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Results tagged with alexandrov-geometry
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user 11142
Alexandrov geometry studies non smooth analogues of Riemannian manifolds with curvature bounded from below or above. It includes spaces with curvature bounded below (briefly $\mathrm{CBB}[\kappa]$) and spaces with curvature bounded above (briefly $\mathrm{CAT}[\kappa]$).
13
votes
Accepted
Is $\partial X$ a sphere for $X$ a complete CAT$(0)$ space?
For a piecewise Euclidean or piecewise hyperbolic metric on a PL manifold, the answer is YES. This is proved (p348 in published version) by M. Davis and T. Januszkiewicz in
M. Davis, T.Januszkiewicz, …
- 63.9k
9
votes
Geodesics on convex hypersufaces
The answer to the question as stated is NO. Indeed, consider (the boundary of) a tetrahedron (or any convex polyhedron). Since the cone angles at the vertices are smaller than $2\pi,$ no geodesic can …
- 96.4k
0
votes
Embedding expanders in CAT(0) spaces
See the recent preprint http://arxiv.org/abs/1306.5434 (Mendel and Naor).
- 96.4k
2
votes
Source for: Geodesics in CAT(0) spaces
The canonical reference is Bridson and Haefliger, Metric Spaces of non-positive curvature
- 96.4k
6
votes
Connecting Lemma in the Alexandrov's existence theorem.
I am not aware of an elementary argument along these lines. A high-powered argument to show a priori connectedness uses Teichmuller theory (the space of polyhedral metrics fibers over teichmuller/modu …
- 96.4k
7
votes
Examples of CAT(0)-groups
The answer is yes, since $M$ is a $CAT(0)$ space, and the group is quasi-isometric to it. See (for example) Jim Cannon's article in Bedford Keane Series:
The theory of negatively curved spaces and gr …
- 96.4k