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Student Mathematical Library 2006; 380 pp; softcover Volume: 16 ISBN10: 0821839888 ISBN13: 9780821839881 List Price: US$54 Institutional Members: US$43.20 All Individuals: US$43.20 Order Code: STML/16.R See also: Mostly Surfaces  Richard Evan Schwartz Curves and Surfaces: Second Edition  Sebastian Montiel and Antonio Ros Differential Geometry: Curves  Surfaces  Manifolds, Third Edition  Wolfgang Kuhnel  Our first knowledge of differential geometry usually comes from the study of the curves and surfaces in \(I\!\!R^3\) that arise in calculus. Here we learn about line and surface integrals, divergence and curl, and the various forms of Stokes' Theorem. If we are fortunate, we may encounter curvature and such things as the SerretFrenet formulas. With just the basic tools from multivariable calculus, plus a little knowledge of linear algebra, it is possible to begin a much richer and rewarding study of differential geometry, which is what is presented in this book. It starts with an introduction to the classical differential geometry of curves and surfaces in Euclidean space, then leads to an introduction to the Riemannian geometry of more general manifolds, including a look at Einstein spaces. An important bridge from the lowdimensional theory to the general case is provided by a chapter on the intrinsic geometry of surfaces. The first half of the book, covering the geometry of curves and surfaces, would be suitable for a onesemester undergraduate course. The local and global theories of curves and surfaces are presented, including detailed discussions of surfaces of rotation, ruled surfaces, and minimal surfaces. The second half of the book, which could be used for a more advanced course, begins with an introduction to differentiable manifolds, Riemannian structures, and the curvature tensor. Two special topics are treated in detail: spaces of constant curvature and Einstein spaces. The main goal of the book is to get started in a fairly elementary way, then to guide the reader toward more sophisticated concepts and more advanced topics. There are many examples and exercises to help along the way. Numerous figures help the reader visualize key concepts and examples, especially in lower dimensions. For the second edition, a number of errors were corrected and some text and a number of figures have been added. Request an examination or desk copy. Readership Undergraduate and graduate students, as well as research mathematicians, interested in differential geometry. Reviews From a review of the German edition: "The book covers all the topics which could be necessary later for learning higher level differential geometry. The material is very carefully sorted and easytoread."  Mathematical Reviews From a review of the first edition: "It is extraordinarily welcome that this comprehensive textbook containing all the high points of differential geometry is now available in an English translation. Required reading for all mathematicians!"  translated from International Mathematical News Table of Contents



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