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BK

2nd ed.

New York : Springer, 2013

xv, 708 s. : il. ; 25 cm

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