Welcome to MathOverflow

Question

6

I am seeking a good introductory reference that could lead to an understanding of the properties of geodesics in complete CAT(0) metric spaces. I am especially interested in learning the differences between geodesics in these spaces and those in an $n$-dimensional Euclidean space with its usual Euclidean metric, which is of course CAT(0). I am ultimately interested in simplicial, cubical, and polyhedral complexes, but I am willing to start anywhere. Thanks for educating me!
alt text
_{[Image from "Shortest path problem in rectangular complexes of global nonpositive curvature" (Elsevier link)]}

flag	asked Jul 14 at 1:19 Joseph O'Rourke 44.2k●1●50●216

4

Have you had a look at Bridson-Haefliger? math.psu.edu/petrunin/papers/scans/books/… – Ian Agol Jul 14 at 1:26

Thanks, Ian, that book (Metric Spaces of NonPositive Curvature, 2009) seems perfect, even freely downloadable as PDF (in comparison to 122 $US to purchase new!)---Thanks! – Joseph O'Rourke Jul 14 at 1:48

2

I, too, would like to better understand CAT(0) metric spaces. In addition to the Bridson-Haefliger book, I have had recommended to me "A course in metric geometry", by Burago, Burago, and Ivanov. – Russ Woodroofe Jul 14 at 5:09

1

See also Ballmann's "Lectures on spaces of nonpositive curvature", math.psu.edu/petrunin/papers/akp-papers/… The advantage over Bridson and Haefliger is that Ballmann's book is shorter and has all the staff that you need. – Misha Jul 14 at 13:55

Patricia Hersh · Answer 1 · 2012-07-14 01:55:05Z UTC

Your pictures remind me of the space of phylogenetric trees (which is cubical and exhibits similar folding), so if you are interested in reading about very concrete such examples, you might take a look at:

L. Billera, S. Holmes, K. Vogtmann, Geometry of the space of phylogenetic trees, Advances in Applied Math, 27 (2001), no. 4, 733-467.
M. Owen and S. Provan, A fast algorithm for computing geodesic distances in tree space, IEEE/ACM Trans. Computational Biology and Bioinformatics, 8: 2-13, 2011.
F. Ardila, M. Owen and S. Sullivant, Geodesics in CAT(0) Cubical Complexes, Advances in Applied Math. 48 (2012), 142-163.

(1) proves uniqueness of geodesics in tree space, (2) gives a polynomial time algorithm to find geodesic distances, and (3) works in more generality.

Igor Rivin · Answer 2 · 2012-07-14 09:22:34Z UTC

0

The canonical reference is Bridson and Haefliger, Metric Spaces of non-positive curvature

answered Jul 14 at 9:22

Igor Rivin
31.3k●3●37●71

	Thanks, Igor, this was also suggested by Ian Agol. – Joseph O'Rourke Jul 14 at 11:50
	Ah, I didn't read the comments :( – Igor Rivin Jul 14 at 13:11

Source for: Geodesics in CAT(0) spaces

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

2 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.