Author Description

This book is intended for a one-year course in Riemannian geometry. It will serve as a single source, introducing students to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialize in Riemannian geometry. Instead of variational techniques, the author uses a unique approach emphasizing distance functions and special coordinate systems. This approach uses elementary calculus together with techniques from differential equations, thereby providing a more direct and elementary route for students. Many of the chapters contain material typically found in specialized texts and never before published together in one source. Key sections include noteworthy coverage of geodesic geometry, Bochner technique, symmetric spaces, holonomy, comparison theory for both Ricci and sectional curvature, and convergence theory. This volume is one of the few published works to combine both the geometric parts of Riemannian geometry and the analytic aspects of the theory, as well as presenting the most up-to-date research including sections on convergence and compactness of families of manifolds. This book will appeal to readers with a knowledge of standard manifold theory, including such topics as tensors and Stokes' theorem. Scattered throughout the text is a variety of exercises that will help to motivate readers to deepen their understanding of the subject.