Introduction to Smooth Manifolds

Description

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. Front Matter....Pages I-XV Smooth Manifolds....Pages 1-31 Smooth Maps....Pages 32-49 Tangent Vectors....Pages 50-76 Submersions, Immersions, and Embeddings....Pages 77-97 Submanifolds....Pages 98-124 Sard’s Theorem....Pages 125-149 Lie Groups....Pages 150-173 Vector Fields....Pages 174-204 Integral Curves and Flows....Pages 205-248 Vector Bundles....Pages 249-271 The Cotangent Bundle....Pages 272-303 Tensors....Pages 304-326 Riemannian Metrics....Pages 327-348 Differential Forms....Pages 349-376 Orientations....Pages 377-399 Integration on Manifolds....Pages 400-439 De Rham Cohomology....Pages 440-466 The de Rham Theorem....Pages 467-489 Distributions and Foliations....Pages 490-514 The Exponential Map....Pages 515-539 Quotient Manifolds....Pages 540-563 Symplectic Manifolds....Pages 564-595 Back Matter....Pages 596-708