This volume is an English translation of Sakai's textbook on Riemannian geometry which was originally written in Japanese and published in 1992. The author's intent behind the original book was to provide to advanced undergraduate and graduate students an introduction to modern Riemannian geometry that could also serve as a reference. The book begins with an explanation of the fundamental notion of Riemannian geometry. Special emphasis is placed on understandability and readability, to guide students who are new to this area. The remaining chapters deal with various topics in Riemannian geometry, with the main focus on comparison methods and their applications. The author has faithfully translated the Japanese edition with the exception of appendix 6on the collapsing of Riemannian manifolds and Gromov's convergence theoremwhich has been considerably revised and expanded, including the addition of a few comments on further developments and corrections of small errors. Readership Advanced undergraduate and graduate students interested in differential geometry. Reviews "A good source for teaching a somewhat advanced class in differential geometry and certainly contains enough material for a oneyear course. [It is] also a good source for the working differential geometer ... a fine book and worthwhile addition to any differential geometer's library."  Bulletin of the AMS "This book on differential geometry packs into about 350 pages a great variety of topicsfrom the basics to spectral geometry and the topology of Riemannian manifolds ... A good text for a graduate course in which students are wellprepared and motivated ... should also be a very good reference for a practicing mathematician interested in Riemannian geometry ... touches on a great many subjects in addition to those it covers in detail."  Mathematical Reviews "The book is wellwritten and will enable the reader to enter areas which are still in rapid progress. The author succeeds very well in explaining the fundamental concepts of Riemannian geometry and why the topics he deals with deserve the attention of the reader ... The book is a valuable addition to the literature and will be a good reference."  Zentralblatt MATH Table of Contents  Preliminaries from manifolds
 Fundamental concepts in Riemannian geometry
 Global concepts in Riemannian geometry
 Comparison theorems and applications
 Curvature and topology of Riemannian manifolds
 Isoperimetric inequality and spectral geometry
 Appendices
 Hints and solutions to problems and exercises
 Bibliography
 Index
