Graduate Studies in Mathematics 2001; 415 pp; hardcover Volume: 33 ISBN10: 0821821296 ISBN13: 9780821821299 List Price: US$50 Member Price: US$40 Order Code: GSM/33
 "Metric geometry" is an approach to geometry based on the notion of length on a topological space. This approach experienced a very fast development in the last few decades and penetrated into many other mathematical disciplines, such as group theory, dynamical systems, and partial differential equations. The objective of this graduate textbook is twofold: to give a detailed exposition of basic notions and techniques used in the theory of length spaces, and, more generally, to offer an elementary introduction into a broad variety of geometrical topics related to the notion of distance, including Riemannian and CarnotCarathéodory metrics, the hyperbolic plane, distancevolume inequalities, asymptotic geometry (large scale, coarse), Gromov hyperbolic spaces, convergence of metric spaces, and Alexandrov spaces (nonpositively and nonnegatively curved spaces). The authors tend to work with "easytotouch" mathematical objects using "easytovisualize" methods. The authors set a challenging goal of making the core parts of the book accessible to firstyear graduate students. Most new concepts and methods are introduced and illustrated using simplest cases and avoiding technicalities. The book contains many exercises, which form a vital part of the exposition. Readership Advanced undergraduates, graduate students, and research mathematicians interested in geometry and specialists in related fields. Reviews "The book is well worth reading. Contributing to this are the many elementary examples with which the authors supplement the text ... Anyone who is intensely concerned with Riemannian geometry will not pass up this book. It is so far without competition and fills a gap in the market."  Translated from Jahresbericht der Deutschen MathematikerVereinigung Table of Contents  Metric Spaces
 Length Spaces
 Constructions
 Spaces of Bounded Curvature
 Smooth Length Structures
 Curvature of Riemannian Metrics
 Space of Metric Spaces
 Largescale Geometry
 Spaces of Curvature Bounded Above
 Spaces of Curvature Bounded Below
 Bibliography
 Index
