Includes References (675-677) and Index

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 contangent 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

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.