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 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 algebra and real analysis.

Contents

Chapter 1: Smooth Manifolds

Chapter 2: Smooth Maps

Chapter 3: Tangent Vectors

Chapter 4: Submersions, Immersions, and Embeddings

Chapter 5: Submanifolds

Chapter 6: Sard’s Theorem

Chapter 7: Lie Groups

Chapter 8: Vector Fields

Chapter 9: Integral Curves and Flows

Chapter 10: Vector Bundles

Chapter 11: The Cotangent Bundle

Chapter 12: Tensors

Chapter 13: Riemannian Metrics

Chapter 14: Differential Forms

Chapter 15: Orientations

Chapter 16: Integration on Manifolds

Chapter 17: De Rham Cohomology

Chapter 18: The de Rham Theorem

Chapter 19: Distributions and Foliations

Chapter 20: The Exponential Map

Chapter 21: Quotient Manifolds

Chapter 22: Symplectic Manifolds

Review

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)

About the Author

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).