Metric Spaces of Non-Positive Curvature

Martin R. Bridson, André Haefliger (auth.)9783540643241, 3540643249

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 .

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Table of contents :
Front Matter….Pages I-XXI
Front Matter….Pages 1-1
Basic Concepts….Pages 2-14
Length Spaces….Pages 15-31
Normed Spaces….Pages 32-46
Some Basic Constructions….Pages 47-55
M κ —Polyhedral Complexes….Pages 56-80
Group Actions and Quasi-Isometries….Pages 81-96
Front Matter….Pages 97-130
Definitions and Characterizations of CAT( κ ) Spaces….Pages 131-156
Convexity and its Consequences….Pages 157-157
Angles, Limits, Cones and Joins….Pages 158-174
The Cartan-Hadamard Theorem….Pages 175-183
M к -Polyhedral Complexes of Bounded Curvature….Pages 184-192
Isometries of CAT(0) Spaces….Pages 193-204
The Flat Torus Theorem….Pages 205-227
The Boundary at Infinity of a CAT(0) Space….Pages 228-243
The Tits Metric and Visibility Spaces….Pages 244-259
Symmetric Spaces….Pages 260-276
Gluing Constructions….Pages 277-298
Simple Complexes of Groups….Pages 299-346
Front Matter….Pages 347-366
δ -Hyperbolic Spaces and Area….Pages 367-396
Non-Positive Curvature and Group Theory….Pages 397-397
Complexes of Groups….Pages 398-437
Groupoids of Local Isometries….Pages 438-518
Back Matter….Pages 519-583
….Pages 584-619