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 linear algebra and real analysis. Preface 1 Smooth Manifolds 2 Smooth Maps 3 Tangent Vectors 4 Submersions Immersions and Embeddings 5 Submanifolds 6 Sards 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.