Source for: Geodesics in CAT(0) spaces
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.
The canonical reference is Bridson and Haefliger, Metric Spaces of non-positive curvature
A more recent reference for geodesics is the following preprint:
Koyo Hayashi. A polynomial time algorithm to compute geodesics in CAT(0) cubical complexes. Oct 2017, arXiv:1710.09932.
Even though the title says "cubical complexes", from the abstract:
Our algorithm is applicable to any CAT(0) space in which geodesics between two close points can be computed, not limited to CAT(0) cubical complexes.