Springer Science & Business Media, 11 janv. 2008 - 418 pages
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 the second 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.
"This is a carefully written and wide-ranging textbook suitable mainly for graduate courses, although some advanced undergraduate courses may benefit from the early chapters. The subject matter is differential topology and geometry, that is, the study of curves, surfaces and manifolds where the assumption of differentiability adds the tools of differentiable and integral calculus to those of topology. Within this area, the book is unusually comprehensive.... The style is clear and precise, and this makes the book a good reference text. There are many good exercises."—The Mathematical Gazette (Review of the Second Edition)
"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 the First Edition)
"The author has very well succeeded in writing an interesting, stimulating and pleasant reading book [with] an intelligent equilibrium between rigor and informal."—Zentralblatt Math (Review of the First Edition)
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The Local Theory of Smooth Functions
The Global Theory of Smooth Functions
Flows and Foliations
Lie Groups and Lie Algebras 161
Covectors and 1Forms
Multilinear Algebra and Tensors
Integration of Forms and de Rham Cohomology
Forms and Foliations
Appendix A Construction of the Universal Covering 369
Ordinary Differential Equations 379
The de Rharn Cohomology Theorem