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Results tagged with alexandrov-geometry
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user 1573
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 $\mathrm{CAT}[\kappa]$).
14
votes
Accepted
Homeomorphism/ homotopy types of non-negatively curved manifolds
As mentioned in comments, the first dimension where an infinite family of pairwise non-homeomorphic closed nonnegatively curved manifolds occurs is $3$ (the lens spaces). The question becomes more cha …
- 29.1k
10
votes
Accepted
Gromov-Hausdorff limits of 2-dimensional Riemannian surfaces
As mentioned in the comments, if the limit is a circle, then by the Yamaguchi fibration theorem, $M_i$ fibers over the circle, and hence it is a torus (or Klein bottle in the non-orientable case).
If …
- 29.1k
10
votes
Accepted
Hausdorff vs Gromov-Hausdorff convergence of convex hypersurfaces
The reference is Lemma 10.2.7 in A course of metric geometry by Burago-Burago-Ivanov. They do it in 3d but it does not matter. The main point is that if two convex bodies are Hausdorff close, then on …
- 29.1k
8
votes
Gromov-Hausdorff limits of 2-dimensional Riemannian surfaces
I remembered another reason why closed surfaces of negative Euler characteristic cannot collapse under a lower bound on sectional curvature.
Much more is true: if a sequence of $n$-dimensional close …
- 29.1k
6
votes
Is there Domain Invariance for Alexandrov spaces?
Here I will clarify the cohomological issues in Sergei's answer above. For applications to Alexandrov spaces scroll to the end of the post.
I will use Alexander-Spanier cohomology with compact suppor …
- 29.1k
4
votes
Accepted
Convergence of extremal subsets in Alexandrov spaces
The limit of extremal subsets is an extremal subset, see Lemma 4.1.3 in Petrunin's Semiconcave functions in Alexandrov geometry. The non-collapsing assumption is not needed.
- 29.1k
3
votes
Soul theorem for non-negativly curved open Alexandrov manifolds?
This is not an answer to your question but there is a soul theorem for Alexandrov spaces due to Perelman (see Section 6 in here) with the conclusion that that the space deformation retracts to a soul. …
- 29.1k
2
votes
Accepted
Hausdorff convergence of submanifolds in $\mathbb{S}^m$
One cannot expect homology equivalence of limits. Here is a standard example for surfaces in $\mathbb R^3$ which can be easily adapted to your situation. Start from the unit sphere in $\mathbb R^3$ an …
- 29.1k
2
votes
Convex hull in CAT(0)
Here is a quote from the paper "Schauder Fixed Point Theorem in Spaces with Global Nonpositive Curvature" by Niculescu and Rovenţa, http://www.hindawi.com/journals/fpta/2009/906727.html, DOI:10.1155/2 …
- 29.1k