Introduction to Smooth Manifolds / by John Lee.

Format:

Book

Author/Creator:

Lee, John, author.

Contributor:

SpringerLink (Online service)

Series:

Graduate texts in mathematics 0072-5285 ; 218.

Graduate Texts in Mathematics, 0072-5285 ; 218

Language:

English

Subjects (All):

Geometry, Differential.

Differential Geometry.

Local Subjects:

Differential Geometry.

Physical Description:

1 online resource.

Edition:

Second edition 2012.

Contained In:

Springer Nature eBook

Place of Publication:

New York, NY : Springer New York : Imprint: Springer, 2012.

System Details:

text file PDF

Summary:

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.

Contents:

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.

Other Format:

Printed edition:

ISBN:

9781441999825

Access Restriction:

Restricted for use by site license.