Cheeger and Ebin take a sensible approach to this new edition of their mid-1970s original: the enormous growth in this field has made attempting to upgrade their first edition unmanageable. Instead, they offer updated and expanded references and a sincere conviction readers can still use significant parts of their findings. They begin with basic concepts and results, such as the exponential map and normal coordinates, the Hopf-Rinow theorem, Jacobi fields, basic index lemmas, the Rauch comparison theorem and theorems to which Cartan was involved. They then move briskly through Toponogov's theorem, homogeneous spaces, Morse theory, closed deodesics and the cut locus, the sphere theorem and its generalizations, the differentiable space theorem, complete manifolds of non-negative curvature, and compact manifolds of non-positive curvature. Annotation ©2008 Book News, Inc., Portland, OR (booknews.com)