Questions tagged [alexandrov-geometry]
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 $\mathrm{CAT}[\kappa]$).
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questions
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Isometric imbedding of a 2-disk into Euclidean 3-space
Let us call a cap the intersection of the boundary of 3-dimensional convex compact set $K$ in $\mathbb{R}^3$ with a half-space bounded by a plane $H$ such that the orthogonal projection to $H$ of this ...
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1answer
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Is a cap an Alexandrov space?
Let us call a cap the intersection of the boundary of 3-dimensional convex compact set $K$ in $\mathbb{R}^3$ with a half-space bounded by a plane $H$ such that the orthogonal projection to $H$ of this ...
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Reference to equivariant Gromov-Hausdorff convergence
I am looking for a reference to the following notions and facts below which, I think, I can prove, but which might be known to experts.
Let us fix a finite group $G$. Consider the class of all compact ...
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Regularity of the distance function to a compact set in an Alexandrov space
Let $X$ be a finitely dimensional Alexandrov space with curvature bounded below. For any compact set $K \subset X$, can we find $0<\epsilon_1<\epsilon$ such that the distance function $f=|\cdot ...
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Essential singularities in an Alexandrov space
For a finitely-dimensional Alexandrov space $X$ with curvature bounded below, a point $p \in X$ is called an essential singular point if $\Sigma_p$ satisfies $\min_{\xi \in\Sigma_p } \max_{\eta \in \...
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Essential singular set of an Alexandrov space
Let $X$ be a locally compact Alexandrov space with curvature bounded below. Suppose $C$ is a closed subset that consists of the essential singular points, where a point $p$ is called an essential ...
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Topology of compact Alexandrov spaces with nonnegative curvature
Let $(X,d)$ be a compact Alexandrov space with nonnegative curvature and $\partial X \ne \emptyset$. If we set $C=\{x\in X \mid d(x, \partial X)=\max_{y \in X} d(y,\partial X)\}$, can we show $B(C,\...
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1answer
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Does codimension-1 collapsing with bounded curvature have boundary?
Let $(M^n,g_i)$ be a sequence of smooth complete Riemannian manifold with $|sec_{g_i}| \le 1$. Suppose $(M_i^n,g_i)$ converges to a limit space $(X^{n-1},d)$ in the Gromov-Hausdorff sense, where the ...
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Rigidity for convex surfaces in elliptic/hyperbolic space
From Alexandrov's work we know that any metric on the sphere with lower curvature bound $\kappa$ (in the sense of Alexandrov) can be realized as a closed convex surface (i.e. boundary of a compact ...
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2answers
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Completion of an Alexandrov space
Let $X$ be an incomplete Alexandrov space with sec $\ge -1$ in the sense that for any point in $X$ there exists a small neighborhood in which the four-points criterion is satisfied.
Suppose $X$ is ...
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Topology of $3$-dimensional noncompact Alexandrov space with sec $\ge 0$
For any noncompact $3$-dimensional smooth manifold $M$ with sec $\ge 0$, it follows from soul's theorem that $M$ is diffeomorphic to $\mathbb R^3, S^2 \times \mathbb R$ or their quotients. Can we have ...
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Semiconcavity estimate for the squared distance on a compact Riemannian manifold
I am currently reading this paper on the Riemannian structure of the Wasserstein space over a compact Riemannian manifold (my question doesn't concern the Wasserstein metric), specifically Section 4.1,...
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0answers
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Nearby convex set in a nearby space
Let $K$ be a convex set in a CAT(0) space $X$. Suppose $X'$ is a CAT(0) space that is very close to $X$.
Is there a convex set $K'\subset X'$ that is close to $K\subset X$?
Two spaces $X$ and $X'$ ...
4
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1answer
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Alexandrov spaces of zero curvature
Let $X$ be an $n$-dimensional Alexandrov space with curvature satisfying both $\ge 0$ and $\le 0$. Can we prove that any tangent cone of $X$ must be isometric to $\mathbb R^{k} \times C(S^{n-k-1}/\...
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Convex separation in CAT(0) spaces
Are there any convex separation theorems for CAT(0) spaces? Particularly, can one separate two disjoint convex sets by a horosphere?
In Riemannian manifolds, I have seen some results on convex ...
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1answer
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Translation lengths in CAT(0) spaces
Let $a,b$ be two loxodromic isometries of a CAT(0) space. Assume that, for every $n \geq 1$, $a^nb$ is also loxodromic. Is it possible for the translation length of $a^nb$ to be bounded independently ...
3
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1answer
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Convex spherical neighborhood in Alexandrov spaces
Let $X$ be an Alexandrov space with curvature $\ge k$. For any point $p \in X$, can we find a constant $r>0$ such that $B(p,r)$ is geodesically convex?
Here, geodesically convex means for every two ...
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2answers
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Toponogov's theorem in the Alexandrov space with respect to a compact set
Let $X$ be an Alexandrov space with curvature $\ge 0$ and $A \subset X$ a compact set. Suppose $p,q\in X$ satisfying $|Ap|=|Aq|=r$ and $p_1,q_1$ are points on the geodesics $Ap,Aq$ respectively such ...
2
votes
1answer
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Uniqueness of geodesics in the Alexandrov space
Let $X$ be an Alexandrov space with curvature bounded below. For any point $p \in X$, we define $C_p$ to be the set that consists of all points $q$ such that there are at least two minimizing ...
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Ball Covering Property in Non-negatively Curved Spaces [Reference Request]
$\DeclareMathOperator{\vol}{Vol}$
Suppose we are working inside a Riemannian $n$-manifold $M$ of non-negative Ricci curvature. In his PhD thesis (see pp.8–9), Zhong-dong Liu presents an incredibly ...
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About the paper "Elements of Morse theory on Alexandrov spaces"
I am learning some structure theorems in Alexandrov geometry and the paper "Elements of Morse theory on Alexandrov spaces" by G. Perelman is frequently quoted. However, I am unable to find ...
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Convexity of the scalar product in the Alexandrov space
Given a point $p$ in an Alexandrov space $A$ with curvature bounded below, we denote by $T_p$ the tangent cone at $p$. For two tangent vectors $u$ and $v$ in $T_p$, we define a scalar product as ...
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Concavity of the distance function to the boundary of Alexandrov space
I was told that the following fact is true.
Let $X$ be a finite dimensional Alexandrov space with non-negative curvature.
Then the function
$$x\mapsto dist(x, \partial X)$$
is concave (namely its ...
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0answers
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When Riemannian manifold with boundary is Alexandrov space?
I am looking for a proof or, better, a reference to a proof of the following known fact.
Let $(M,g)$ be a smooth Riemannian manifold with boundary. Assume the sectional curvature of $M$ is at least $\...
7
votes
1answer
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Geodesic line with endpoints in interior of Riemannian manifold or Alexandrov space
Let $X$ be a finite dimensional Alexandrov space with curvature bounded below and non-empty boundary. Let $\gamma$ be a shortest geodesic path in $X$ whose endpoints belong to the interior of $X$.
...
2
votes
1answer
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Tangent cone of metric graph
I am starting to study some lecture notes about metric geometry and I would appreciate it if someone could some questions regarding the notion of the tangent cone.
Consider 3 half lines joined by ...
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0answers
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CBB inequality and two comparison triangles / 4-point condition for CBB spaces
Assume $(X,d)$ is a CBB($\kappa$) space with $\kappa > 0$. (That is we can find comparison triangles in the model space $(M_\kappa^2, \bar{d})$ and the reverse of the CAT inequality holds; ...
2
votes
1answer
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gradient curve $\gamma$ defined on $(-T,0]$, can't be extended from $\gamma(-T)$?
Let $f$ be a semi-concave function on an Alexandrov space $X$.
Denote $\gamma_p(t)$ the $f$-gradient curve with $\gamma_p(0)=p$, i.e.
$$
\gamma^+_p(t)=\nabla_{\gamma_p(t)}f.
$$
If $X$ is a Riemannian ...
2
votes
1answer
186 views
Unbounded curvature implies infinite diameter on complete metric spaces
I recently asked this question Unbounded sectional curvature implies infinite diameter?.
I would like now to ask something similar, but in another context.
Suppose you have a complete metric space $(...
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votes
0answers
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Estimate of volume of a ball on the boundary of Riemannian manifold
Let $M^n$ be a smooth compact Riemannian manifold with geodesically locally convex boundary and sectional curvature at least $-1$. Let $x\in M$ and $\varepsilon\in (0,1)$.
Does there exist a ...
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0answers
65 views
A Lipschitz limit of Riemannian metrics with curvature in $[-1,1]$
Let $(M,g)$ be a compact manifold with a metric $g$ (not necessarily Riemannian one). Suppose that $(M,g)$ is a Lipschitz limit of a sequence of Riemannian manifolds $(M_n,g_n)$ with the sectional ...
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0answers
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When is the quotient of a geodesic space again a geodesic space?
I asked this on the math.stackexchange forum about a week ago but did not get any answers so I figured I might try it here as well. This is a straight up copy paste from my question here.
I am ...
6
votes
1answer
120 views
Continuity of volume of boundary of Riemannian manifolds in the Gromov-Hausdorff sense
Let $\{X_i^n\}$ be a sequence of smooth compact Riemannian $n$-dimensional manifolds with boundary. Assume that this sequence has uniformy bounded below sectional curvature, and each $X_i$ is ...
2
votes
1answer
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Measure of the boundary of Alexandrov space
Let $X$ be a compact $n$-dimensional Alexandrov space with curvature bounded below. Let $\partial X$ denote its boundary in the sense of the theory of Alexandrov spaces.
Is it true that if $\...
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votes
0answers
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Dimension of Alexandrov space which is homeomorphic to a manifold
Let $M^n$ be a smooth manifold of dimension $n$. Let $M$ given a metric with curvature bounded below in the sense of Alexandrov which induces the original topology of $M$.
It is true that the ...
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1answer
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Neighborhood of $(n,\delta)$-strained point in Alexandrov space homeomorphic to $\mathbb{R}^n$, how big is $\delta$?
Let $M$ be an $n$-dimensional Alexandrov space with curvature $\geqslant k$. A point $p\in M$ is said to be an $(n,\delta)$ strained point if there are $n$ pairs of points $a_i, b_i$ such that
$$
\...
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Pointed version of Perelman stability theorem
I am wondering if there is a version of the Perelman stability theorem which says approximately the following:
Let $\{(X_i,p_i)\}$ be a sequence of pointed $n$-dimensional complete Alexandrov spaces ...
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votes
1answer
525 views
Why 2-tori with Gauss curvature $\geq -1$ cannot collapse to segment?
Let $\{(\mathbb{T}^2,g_i)\}_{i=1}^\infty$ be a sequence of 2-dimensional tori with smooth Riemannian metrics with Gauss curvature at least $-1$. It was explained in the final answer to the post Gromov-...
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Approximation of 2-dimensional Alexandrov spaces
Consider a closed 2-dimensional surface (not necessarily orientable) with a metric with curvature at least -1 in the sense of Alexandrov. It it true that on this surface there is a sequence of ...
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1answer
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A.D. Alexandrov imbedding theorem for metrics with symmetry
A well known theorem due to A.D. Alexandrov says that any metric on the 2-sphere $S^2$ with curvature at least -1 (in the sense of Alexandrov) can be isometrically realized either as convex surface in ...
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1answer
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Inequalities on 4 points in metric spaces with curvature bounded below (CBB)
Let $\kappa > 0$. I have a space $(X,d)$ and take 4 points $w,x,y,z \in X$. I then choose comparison points in the model space $(M_\kappa^2,\bar{d})$ as follows:
Take the comparison triangle $\...
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1answer
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Reference request: metric spaces with curvature bounded from below (CBB) spaces
What is the/a main reference book for spaces with curvature bounded from below (CBB spaces/spaces with curvature $\geq \kappa$ in the sense of Alexandrov)? Looking for an up to date reference.
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1answer
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In a manifold, $\angle xpy>\frac{\pi}{2}$, for $q$ on $px$ or $py$, $B_q(r)$ homeomorphic to $B_p(r)$?
Let M be an n-dimensional Riemannian manifold without boundary, with sectional curvature $\geqslant -1$. For a point $p\in M$, suppose there exist $l, \delta>0$, $x,y \in M$ with $d(p,x),d(p,y)>...
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0answers
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Non-collapsed Alexandrov spaces, level surface of regular map homeo to its lifting?
Let $X_i$ be n dimensional, no boundary Alexandrov spaces with curvature $\geqslant -1$ and diameter $\leqslant D$. Suppose that $X_i$ converge to an n dimensional Alexandrov space $X$. Then by ...
2
votes
1answer
157 views
Is there Brownian motion on Alexandrov spaces?
It is well known that there is a notion of Brownian motion on smooth Riemannian manifolds.
I am wondering if there is a more general notion of Brownian motion on finite dimensional Alexandrov ...
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votes
2answers
517 views
Gluing hexagons to get a locally CAT(0) space
I believe that there are four ways to glue (all) the edges of a regular Euclidean hexagon to get a locally CAT(0) space:
The first two give the torus and the Klein bottle, respectively. What are the ...
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votes
1answer
234 views
Does geometrization of Alexandrov 3-spaces follow from that of 3-orbifolds?
Galaz-Garcia and Guijarro proved the geometrization of closed (compact, boundaryless) Alexandrov 3-spaces. Part of the strategy was to use the so-called ramified double cover $\tilde{X}$ of the space $...
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votes
1answer
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Contractibility of balls in Alexandrov spaces
Let $X$ be a compact finite dimensional Alexandrov space with curvature bounded below.
Does there exist $\varepsilon_0>0$ (depending on $X$) such that for any $\varepsilon \in (0,\varepsilon_0)$...
6
votes
1answer
235 views
Discrete approximations of Riemannian manifolds
MSE crosspost
It's known (due to Perelman) that in class of Alexandrov spaces of fixed dimension and bounded from below curvature Gromov-Hausdorff distance separates homeomorphism types — every $\...
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votes
1answer
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Hausdorff convergence of submanifolds in $\mathbb{S}^m$
Let $\{X_i^n\}_{i\in \mathbb{N}}$ and $\{Y_i^n\}_{i\in \mathbb{N}}$ be sequences of connected closed submanifolds of $\mathbb{S}^{n+2}$, with $n> 5$. Suppose that $\{X_i^n\}_{i\in \mathbb{N}}$ (...
