Introduction to Smooth Manifolds
(Sprache: Englisch)
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...
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Klappentext zu „Introduction to Smooth Manifolds “
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 fewnew 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.
Inhaltsverzeichnis zu „Introduction to Smooth Manifolds “
- Preface1 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
Autoren-Porträt von John Lee
John M. Lee is Professor of Mathematics at the University of Washington in Seattle, where he regularly teaches graduate courses on the topology and geometry of manifolds. He was the recipient of the American Mathematical Society's Centennial Research Fellowship and he is the author of four previous Springer books: the first edition (2003) of Introduction to Smooth Manifolds, the first edition (2000) and second edition (2010) of Introduction to Topological Manifolds, and Riemannian Manifolds: An Introduction to Curvature (1997).
Bibliographische Angaben
- Autor: John Lee
- 2014, 2. Aufl., XVI, 708 Seiten, Masse: 15,3 x 23,4 cm, Kartoniert (TB), Englisch
- Verlag: Springer, Berlin
- ISBN-10: 1489994750
- ISBN-13: 9781489994752
Sprache:
Englisch
Pressezitat
From the reviews of the second edition:"It starts off with five chapters covering basics on smooth manifolds up to submersions, immersions, embeddings, and of course submanifolds. ... the book under review is laden with excellent exercises that significantly further the reader's understanding of the material, and Lee takes great pains to motivate everything well all the way through ... . a fine graduate-level text for differential geometers as well as people like me, fellow travelers who always wish they knew more about such a beautiful subject." (Michael Berg, MAA Reviews, October, 2012)
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