Inhaltsangabe
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. Along the way, the book introduces students to some of the most important exemples of geometric structures that manifolds can carry, such as Riemannian metrics, symplectic structures, and foliations. The book is aimed at students who already have a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis.
Críticas
From the reviews: "This book offers a concise, clear, and detailed introduction to analysis on manifolds and elementary differential geometry. ... Some of the prerequisites are reviewed in an appendix. For the ambitious reader, lots of exercises and problems are provided." (A. Cap, Monatshefte für Mathematik, Vol. 145 (4), 2005) "The title of this 600 pages book is self-explaining. And in fact the book could have been entitled ‘A smooth introduction to manifolds’. ... Also the notations are light and as smooth as possible, which is nice. ... The comprehensive theoretical matter is illustrated with many figures, examples, exercises and problems. Some of these exercises are quite deep ... ." (Pascal Lambrechts, Bulletin of the Belgian Mathematical Society, Vol. 11 (3), 2004) "It introduces and uses all of the standard tools of smooth manifold theory and offers the proofs of all its fundamental theorems. ... This is a clearly and carefully written book in the author’s usual elegant style. The exposition is crisp and contains a lot of pictures and intuitive explanations of how one should think geometrically about some abstract concepts. It could profitably be used by beginning graduate students who want to undertake a deeper study of specialized applications of smooth manifold theory." (Mircea Craioveanu, Zentralblatt MATH, Vol. 1030, 2004) "This text provides an elementary introduction to smooth manifolds which can be understood by junior undergraduates. ... There are 157 illustrations, which bring much visualisation, and the volume contains many examples and easy exercises, as well as almost 300 ‘problems’ that are more demanding. The subject index contains more than 2700 items! ... The pedagogic mastery, the long-life experience with teaching, and the deep attention to students’ demands make this book a real masterpiece that everyone should have in their library." (EMS Newsletter, June, 2003) "Prof. Lee has written the definitive modern introduction to manifolds. ... The material is very well motivated. He writes in a rigorous yet discursive style, full of examples, digressions, important results, and some applications. ... The exercises appearing in the text and at the end of the chapters are an excellent mix ... . it would make an ideal text for a comprehensive graduate-level course in modern differential geometry, as well as an excellent reference book for the working (applied) mathematician." (Peter J. Oliver, SIAM Review, Vol. 46 (1), 2004)
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