"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, and a further (optional) development of Lie theory than is customary in textbooks at this level. New to this edition is a detailed treatment of covering spaces and the fundamental group.

Students, teachers and professionals in mathematics and mathematical physics should find this a most stimulating and useful text.

Klappentext

This book is based on the full year Ph.D. qualifying course on differentiable manifolds, global calculus, differential geometry, and related topics, given by the author at Washington University several times over a twenty year period. It is addressed primarily to second year graduate students and well prepared first year students. Presupposed is a good grounding in general topology and modern algebra, especially linear algebra and the analogous theory of modules over a commutative, unitary ring. Although billed as a "first course" , the book is not intended to be an overly sketchy introduction. Mastery of this material should prepare the student for advanced topics courses and seminars in differen­ tial topology and geometry. There are certain basic themes of which the reader should be aware. The first concerns the role of differentiation as a process of linear approximation of non­ linear problems. The well understood methods of linear algebra are then applied to the resulting linear problem and, where possible, the results are reinterpreted in terms of the original nonlinear problem. The process of solving differential equations (i. e., integration) is the reverse of differentiation. It reassembles an infinite array of linear approximations, result­ ing from differentiation, into the original nonlinear data. This is the principal tool for the reinterpretation of the linear algebra results referred to above.

Inhalt

1. Topological Manifolds.- 2. Local Theory.- 3. Global Theory.- 4. Flows and Foliation.- 5. Lie Groups.- 6. Covectors and 1-Forms.- 7. Multilinear Algebra.- 8. Integration and Cohomology.- 9. Forms and Foliations.- 10. Riemannian Geometry.- Appendix A. Vector Fields on Spheres.- Appendix B. Inverse Function Theorem.- Appendix C. Ordinary Differential Equations.- C.1. Existence and uniqueness of solutions.- C.2. A digression concerning Banach spaces.- C.3. Smooth dependence on initial conditions.- C.4. The Linear Case.- Appendix D. Sard's Theorem.- Appendix E. de Rham-?ech Theorem.- E.1. ?ech cohomology.- E.2. The de Rham-?ech complex.