-604)BV000000067 160 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006613644&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Lang, Serge 1927-2005 Differential and Riemannian manifolds Graduate texts in mathematics Riemann, Variétés de ram Variétés différentiables ram calcul forme inriac champ vecteur inriac espace Banach inriac espace Hilbert inriac géométrie différentielle inriac théorème Cartan-Hadamard inriac théorème Stokes inriac topologie différentielle inriac variété différentielle inriac variété riemannienne inriac équation différentielle inriac Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 gnd Differentialtopologie (DE-588)4012255-4 gnd Mannigfaltigkeit (DE-588)4037379-4 gnd Riemannscher Raum (DE-588)4128295-4 gnd Differentialgeometrie (DE-588)4012248-7 gnd |
| subject_GND | (DE-588)4012269-4 (DE-588)4012255-4 (DE-588)4037379-4 (DE-588)4128295-4 (DE-588)4012248-7 |
| title | Differential and Riemannian manifolds |
| title_auth | Differential and Riemannian manifolds |
| title_exact_search | Differential and Riemannian manifolds |
| title_full | Differential and Riemannian manifolds Serge Lang |
| title_fullStr | Differential and Riemannian manifolds Serge Lang |
| title_full_unstemmed | Differential and Riemannian manifolds Serge Lang |
| title_short | Differential and Riemannian manifolds |
| title_sort | differential and riemannian manifolds |
| topic | Riemann, Variétés de ram Variétés différentiables ram calcul forme inriac champ vecteur inriac espace Banach inriac espace Hilbert inriac géométrie différentielle inriac théorème Cartan-Hadamard inriac théorème Stokes inriac topologie différentielle inriac variété différentielle inriac variété riemannienne inriac équation différentielle inriac Differenzierbare Mannigfaltigkeit (DE-588)4012269-4 gnd Differentialtopologie (DE-588)4012255-4 gnd Mannigfaltigkeit (DE-588)4037379-4 gnd Riemannscher Raum (DE-588)4128295-4 gnd Differentialgeometrie (DE-588)4012248-7 gnd |
| topic_facet | Riemann, Variétés de Variétés différentiables calcul forme champ vecteur espace Banach espace Hilbert géométrie différentielle théorème Cartan-Hadamard théorème Stokes topologie différentielle variété différentielle variété riemannienne équation différentielle Differenzierbare Mannigfaltigkeit Differentialtopologie Mannigfaltigkeit Riemannscher Raum Differentialgeometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006613644&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000067 |
| work_keys_str_mv | AT langserge differentialandriemannianmanifolds |