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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!
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           [Image from "Shortest path problem in rectangular complexes of global nonpositive curvature" (Elsevier link)]

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Have you had a look at Bridson-Haefliger? math.psu.edu/petrunin/papers/scans/books/bridson.haefliger.pdf –  Ian Agol Jul 14 '12 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 '12 at 1:48
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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 '12 at 5:09
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See also Ballmann's "Lectures on spaces of nonpositive curvature", math.psu.edu/petrunin/papers/akp-papers/ballmann-lect.pdf The advantage over Bridson and Haefliger is that Ballmann's book is shorter and has all the staff that you need. –  Misha Jul 14 '12 at 13:55

2 Answers 2

up vote 7 down vote accepted

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:

  1. L. Billera, S. Holmes, K. Vogtmann, Geometry of the space of phylogenetic trees, Advances in Applied Math, 27 (2001), no. 4, 733-467.
  2. 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.
  3. 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.

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Great references, Patricia, directly hitting my interests---Thanks! –  Joseph O'Rourke Jul 14 '12 at 1:56

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

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Thanks, Igor, this was also suggested by Ian Agol. –  Joseph O'Rourke Jul 14 '12 at 11:50
    
Ah, I didn't read the comments :( –  Igor Rivin Jul 14 '12 at 13:11

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