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Results tagged with alexandrov-geometry
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user 68708
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 $\mathrm{CAT}[\kappa]$).
1
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Why 2-tori with Gauss curvature $\geq -1$ cannot collapse to segment?
This is not quite an answer, bur certainly too long to be a comment! Here are two sketchy ideas.
By following the arguments of Shioya and Yamaguchi, you could make rigorous this argument that the onl …
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3
votes
Accepted
Gluing Alexandrov spaces along parts of boundary
In your specific question, the answer is no. Take two triangles, as two 2-dimensional Alexandrov surfaces. Suppose they each have a side of a given length. Then these sides can be your $E_1$ and $E_2$ …
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1
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Does Alexandrov space satisfy a reverse doubling condition?
The answer is no. In fact, it is not even true for Riemannian manifolds.
Consider a flat Riemannian manifold $(M,g)$ of dimension $m$ with diameter $d$ realized by the points $x$ and $y$. Let the vol …
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2
votes
Accepted
A property of concave functions on Alexandrov spaces
The base of Perelman's induction for Main Theorem 1.4 (is this what you're talking about?) is in fact a collection of trivial statements about admissible maps $X^n \to \mathbb{R}^{n+1}$.
To move to t …
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5
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Accepted
Is the boundary of Alexandrov space again an Alexandrov space?
This is an open problem.
It is a special case of the following question:
Is it true that every extremal subset is again an Alexandrov space?
The answer to this question is "No". Petrunin has …
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3
votes
Gradient of distance function at cut points on Alexandrov spaces
The distance function is semi-concave on all of $M \setminus K$. The additional restriction is unnecessary.
The easiest example I can think of for a gradient strictly between zero and one is the cone …
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4
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Geodesics on convex hypersufaces
As for question 2, the tangent cone of $M$ can be defined as the cone on the space of directions, and the space of directions is the completion of the space of geodesic directions. Therefore every geo …
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3
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Why finite dimensional MCS-space implies $\mathbb{R}^m\times cone$ locally?
I'm not sure whether it's generally the case that when you split off an $\mathbb{R}^m$ factor from a cone on an MCS space, the other factor is also a cone on an MCS space. Except, of course, if $m=0$. …
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5
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Accepted
Codimension of the set of topologically singular points of an Alexandrov space.
Your intuition is correct. Since the Alexandrov space is locally homeomorphic to the cone on the space of directions, your induction hypothesis implies that the statement is true locally, and hence gl …
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