Introduction to Smooth Manifolds

Media type: E-Book

Title: Introduction to Smooth Manifolds

Contains: Introduction to Smooth Manifolds; Preface; Prerequisites; Exercises and Problems; About the Second Edition; Acknowledgments; Contents; Chapter 1: Smooth Manifolds; Topological Manifolds; Coordinate Charts; Examples of Topological Manifolds; Topological Properties of Manifolds; Connectivity; Local Compactness and Paracompactness; Fundamental Groups of Manifolds; Smooth Structures; Local Coordinate Representations; Examples of Smooth Manifolds; The Einstein Summation Convention; More Examples; Manifolds with Boundary; Smooth Structures on Manifolds with Boundary; Problems
Chapter 2: Smooth MapsSmooth Functions and Smooth Maps; Smooth Functions on Manifolds; Smooth Maps Between Manifolds; Diffeomorphisms; Partitions of Unity; Applications of Partitions of Unity; Problems; Chapter 3: Tangent Vectors; Tangent Vectors; Geometric Tangent Vectors; Tangent Vectors on Manifolds; The Differential of a Smooth Map; Computations in Coordinates; The Differential in Coordinates; Change of Coordinates; The Tangent Bundle; Velocity Vectors of Curves; Alternative Deﬁnitions of the Tangent Space; Tangent Vectors as Derivations of the Space of Germs
Tangent Vectors as Equivalence Classes of CurvesTangent Vectors as Equivalence Classes of n-Tuples; Categories and Functors; Problems; Chapter 4: Submersions, Immersions, and Embeddings; Maps of Constant Rank; Local Diffeomorphisms; The Rank Theorem; The Rank Theorem for Manifolds with Boundary; Embeddings; Submersions; Smooth Covering Maps; Problems; Chapter 5: Submanifolds; Embedded Submanifolds; Slice Charts for Embedded Submanifolds; Level Sets; Immersed Submanifolds; Restricting Maps to Submanifolds; Uniqueness of Smooth Structures on Submanifolds; Extending Functions from Submanifolds
The Tangent Space to a SubmanifoldSubmanifolds with Boundary; Problems; Chapter 6: Sard's Theorem; Sets of Measure Zero; Sard's Theorem; The Whitney Embedding Theorem; The Whitney Approximation Theorems; Tubular Neighborhoods; Smooth Approximation of Maps Between Manifolds; Transversality; Problems; Chapter 7: Lie Groups; Basic Deﬁnitions; Lie Group Homomorphisms; The Universal Covering Group; Lie Subgroups; Group Actions and Equivariant Maps; Equivariant Maps; Semidirect Products; Representations; Problems; Chapter 8: Vector Fields; Vector Fields on Manifolds; Local and Global Frames
Vector Fields as Derivations of Cinfty(M)Vector Fields and Smooth Maps; Vector Fields and Submanifolds; Lie Brackets; The Lie Algebra of a Lie Group; Induced Lie Algebra Homomorphisms; The Lie Algebra of a Lie Subgroup; Problems; Chapter 9: Integral Curves and Flows; Integral Curves; Flows; The Fundamental Theorem on Flows; Complete Vector Fields; Flowouts; Regular Points and Singular Points; Flows and Flowouts on Manifolds with Boundary; Lie Derivatives; Commuting Vector Fields; Commuting Frames; Time-Dependent Vector Fields; First-Order Partial Differential Equations; Linear Equations
Quasilinear Equations

Contributor: Lee, John M. [Author]

imprint: New York, NY [u.a.]: Springer, 2012

Issue: 2nd ed. 2012

Extent: Online-Ressource (XV, 708 p. 150 illus, digital)

Language: English

DOI: 10.1007/978-1-4419-9982-5

ISBN: 9781441999825

Identifier:

RVK notation: SK 350 : Topologie und Geometrie von Mannigfaltigkeiten, Katastrophentheorie
SI 990 : Sonstige (CSN + Bandzählung)

Keywords: Glatte Mannigfaltigkeit
Glatte Kurve
Glatte Fläche
Glatte Mannigfaltigkeit
Glatte Kurve
Glatte Fläche

Origination:

Footnote: Description based upon print version of record

Description: Preface -- 1 Smooth Manifolds -- 2 Smooth Maps -- 3 Tangent Vectors -- 4 Submersions, Immersions, and Embeddings -- 5 Submanifolds -- 6 Sard's Theorem -- 7 Lie Groups -- 8 Vector Fields -- 9 Integral Curves and Flows -- 10 Vector Bundles -- 11 The Cotangent Bundle -- 12 Tensors -- 13 Riemannian Metrics -- 14 Differential Forms -- 15 Orientations -- 16 Integration on Manifolds.- 17 De Rham Cohomology.- 18 The de Rham Theorem -- 19 Distributions and Foliations.- 20 The Exponential Map.- 21 Quotient Manifolds.- 22 Symplectic Manifolds -- Appendix A: Review of Topology -- Appendix B: Review of Linear Algebra -- Appendix C: Review of Calculus -- Appendix D: Review of Differential Equations -- References -- Notation Index -- Subject Index.

This book is an introductory graduate-level textbook on the theory of smooth manifolds. Its goal is to familiarize students with the tools they will need in order to use manifolds in mathematical or scientific research—smooth structures, tangent vectors and covectors, vector bundles, immersed and embedded submanifolds, tensors, differential forms, de Rham cohomology, vector fields, flows, foliations, Lie derivatives, Lie groups, Lie algebras, and more. The approach is as concrete as possible, with pictures and intuitive discussions of how one should think geometrically about the abstract concepts, while making full use of the powerful tools that modern mathematics has to offer. This second edition has been extensively revised and clarified, and the topics have been substantially rearranged. The book now introduces the two most important analytic tools, the rank theorem and the fundamental theorem on flows, much earlier so that they can be used throughout the book. A few new topics have been added, notably Sard’s theorem and transversality, a proof that infinitesimal Lie group actions generate global group actions, a more thorough study of first-order partial differential equations, a brief treatment of degree theory for smooth maps between compact manifolds, and an introduction to contact structures. Prerequisites include a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis.

Introduction to Smooth Manifolds - [2nd ed. 2012]

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