|To overcome the daunting aspects of complicated formulas in Riemannian geometry, Morgan opens his beginner's guide with m-dimensional submanifolds of R[superscript n] rather than abstract Riemannian manifolds. This teacher establishes the basic material early to describe the most important geometric features of curved objects from the plebeian racetrack to the grand structure of the universe. In his short text, he then moves rapidly to address more complex topics like hyperbolic geometry and the Gauss-Bonnet theorem. This second edition contains a wealth of examples and exercises and includes new material on subjects ranging from isoperimetric problems to Einstein's original paper on general relativity. It concludes with a discussion of global geometry and current research (some by undergraduates) on energy minimizing curves and more.