This text for graduate students and research mathematicians progresses from differentiable manifolds and the tangent structure through the local Frobenius theorem, covariant derivatives, and Riemannian and semi-Riemannian geometry. An unusual feature of the book is the inclusion of an early chapter on the differential geometry of hypersurfaces in Euclidean space. The text includes worked examples and chapter problems ranging in difficulty, and 30 pages of appendices on category theory, topology, calculus theorems, and modules and multilinearity. The first chapters of the book are suitable for a one-semester course on manifolds. There is ample material for a year-long course on manifolds and geometry. Lee is affiliated with the Department of Mathematics and Statistics at Texas Tech University. Annotation ©2010 Book News, Inc., Portland, OR (booknews.com)