# 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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### Connectedness of fibers of almost Riemannian submersions

EDIT: Let $M,N$ be compact connected smooth Riemannian manifolds. Let us assume that $N$ is closed, while $M$ might have a geodesically convex boundary.
Given $f\colon M\to N$ be an $\varepsilon$-...

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### Nested convex hulls in Hadamard manifold

Let $F$ be a finite set in a Hadamard manifold $H$, and $W\supset F$ is its neighborhood.
Is it true that the closure of the convex hull of $F$ lies in the interior of the convex hull of $W$?
...

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### Amalgamated product acting on CAT(0) cube complex

I was reading the following result from the book Metric spaces of non-positive curvature by Bridson and Haefliger.
Result:
Let $F_0,F_1$ and $H$ be groups acting properly
by isometries on complete $...

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### Does the Alexandrov angle define convex functions along geodesics in CAT(0) spaces?

Let $X$ be a CAT(0) space and suppose $a,b,c,d\in X$ satisfy
$$
\max\{\angle_a(b,c),\angle_a(b,d)\}<\frac{\pi}{2}.
$$
Let $\gamma:[0,\ell]\to X$ be the geodesic with $\gamma(0)=c$ and $\gamma(\ell)...

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### non-negative curvature condition for polyhedral manifolds

A polyhedral manifold P, i.e, a topological manifold with a triangulation where each simplex is isometric to a simplex in Euclidean space (other constant curvature spaces are allowed), is said to have ...

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### Intersection of conical neighbourhoods on a polyhedral space

Let $P$ be a non-negatively curved (in the Alexandrov sense) polyhedral space (of dimension 3, say), $p,q\in P$ be vertices, and let $e$ be an edge connecting $p$ and $q$. Assume $e$ has cone angle $0&...

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### When is the angular metric on the space of directions intrinsic?

Suppose we have a point $p$ in an Alexandrov space $X$ of curvature bounded below and let $\Sigma_pX$ denote the space of directions of $X$ at $p$. What conditions on $X$ are necessary to ensure that $...

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### Tangent cone of a proper CAT(0) is a proper CAT(0) space

Let $(X,d)$ be a proper CAT$(0)$ space. Let $x\in X$ and let $T_x X$ be the tangent cone of $X$ at $x$ equipped with its usual distance denoted $d_x$. It is a known fact that $(T_x X, d_x)$ is a ...

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### An inequality in Perelman's paper "Elements of Morse theory on Aleksandrov spaces"

I am trying to understand Perelman's paper "Elements of Morse theory on Aleksandrov spaces", St. Petersburg Math. J. 5 (1994), no. 1, 205–213. A version in Russian is here.
Let $\Sigma^n$ be ...

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### Question on G. Perelman's paper "Elements of Morse theory on Aleksandrov spaces"

I am reading Perelman's paper "Elements of Morse theory on Aleksandrov spaces", St. Petersburg Math. J. 5 (1994), no. 1, 205–213. Here is a Russian version (I cold not find the English one)....

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### Example of CAT($k$) space [closed]

Good time of day. I repeat the question from MSE (https://math.stackexchange.com/questions/4464888/question-about-example-of-catk-space) because no response has been received.Question is the following:...

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### Isometric classification of 1-dimensional Alexandrov spaces

It is well known and easy to see (modulo standard basic facts) that any compact 1-dimensional Alexandrov space with curvature bounded from below is isometric either to a circle or to a segment.
I am ...

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### Convergence of extremal subsets in Alexandrov spaces

Let $\{X_i^n\}$ be a sequence of $n$-dimensional Alexandrov spaces with curvature uniformly bounded from below which converges in the Gromov-Hausdorff sense to a compact $n$-dimensional Alexandrov ...

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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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### 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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### 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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### 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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### 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 $X'$ ...

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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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### 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 ...

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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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### An analogue of the Milnor-Švarc lemma for Busemann boundaries

The Milnor-Švarc lemma, is, without doubt, regarded as one of the most important statements in geometric group theory. (Edit) One of the corollaries of this lemma says that if a hyperbolic group $G$ ...

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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 ...

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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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### 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 $\...

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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 $X$.
...

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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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### Length and curvature for closed curves in negatively curved spaces

In the Euclidean plane, for a closed smooth curve of length $\ell$ whose curvature is bounded above by $\epsilon$ we have the inequality
$$ \ell \ge 2\pi \epsilon^{-1} $$
which follows from the fact ...

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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; ...

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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 ...

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### 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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### 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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### 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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### 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 ...

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### 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 ...

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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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### 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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### 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
$$
\...