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

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.