Metric Spaces of Non-Positive Curvature

1. Front Matter

   Pages I-XXI

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2. Geodesic Metric Spaces

   1. Front Matter

      Pages 1-1

      PDF

   2. Basic Concepts

      • Martin R. Bridson, André Haefliger

      Pages 2-14

   3. The Model Spaces M ⁿ _κ

      • Martin R. Bridson, André Haefliger

      Pages 15-31

   4. Length Spaces

      • Martin R. Bridson, André Haefliger

      Pages 32-46

   5. Normed Spaces

      • Martin R. Bridson, André Haefliger

      Pages 47-55

   6. Some Basic Constructions

      • Martin R. Bridson, André Haefliger

      Pages 56-80

   7. More on the Geometry of M ⁿ _κ

      • Martin R. Bridson, André Haefliger

      Pages 81-96

   8. M _κ —Polyhedral Complexes

      • Martin R. Bridson, André Haefliger

      Pages 97-130

   9. Group Actions and Quasi-Isometries

      • Martin R. Bridson, André Haefliger

      Pages 131-156

3. CAT(κ) Spaces

   1. Front Matter

      Pages 157-157

      PDF

   2. Definitions and Characterizations of CAT(κ) Spaces

      • Martin R. Bridson, André Haefliger

      Pages 158-174

   3. Convexity and its Consequences

      • Martin R. Bridson, André Haefliger

      Pages 175-183

   4. Angles, Limits, Cones and Joins

      • Martin R. Bridson, André Haefliger

      Pages 184-192

   5. The Cartan-Hadamard Theorem

      • Martin R. Bridson, André Haefliger

      Pages 193-204

   6. M _к -Polyhedral Complexes of Bounded Curvature

      • Martin R. Bridson, André Haefliger

      Pages 205-227

   7. Isometries of CAT(0) Spaces

      • Martin R. Bridson, André Haefliger

      Pages 228-243

   8. The Flat Torus Theorem

      • Martin R. Bridson, André Haefliger

      Pages 244-259

   9. The Boundary at Infinity of a CAT(0) Space

      • Martin R. Bridson, André Haefliger

      Pages 260-276

   10. The Tits Metric and Visibility Spaces

       • Martin R. Bridson, André Haefliger

       Pages 277-298

The purpose of this book is to describe the global properties of complete simply­ connected spaces that are non-positively curved in the sense of A. D. Alexandrov and to examine the structure of groups that act properly on such spaces by isometries. Thus the central objects of study are metric spaces in which every pair of points can be joined by an arc isometric to a compact interval of the real line and in which every triangle satisfies the CAT(O) inequality. This inequality encapsulates the concept of non-positive curvature in Riemannian geometry and allows one to reflect the same concept faithfully in a much wider setting - that of geodesic metric spaces. Because the CAT(O) condition captures the essence of non-positive curvature so well, spaces that satisfy this condition display many of the elegant features inherent in the geometry of non-positively curved manifolds. There is therefore a great deal to be said about the global structure of CAT(O) spaces, and also about the structure of groups that act on them by isometries - such is the theme of this book. 1 The origins of our study lie in the fundamental work of A. D. Alexandrov .

• Connected space
• Group theory
• Non-positive curvature
• complexes of groups
• geodesics
• groups of isometries
• hyperbolic

"This book is beautifully and clearly written and contains many illustrations and examples but also many deep results.
It is my opinion that this book will become a standard work in mathematical literature and will be used by many people, from undergraduates to specialists."
K. Dekimpe in "Nieuw Archief voor Wiskunde", June 2001

"In conclusion, it can be said that the book is an indispensable reference and a very useful tool for graduate students who want to learn this theory as well as for researchers working in the subject. The exposition is clear, the proofs are complete, and some of the advanced results that are discussed are original. Every section of the book contains interesting historical remarks and comments."

A. Papadopoulos in "Zentralblatt für Mathematik und ihre Grenzgebiete", 2002

 

• Mathematical Institute, University of Oxfod, Oxford, Great Britain

  Martin R. Bridson

• Section de Mathématiques, Université de Genève, Genève 24, Switzerland

  André Haefliger

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