Intended for a one year course, this text serves as a single source, introducing readers 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. This is one of the few Works to combine both the geometric parts of  Riemannian geometry and the analytic aspects of the theory. The book will appeal to a readership that have a basic knowledge of standard manifold theory, including tensors, forms, and Lie groups.Important revisions to the third edition include:a substantial addition of unique and enriching exercises scattered throughout the text;inclusion of an increased number of coordinate calculations of connection and curvature;addition of general formulas for curvature on Lie Groups and submersions;integration of variational calculus into the text allowing for an early treatment of the Sphere theorem using a proof by Berger;incorporation of several recent results about manifolds with positive curvature;presentation of a new simplifying approach to the Bochner technique for tensors with application to bound topological quantities with general lower curvature bounds.From reviews of the first edition:"The book can be highly recommended to all mathematicians who want to get a more profound idea about the most interesting achievements in Riemannian geometry. It is one of the few comprehensive sources of this type."?Bernd Wegner, ZbMATH

"This is a very advanced textbook on metric and algebraic proofs of critical theorems in the field of metric spaces involving manifolds and other 3D structures. ? First, definitions, theorems, proofs, and exercises abound throughout every section of this 500 page mathematics book. The history of development in the area is comprehensive. ? The experts will find this a useful research tool. ? I recommend this book for researchers having a strong background to begin with." (Joseph J. Grenier, Amazon.com, June, 2016)