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BK
2nd ed.
New York : Springer, 2013
xv, 708 s. : il. ; 25 cm

objednat
ISBN 978-1-4419-9981-8 (váz.)
Graduate text in mathematics, 218 ISSN 0072-5285 ;
Obsahuje bibliografii na s. 675-677 a rejstříky
000246063
Contents // 1 Smooth Manifolds ... 1 // Topological Manifolds... // Smooth Structures... 10 // Examples of Smooth Manifolds... 17 // Manifolds with Boundary... 24 // Problems... 29 // 2 Smooth Maps... 22 // Smooth Functions and Smooth Maps... 32 // Partitions of Unity... 4 // Problems... 4 // 3 Tangent Vectors... 50 // Tangent Vectors... 51 // The Differential of a Smooth Map... 55 // Computations in Coordinates... 60 // The Tangent Bundle... 65 // Velocity Vectors of Curves... 68 // Alternative Definitions of the Tangent Space... 71 // Categories and Functors... 73 // Problems... 75 // 4 Submersions, Immersions, and Embeddings... 77 // Maps of Constant Rank... 77 // Embeddings... 85 // Submersions... 88 // Smooth Covering Maps... 91 // Problems... 95 // 5 Submanifolds... 98 // Embedded Submanifolds ... 98 // Immersed Submanifolds...108 // xi // xjj Contents // Restricting Maps to Submanifolds...112 // The Tangent Space to a Submanifold...115 // Submanifolds with Boundary...120 // Problems... // 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...142 // 7 Lie Groups ...150 // Basic Definitions...151 // Lie Group Homomorphisms...153 // Lie Subgroups...156 // Group Actions and Equi variant Maps...161 // Problems...121 // 8 Vector Fields ...124 // Vector Fields on Manifolds...124 // Vector Fields and Smooth Maps...181 // Lie Brackets...185
// The Lie Algebra of a Lie Group...189 // Problems...19 // 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... .7 // 97? // Covectors ... z/z // The Differential of a Function...280 // Pullbacks of Covector Fields ...284 // Line Integrals ...287 // Conservative Covector Fields...292 // Problems...299 // 12 Tensors ... // Multilinear Algebra . // Symmetric and Alternating Tensors...313 // Tensors and Tensor Fields on Manifolds ...316 // Problems...224 // 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 // Appendix ? 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 // AppendixC 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 // Appendix D Review of Differential Equations...663 // Existence, Uniqueness, and Smoothness ...663 // Simple Solution Techniques...672 // References...675 // Notation Index...678 // Subject Index...683

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