An invitation to Alexandrov geometry: CAT(0) spaces


Here you can download our book

The source files can be downloaded from the project page on GitHub; the final publication is available at Springer via DOI: 10.1007/978-3-030-05312-3. An older (CC BY-SA)-version is available at arXiv:1701.03483.

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 to undergraduate students at different occasions at the MASS program at Penn State University and the Summer School “Algebra and Geometry” in Yaroslavl.

Chapter 1 includes the necessary preliminaries.

In Chapter 2, we discuss the Reshetnyak gluing theorem and apply it to a problem in billiards which was solved by Dmitri Burago, Serge Ferleger and Alexey Kononenko.

In Chapter 3, we discuss the Hadamard–Cartan globalization theorem, and apply it to the construction of exotic aspherical manifolds introduced by Michael Davis.

In Chapter 4, we discuss examples of Alexandrov spaces with curvature bounded above. This chapter is based largely on work of Samuel Shefel on nonsmooth saddle surfaces.

Misprints in the published version