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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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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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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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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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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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 ...
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 ...