An invitation to Alexandrov geometry: CAT(0) spaces

by Stephanie Alexander, Vitali Kapovitch, and Anton Petrunin

Links: Published version - Erratum - Slightly updated version - Old CC BY-SA version - Source files at GitHub - Draft of its second edition

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 third author.

Here is the list of topics in the second edition.

• Reshetnyak gluing theorem and its 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.