Manifolds and differential geometry:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Math. Soc.
2009
|
| Series: | Graduate studies in mathematics
107 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke |
| Physical Description: | XIV, 671 S. graph. Darst. |
| ISBN: | 9780821848159 9781470469825 |
Staff View
MARC
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| 100 | 1 | |a Lee, Jeffrey M. |d 1956- |e Verfasser |0 (DE-588)14026714X |4 aut | |
| 245 | 1 | 0 | |a Manifolds and differential geometry |c Jeffrey M. Lee |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 2009 | |
| 300 | |a XIV, 671 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate studies in mathematics |v 107 | |
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
| 650 | 4 | |a Geometry, Differential | |
| 650 | 4 | |a Topological manifolds | |
| 650 | 4 | |a Riemannian manifolds | |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-017737052 | |
Record in the Search Index
| _version_ | 1821138765935017984 |
|---|---|
| adam_text |
Contents
Preface
xi
Chapter
1.
Differentiable Manifolds
1
§1.1.
Preliminaries
2
§1.2.
Topological Manifolds
6
§1.3.
Charts, Atlases and Smooth Structures
11
§1.4.
Smooth Maps and Diffeomorphisms
22
§1.5.
Cut-off Functions and Partitions of Unity
28
§1.6.
Coverings and Discrete Groups
31
§1.7.
Regular Submanifolds
46
§1.8.
Manifolds with Boundary
48
Problems
51
Chapter
2.
The Tangent Structure
55
§2.1.
The Tangent Space
55
§2.2.
Interpretations
65
§2.3.
The Tangent Map
66
§2.4.
Tangents of Products
72
§2.5.
Critical Points and Values
74
§2.6.
Rank and Level Set
78
§2.7.
The Tangent and Cotangent Bundles
81
§2.8.
Vector Fields
87
§2.9.
1-Forms
110
§2.10.
Line Integrals and Conservative Fields
116
Contents
§2.11.
Moving
Erames 12°
Problems
■'•22
Chapter
3.
Immersion and Submersion
127
§3.1.
Immersions
127
§3.2.
Immersed and Weakly Embedded Submanifolds
130
§3.3.
Submersions
138
Problems
140
Chapter
4.
Curves and Hypersurfaces in Euclidean Space
143
§4.1.
Curves
145
§4.2.
Hypersurfaces
152
§4.3.
The Levi-Civita Covariant Derivative
165
§4.4.
Area and Mean Curvature
178
§4.5.
More on Gauss Curvature
180
§4.6.
Gauss Curvature Heuristics
184
Problems
187
Chapter
5.
Lie Groups
189
§5.1.
Definitions and Examples
189
§5.2.
Linear Lie Groups
192
§5.3.
Lie Group Homomorphisms
201
§5.4.
Lie Algebras and Exponential Maps
204
§5.5.
The Adjoint Representation of a Lie Group
220
§5.6.
The Maurer-Cartan Form
224
§5.7.
Lie Group Actions
228
§5.8.
Homogeneous Spaces
240
§5.9.
Combining Representations
249
Problems
253
Chapter
6.
Fiber Bundles
257
§6.1.
General Fiber Bundles
257
§6.2.
Vector Bundles
270
§6.3.
Tensor Products of Vector Bundles
282
§6.4.
Smooth Functors
283
§6.5.
Hom
285
§6.6.
Algebra Bundles
287
§6.7.
Sheaves
288
Contents
vii
§6.8. Principal and Associated
Bundles
291
Problems 303
Chapter
7. Tensors 307
§7.1.
Some Multilinear Algebra
308
§7.2.
Bottom-Up Approach to Tensor Fields
318
§7.3.
Тор
-Down
Approach to Tensor Fields
323
§7.4.
Matching the Two Approaches to Tensor Fields
324
§7.5.
Tensor Derivations
327
§7.6.
Metric Tensors
331
Problems
342
Chapter
8.
Differential Forms
345
§8.1.
More Multilinear Algebra
345
§8.2.
Differential Forms
358
§8.3.
Exterior Derivative
363
§8.4.
Vector-Valued and Algebra-Valued Forms
367
§8.5.
Bundle-Valued Forms
370
§8.6.
Operator Interactions
373
§8.7.
Orientation
375
§8.8.
Invariant Forms
384
Problems
388
Chapter
9.
Integration and Stokes' Theorem
391
§9.1.
Stokes' Theorem
394
§9.2.
Differentiating Integral Expressions; Divergence
397
§9.3.
Stokes' Theorem for Chains
400
§9.4.
Differential Forms and Metrics
404
§9.5.
Integral Formulas
414
§9.6.
The Hodge Decomposition
418
§9.7.
Vector Analysis on R3
425
§9.8.
Electromagnetism
429
§9.9.
Surface Theory Redux
434
Problems
437
Chapter
10. De Rham
Cohomoiogy
441
§10.1.
The Mayer-Vietoris Sequence
447
§10.2.
Homotopy
Invariance
449
viii Contents
§10.3.
Compactly Supported Cohomology
456
§10.4.
Poincaré
Duality
460
Problems
465
Chapter
11.
Distributions and Frobenius' Theorem
467
§11.1.
Definitions
468
§11.2.
The Local Probenius Theorem
471
§11.3.
Differential Forms and Integrability
473
§11.4.
Global Probenius Theorem
478
§11.5.
Applications to Lie Groups
484
§11.6.
Fundamental Theorem of Surface Theory
486
§11.7.
Local Fundamental Theorem of Calculus
494
Problems
498
Chapter
12.
Connections and Covariant Derivatives
501
§12.1.
Definitions
501
§12.2.
Connection Forms
506
§12.3.
Differentiation Along a Map
507
§12.4. Ehresmann
Connections
509
§12.5.
Curvature
525
§12.6.
Connections on Tangent Bundles
530
§12.7.
Comparing the Differential Operators
532
§12.8.
Higher Covariant Derivatives
534
§12.9.
Exterior Covariant Derivative
536
§12.10.
Curvature Again
540
§12.11.
The
Bianchi
Identity
541
§12.12.
G-Connections
542
Problems
544
Chapter
13.
Riemannian and Semi-Riemannian Geometry
547
§13.1.
Levi-Civita Connection
550
§13.2.
Riemann Curvature Tensor
553
§13.3.
Semi-Riemannian Submanifolds
560
§13.4.
Geodesies
567
§13.5.
Riemannian Manifolds and Distance
585
§13.6.
Lorentz
Geometry
588
§13.7.
Jakobi
Fields
594
Contents ix
§13.8. First and
Second
Variation
of Arc Length
599
§13.9.
More Riemannian Geometry
612
§13.10.
Cut Locus
617
§13.11.
Rauch's Comparison Theorem
619
§13.12. Weitzenböck
Formulas
623
§13.13.
Structure of General Relativity
627
Problems
634
Appendix A. The Language of Category Theory
637
Appendix B. Topology
643
ŞB.1.
The Shrinking Lemma
643
§B.2. Locally Euclidean Spaces
645
Appendix C. Some Calculus Theorems
647
Appendix D. Modules and Multilinearity
649
§D.l. R-Algebras
660
Bibliography
663
Index
667 |
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| isbn | 9780821848159 9781470469825 |
| language | English |
| lccn | 2009012421 |
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| spelling | Lee, Jeffrey M. 1956- Verfasser (DE-588)14026714X aut Manifolds and differential geometry Jeffrey M. Lee Providence, RI American Math. Soc. 2009 XIV, 671 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 107 Hier auch später erschienene, unveränderte Nachdrucke Geometry, Differential Topological manifolds Riemannian manifolds Topologische Mannigfaltigkeit (DE-588)4185712-4 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Riemannscher Raum (DE-588)4128295-4 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 s Topologische Mannigfaltigkeit (DE-588)4185712-4 s Riemannscher Raum (DE-588)4128295-4 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-1170-1 Graduate studies in mathematics 107 (DE-604)BV009739289 107 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017737052&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lee, Jeffrey M. 1956- Manifolds and differential geometry Graduate studies in mathematics Geometry, Differential Topological manifolds Riemannian manifolds Topologische Mannigfaltigkeit (DE-588)4185712-4 gnd Differentialgeometrie (DE-588)4012248-7 gnd Riemannscher Raum (DE-588)4128295-4 gnd |
| subject_GND | (DE-588)4185712-4 (DE-588)4012248-7 (DE-588)4128295-4 |
| title | Manifolds and differential geometry |
| title_auth | Manifolds and differential geometry |
| title_exact_search | Manifolds and differential geometry |
| title_full | Manifolds and differential geometry Jeffrey M. Lee |
| title_fullStr | Manifolds and differential geometry Jeffrey M. Lee |
| title_full_unstemmed | Manifolds and differential geometry Jeffrey M. Lee |
| title_short | Manifolds and differential geometry |
| title_sort | manifolds and differential geometry |
| topic | Geometry, Differential Topological manifolds Riemannian manifolds Topologische Mannigfaltigkeit (DE-588)4185712-4 gnd Differentialgeometrie (DE-588)4012248-7 gnd Riemannscher Raum (DE-588)4128295-4 gnd |
| topic_facet | Geometry, Differential Topological manifolds Riemannian manifolds Topologische Mannigfaltigkeit Differentialgeometrie Riemannscher Raum |
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