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.
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.
Misprints in the published version
- The numbers in some citations are shifted by 1; at least one citation is missing.
- page 34--35, the patchwork along a curve (3.2.2) has to be formulated for local geodesics only; minor changes in the proof are required.
- page 44, line \(-4\). The following
"By the globalization theorem (3.3.1), any proper length CAT(0) space is contractible.
Therefore all proper..."
has to be changed to
"By 2.2.6, any proper length CAT(0) space is contractible. Therefore, by the globalization theorem (3.3.1), all proper..."
- page 53, line 15: \(\hat{\varphi^{-1}}(W)\) \(\to\) \(\hat\varphi^{-1}(W)\)
- page 63, lines 8--9 (Exercise 4.8.1): closed set bounded by a Lipschitz hypersurface \(\to\) closed subgraph of a Lipschitz function \(f\colon \mathbb{R}^{m-1}\to\mathbb{R}\).