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Graduate Studies in Mathematics
2001; 415 pp; hardcover
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 Carnot-Carathéodory metrics, the hyperbolic plane, distance-volume inequalities, asymptotic geometry (large scale, coarse), Gromov hyperbolic spaces, convergence of metric spaces, and Alexandrov spaces (non-positively and non-negatively curved spaces). The authors tend to work with "easy-to-touch" mathematical objects using "easy-to-visualize" methods.
The authors set a challenging goal of making the core parts of the book accessible to first-year 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.
Advanced undergraduates, graduate students, and research mathematicians interested in geometry and specialists in related fields.
"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 Mathematiker-Vereinigung
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