"This textbook, probably the best introduction to differential geometry to be published since Eisenhart's, greatly benefits from the author's knowledge of what to avoid, something that a beginner is likely to miss. The presentation is smooth, the choice of topics optimal, and the book can be profitably used for self teaching." --- The Bulletin of Mathematical Books (review of 1st edition) "A thorough, modern, and lucid treatment of the differential topology, geometry, and global analysis needed to begin advanced study of research in these areas." --- Choice (review of 1st edition) "Probably the most outstanding novelty...is the appropriate selection of topics." --- Mathematical Reviews (review of 1st edition) The basics of differentiable manifolds, global calculus, differential geometry, and related topics constitute a core of information essential for the first or second year graduate student preparing for advanced courses and seminars in differential topology and geometry. {Differentiable Manifolds} is a text designed to cover this material in a careful and sufficiently detailed manner, presupposing only a good foundation in general topology, calculus, and modern algebra. This second edition contains a significant amount of new material, which, in addition to classroom use, will make it a useful reference text. Topics that can be omitted safely in a first course are clearly marked, making this edition easier to use for such a course, as well as for private study by non-specialists wishing to survey the field. The themes of linearization, (re) integration, and global versus local calculus are emphasized throughout. Additional features include a treatment of the elements of multivariable calculus, formulated to adapt readily to the global context, an exploration of bundle theory, TOC:Preface to the Second Edition * Topological Manifolds * The Local Theory of Smooth Functions * The Global Theory of Smooth Functions * Flows and Foliations * Lie Groups and Lie Algebras * Covectors and 1-- Forms * Multilinear Algebra and Tensors * Integration of Forms and de Rham Cohomology * Forms and Foliations * Riemannian Geometry * Principal Bundles * Appendix A. Construction of the Universal Covering * Appendix B. Inverse Function Theorem * Appendix C. Ordinary Differential Equations * Appendix D. The de Rham Cohomology Theorem * Bibliography * Index
