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Results tagged with alexandrov-geometry
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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 $\mathrm{CAT}[\kappa]$).
6
votes
Accepted
Alexandrov spaces of zero curvature
No, that's too much to ask for. If you take any convex cone in $\mathbb R^n$ (any such cone is both $CAT(0)$ and Alexandrov of $curv\ge 0$) then the tangent space at the origin is just the cone itself …
- 7,842
1
vote
gradient curve $\gamma$ defined on $(-T,0]$, can't be extended from $\gamma(-T)$?
The answer depends on your definition of semi-concave functions. If you only require them to be semi-concave on geodesics then an obvious example is given by $X$ equal to the closed unit ball in $\mat …
- 7,842
3
votes
Isometric classification of 1-dimensional Alexandrov spaces
I don't think anyone bothered to write a formal proof down but this is indeed quite easy. Tangent spaces (which are always metric cones) can only be lines or half lines. From this it's immediate that …
- 7,842
12
votes
Accepted
Smoothability of compact Alexandrov surfaces with curvature bounded from below
Edit: Addressing Igor's comment I'd like to correct the references I gave. The correct reference for the exact argument I sketch should be the original book by Alexandrov "Intrinsic Geometry of Convex …
- 7,842