An invitation to Alexandrov geometry: CAT(0) spaces

by Stephanie Alexander, Vitali Kapovitch, and Anton Petrunin

Links: Second edition (2026) - First edition (2019) - Erratum to first edition - arXiv - Source files

The idea is to demonstrate the beauty and power of Alexandrov geometry by reaching interesting applications and theorems with a minimum of preparation. These notes arise as an offshoot of the book on Alexandrov geometry we have been writing for a number of years. The notes were shaped in a number of lectures given by the authors.

Here is the list of topics in the second edition.

• Reshetnyak gluing theorem.
• An application to a problem in billiards which was solved by Dmitri Burago, Serge Ferleger and Alexey Kononenko.
• Reshetnyak majorization theorem.
• Hadamard–Cartan globalization.
• Polyhedral spaces and the construction of exotic aspherical manifolds introduced by Michael Davis.
• Two-convexity and saddle surfaces; this part is based on work of Samuel Shefel.
• Barycenters and dimension; this part is based on work of Bruce Kleiner.