This volume discusses CAT(0) spaces metric spaces with nonpositive curvature in the sense of Alexandrov which may be viewed as a non-linear generalization of Hilbert spaces. The book showcases the beauty and power of Alexandrov geometry by reaching interesting applications and theorems with a minimum of preparation.

This thoroughly revised and updated edition expands the 2019 SpringerBriefs in Mathematics volume. Drawing on extensive teaching experience, the authors have added two major topics, introduced numerous new exercises, corrected errors, simplified several proofs, and reorganized the material to better support teaching needs. The presentation is now more accessible and elementary.

Primarily intended for graduate students and motivated undergraduates, the book includes numerous exercises ranging from routine to advanced, together with semisolutions and hints. It is well suited for self-study by advanced undergraduates, graduate students, and researchers.

From the reviews of the first edition:
"In the preface of this book, the authors state a Manifesto of Alexandrov geometry, under the slogan back to Euclid: Alexandrov geometry may be viewed as a direct generalization of the axiomatic system of Euclid, with some of the equalities changed to inequalities. Indeed, metric geometry permits to formulate curvature bounds merely in terms of distance relations, e.g. via triangle comparison. This monograph is a brief and well crafted introduction into this highly active field. [...] Throughout the text there are numerous exercises of varying difficulty, as well as hints for the solutions in a separate section called Semisolutions. This book is a pleasure to read and is highly recommended as a concise and stimulating introduction to metric geometry."
Monatshefte für Mathematik 2021

Provides accessible lecture notes on CAT(0) spaces, suitable for self-study Leads quickly from basics to major results, and contains many exercises Emphasizes concrete geometric applications

Autorentext

Stephanie B. Alexander (1941–2023) was a professor at the University of Illinois at Urbana-Champaign and a Fellow of the American Mathematical Society, recognized for her contributions to differential and metric geometry, in particular to Alexandrov geometry.

Vitali Kapovitch is a professor of mathematics at the University of Toronto whose research centers on global Riemannian geometry, comparison geometry, Alexandrov spaces and spaces with lower Ricci curvature bounds.

Anton Petrunin is a professor of mathematics at Penn State University working in metric, Riemannian, and Alexandrov geometry.

Klappentext

This volume discusses CAT(0) spaces – metric spaces with nonpositive curvature in the sense of Alexandrov – which may be viewed as a non-linear generalization of Hilbert spaces. The book showcases the beauty and power of Alexandrov geometry by reaching interesting applications and theorems with a minimum of preparation. 

This thoroughly revised and updated edition expands the 2019 SpringerBriefs in Mathematics volume. Drawing on extensive teaching experience, the authors have added two major topics, introduced numerous new exercises, corrected errors, simplified several proofs, and reorganized the material to better support teaching needs. The presentation is now more accessible and elementary.

 Primarily intended for graduate students and motivated undergraduates, the book includes numerous exercises ranging from routine to advanced, together with “semisolutions” and hints. It is well suited for self-study by advanced undergraduates, graduate students, and researchers.

 From the reviews of the first edition:
"In the preface of this book, the authors state a “Manifesto of Alexandrov geometry”, under the slogan “back to Euclid”: Alexandrov geometry may be viewed as a direct generalization of the axiomatic system of Euclid, with some of the equalities changed to inequalities. Indeed, metric geometry permits to formulate curvature bounds merely in terms of distance relations, e.g. via triangle comparison. This monograph is a brief and well crafted introduction into this highly active field. [...]  Throughout the text there are numerous exercises of varying difficulty, as well as hints for the solutions in a separate section called “Semisolutions”. This book is a pleasure to read and is highly recommended as a concise and stimulating introduction to metric geometry."
Monatshefte für Mathematik 2021

Inhalt

Preface.- 1 Preliminaries.-A. Metric spaces 5; B. Geodesics, triangles, and hinges 6; C. Length.-spaces 7; D. Constructions 10; E. Model angles and triangles 11;.-F. Angles and the first variation 13; G. Space of directions and tan-gent space 15; H. Hausdorff convergence 16; I. GromovHausdorffconvergence 17; J. Remarks 19.2 GluingA. The 4-point condition 21; B. Geodesics 22; C. Thin triangles 23; D.Inheritance lemma 25; E. Reshetnyak's gluing 27; F. Comments 28.3 BilliardsA. Puff pastry 29; B. Wide corners 33; C. Billiards 34; D. Comments 37.4 MajorizationA. Formulation 39; B. Triangles 40; C.Polygons 44; D. General case 45; E. Comments 46.5 GlobalizationA. Locally CAT spaces 47; B. Space of local geodesic paths 47; C. Globalization 50; D. Remarks 53.6 Polyhedral spacesA. Products, cones, and suspension 55; B. Polyhedral spaces 57; C. CAT criterion 58; D. Flag complexes 59; E. Cubical complexes 61; F. Exotic aspherical manifolds 63; G. Remarks 65.7 SubsetsA. Motivating examples 69; B. Two-convexity 71; C. Sets with smooth boundary 74; D. Open plane sets 76; E. Shefel's theorem 78;F. Polyhedral case 80; G. Two-convex hulls 81; H. Proof of Shefel's theorem 83; I. Remarks 84.
8 BarycentersA. Definition 87; B. Barycentric simplex 88; C. Convexity of up-set 89; D. Nondegenerate simplex 91; E. bi-Hölder embedding 91; F. Topological dimension 92; G. Dimension theorem 93; H. Hausdorff dimension 96; I. Remarks 96.Semisolutions.- Index.