Introduction to Smooth Manifolds

Manifolds are everywhere. These generalizations of curves and surfaces to arbitrarily many dimensions provide the mathematical context for under­ standing "space" in all of its manifestations. Today, the tools of manifold theory are indispensable in most major subfields of pure mathematics, and outside of pure mathematics they are becoming increasingly important to scientists in such diverse fields as genetics, robotics, econometrics, com­ puter graphics, biomedical imaging, and, of course, the undisputed leader among consumers (and inspirers) of mathematics-theoretical physics. No longer a specialized subject that is studied only by differential geometers, manifold theory is now one of the basic skills that all mathematics students should acquire as early as possible. Over the past few centuries, mathematicians have developed a wondrous collection of conceptual machines designed to enable us to peer ever more deeply into the invisible world of geometry in higher dimensions. Once their operation is mastered, these powerful machines enable us to think geometrically about the 6-dimensional zero set of a polynomial in four complex variables, or the lO-dimensional manifold of 5 x 5 orthogonal ma­ trices, as easily as we think about the familiar 2-dimensional sphere in ]R3.

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. Leehas 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)