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.
