Introduction to smooth manifolds:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York
Springer
[2013]
|
| Edition: | Second edition |
| Series: | Graduate texts in mathematics
218 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverzeichnis Seite 675 - 677 |
| Physical Description: | XV, 708 Seiten Diagramme. - Illustrationen, Diagramme |
| ISBN: | 9781441999818 9781489994752 |
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MARC
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| 100 | 1 | |a Lee, John M. |d 1950- |e Verfasser |0 (DE-588)122260880 |4 aut | |
| 245 | 1 | 0 | |a Introduction to smooth manifolds |c John M. Lee |
| 250 | |a Second edition | ||
| 264 | 1 | |a New York |b Springer |c [2013] | |
| 264 | 4 | |c © 2013 | |
| 300 | |a XV, 708 Seiten |b Diagramme. - Illustrationen, Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a Graduate texts in mathematics |v 218 | |
| 500 | |a Literaturverzeichnis Seite 675 - 677 | ||
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Record in the Search Index
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| adam_text | Contents 1 2 3 Topological Manifolds................................................................................... Smooth Structures......................................................................................... Examples of Smooth Manifolds................................................................... Manifolds with Boundary............................................................................ Problems......................................................................................................... 1 2 10 17 24 29 Smooth Maps..................................................................................................... 32 Smooth Functions and Smooth Maps......................................................... Partitions of Unity......................................................................................... Problems......................................................................................................... 32 40 48 Tangent Vectors.................................................................................................. 50 51 55 60 65 68 71 73 75 Smooth Manifolds ........................................................................................... Tangent Vectors............................................................................................ The Differential of a Smooth Map................................................................ Computations in Coordinates...................................................................... The Tangent
Bundle...................................................................................... Velocity Vectors of Curves............................................................................ Alternative Definitions of the Tangent Space............................................. Categories and Functors............................................................................... Problems......................................................................................................... 4 Submersions, Immersions, and Embeddings............................................. Maps of Constant Rank............................................................................... Embeddings.................................................................................................. Submersions.................................................................................................. Smooth Covering Maps............................................................................... Problems......................................................................................................... 5 77 77 85 88 91 95 98 Embedded Submanifolds ............................................................................ 98 Immersed Submanifolds.................................................................................. 108 Submanifolds..................................................................................................... xi
xii Contents Restricting Maps to Submanifolds ................................................................... 112 The Tangent Space to a Submanifold............................................................ 115 Submanifolds with Boundary......................................................................... 120 Problems............................................................................................................ 123 6 Sard’s Theorem............................................................................................... 125 Sets of Measure Zero ...................................................................................... 125 Sard’s Theorem............................................................................................... 129 The Whitney Embedding Theorem................................................................131 The Whitney Approximation Theorems......................................................... 136 Transversality...................................................................................................143 Problems............................................................................................................ 147 7 Lie Groups...................................................................................................... 150 Basic Definitions............................................................................................... 151 Lie Group Homomorphisms ............................................................................ 153 Lie
Subgroups.................................................................................................. 156 Group Actions and Equivariant Maps............................................................ 161 Problems............................................................................................................ 171 8 Vector Fields...................................................................................................174 Vector Fields on Manifolds............................................................................ 174 Vector Fields and Smooth Maps...................................................................... 181 Lie Brackets...................................................................................................... 185 The Lie Algebra of a Lie Group...................................................................... 189 Problems............................................................................................................ 199 9 Integral Curves and Flows............................................................................ 205 Integral Curves.................................................................................................. 206 Flows.................................................................................................................. 209 Flowouts............................................................................................................217 Flows and Flowouts on Manifolds with Boundary...................................... 222 Lie
Derivatives.................................................................................................. 227 Commuting Vector Fields............................................................................... 231 Time-Dependent Vector Fields ..................................................................... 236 First-Order Partial Differential Equations......................................................239 Problems........................................................................................................... 245 10 Vector Bundles............................................................................................... 249 Vector Bundles.................................................................................................. 249 Local and Global Sections of Vector Bundles............................................... 255 Bundle Homomorphisms ............................................................................... 261 Subbundles........................................................................................................ 264 Fiber Bundles.................................................................................................. 268 Problems........................................................................................................... 268
Contents xiii 11 The Cotangent Bundle................................................................................. 272 Covectors ........................................................................................................ 272 The Differential of a Function........................................................................ 280 Pullbacks of Covector Fields ........................................................................ 284 Line Integrals .................................................................................................. 287 Conservative Covector Fields........................................................................ 292 Problems........................................................................................................... 299 12 Tensors ........................................................................................................... 304 Multilinear Algebra........................................................................................ 305 Symmetric and Alternating Tensors............................................................... 313 Tensors and Tensor Fields on Manifolds ..................................................... 316 Problems........................................................................................................... 324 13 Riemannian Metrics.....................................................................................327 Riemannian Manifolds..................................................................................... 327 The Riemannian Distance
Function............................................................... 337 The Tangent-Cotangent Isomorphism............................................................ 341 Pseudo-Riemannian Metrics............................................................................343 Problems........................................................................................................... 344 14 Differential Forms........................................................................................ 349 The Algebra of Alternating Tensors............................................................... 350 Differential Forms on Manifolds .................................................................. 359 Exterior Derivatives........................................................................................ 362 Problems........................................................................................................... 373 15 Orientations..................................................................................................... 377 Orientations of Vector Spaces........................................................................ 378 Orientations of Manifolds...............................................................................380 The Riemannian Volume Form..................................................................... 388 Orientations and Covering Maps .................................................................. 392
Problems........................................................................................................... 397 16 Integration on Manifolds...................................................... 400 The Geometry of Volume Measurement........................................................ 401 Integration of Differential Forms.................................................................. 402 Stokes’s Theorem........................................................................................... 411 Manifolds with Corners..................................................................................415 Integration on Riemannian Manifolds........................................................... 421 Densities........................................................................................................... 427 Problems........................................................................................................... 434 17 De Rham Cohomology................................................................................. 440 The de Rham Cohomology Groups...............................................................441 Homotopy Invariance.................................................................................... 443 The Mayer-Vietoris Theorem........................................................................ 448 Degree Theory................................................................................................. 457
xiv Contents Proof of the Mayer-Vietoris Theorem............................................................ 460 Problems............................................................................................................464 18 The de Rham Theorem.................................................................................. 467 Singular Homology.........................................................................................467 Singular Cohomology..................................................................................... 472 Smooth Singular Homology............................................................................ 473 The de Rham Theorem..................................................................................... 480 Problems........................................................................................................... 487 19 Distributions and Foliations.........................................................................490 Distributions and Involutivity.........................................................................491 The Frobenius Theorem.................................................................................. 496 Foliations........................................................................................................... 501 Lie Subalgebras and Lie Subgroups............................................................... 505 Overdetermined Systems of Partial Differential Equations......................... 507
Problems........................................................................................................... 512 20 The Exponential Map .................................................................................. 515 One-Parameter Subgroups and the Exponential Map.................................. 516 The Closed Subgroup Theorem..................................................................... 522 Infinitesimal Generators of Group Actions.................................................. 525 The Lie Correspondence.................................................................................. 530 Normal Subgroups............................................................................................533 Problems........................................................................................................... 536 21 Quotient Manifolds........................................................................................ 540 Quotients of Manifolds by Group Actions..................................................... 541 Covering Manifolds........................................................................................ 548 Homogeneous Spaces..................................................................................... 550 Applications to Lie Theory ............................................................................555 Problems........................................................................................................... 560 22 Symplectic
Manifolds..................................................................................... 564 Symplectic Tensors ........................................................................................ 565 Symplectic Structures on Manifolds............................................................... 567 The Darboux Theorem..................................................................................... 571 Hamiltonian Vector Fields............................................................................... 574 Contact Structures........................................................................................... 581 Nonlinear First-Order PDEs........................................................................... 585 Problems........................................................................................................... 590 Appendix A Review of Topology...................................................................596 Topological Spaces ........................................................................................ 596 Subspaces, Products, Disjoint Unions, and Quotients.................................. 601 Connectedness and Compactness.................................................................. 607 Homotopy and the Fundamental Group........................................................ 612 Covering Maps..................................................................................................615
Contents XV Review of Linear Algebra........................................................ 617 Vector Spaces.................................................................................................. 617 Linear Maps..................................................................................................... 622 The Determinant............................................................................................... 628 Inner Products and Norms............................................................................... 635 Direct Products and Direct Sums.................................................................. 638 Appendix B Appendix C Review of Calculus ................................................................. 642 Total and Partial Derivatives............................................................................ 642 Multiple Integrals............................................................................................649 Sequences and Series of Functions............................................................... 656 The Inverse and Implicit Function Theorems............................................... 657 Review of Differential Equations............................................ 663 Existence, Uniqueness, and Smoothness ..................................................... 663 Simple Solution Techniques............................................................................ 672 Appendix D References........................................................................................................ 675
Notation Index.................................................................................................. 678 Subject Index..................................................................................................... 683
|
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| author | Lee, John M. 1950- |
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| discipline | Mathematik |
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| spelling | Lee, John M. 1950- Verfasser (DE-588)122260880 aut Introduction to smooth manifolds John M. Lee Second edition New York Springer [2013] © 2013 XV, 708 Seiten Diagramme. - Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 218 Literaturverzeichnis Seite 675 - 677 Glatte Kurve (DE-588)4157470-9 gnd rswk-swf Glatte Mannigfaltigkeit (DE-588)4157471-0 gnd rswk-swf Glatte Fläche (DE-588)4157467-9 gnd rswk-swf Glatte Mannigfaltigkeit (DE-588)4157471-0 s Glatte Kurve (DE-588)4157470-9 s Glatte Fläche (DE-588)4157467-9 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4419-9982-5 Graduate texts in mathematics 218 (DE-604)BV000000067 218 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=025300867&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lee, John M. 1950- Introduction to smooth manifolds Graduate texts in mathematics Glatte Kurve (DE-588)4157470-9 gnd Glatte Mannigfaltigkeit (DE-588)4157471-0 gnd Glatte Fläche (DE-588)4157467-9 gnd |
| subject_GND | (DE-588)4157470-9 (DE-588)4157471-0 (DE-588)4157467-9 |
| title | Introduction to smooth manifolds |
| title_auth | Introduction to smooth manifolds |
| title_exact_search | Introduction to smooth manifolds |
| title_full | Introduction to smooth manifolds John M. Lee |
| title_fullStr | Introduction to smooth manifolds John M. Lee |
| title_full_unstemmed | Introduction to smooth manifolds John M. Lee |
| title_short | Introduction to smooth manifolds |
| title_sort | introduction to smooth manifolds |
| topic | Glatte Kurve (DE-588)4157470-9 gnd Glatte Mannigfaltigkeit (DE-588)4157471-0 gnd Glatte Fläche (DE-588)4157467-9 gnd |
| topic_facet | Glatte Kurve Glatte Mannigfaltigkeit Glatte Fläche |
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