This is a systematic presentation of results concerning the isometric embedding of Riemannian manifolds in local and global Euclidean spaces, especially focused on the isometric embedding of surfaces in a Euclidean space R3 and primarily employing partial differential equation techniques for proving results. The authors begin their treatment with discussion of fundamental results of the isometric embedding of Riemannian manifolds in Euclidean spaces, including the Janet-Cartan Theorem and Nash Embedding Theorem. Next, looking at the local isometric embedding of surfaces in R3, they discuss metrics with Gauss curvature that is everywhere positive, negative, nonnegative, nonpositive, as well as the case of the mixed sign. Finally, the explore the global isometric embedding of surfaces in R3, focusing on metrics on S2 with positive Gauss curvature and complete metrics in R2 with negative Gauss curvature. Annotation ©2007 Book News, Inc., Portland, OR (booknews.com)