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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Are all these groups CAT(0) groups?
Given a geodesic metric space $X$ together with a choice of midpoints
$m:X\times X\rightarrow X$ (i.e. $d(m(x,y),x)=d(m(x,y),y)=d(x,y)/2$).
Assume furthermore, that the following nonpositive curvature ...
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Is it overkill to invoke Kirszbraun theorem to prove the following fact ?
Given a small enough convex triangle $(abc)$ in a (smooth or Alexandrov) surface $(X,d)$ of curvature greater than $-1$, let $(\overline{abc})$ be its comparison triangle in $\mathbb{H}^2$. Then there ...
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Generalized flag complex?
Assume we glue an $n$-dimensional simplicial complex $K$
from copies of an $n$-simplex $\Delta$ with fixed spherical metric.
We may think that $\Delta$ has colored vertices
and we glue so that the ...
7
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Gromov's compactness theorem for manifolds with boundary
The Gromov's compactness theorem says that if $\{M_i^n\}$ is a sequence of closed Riemannian manifolds of dimension $n$ with uniformly bounded diameter and uniformly bounded from below Ricci curvature ...
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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 $X'$ ...
6
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Is there a Bishop-Gromov inequality for manifolds with boundary?
EDIT. Let $M^n$ be a smooth compact Riemannian manifold with smooth boundary.
Assume in addition that near the boundary $M$ is locally geodesically convex.
Assume that the Ricci curvature satisfies $...
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Yamaguchi submersion theorem
Let me remind first a theorem of Yamaguchi (1996).
Given $n\in \mathbb{N}, \mu_0>0$. Then there exist $\delta_n>0$ and $\epsilon_n(\mu_0)>0$ with the following property. Let $X$ be an $n$-...
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Historical perspectives on CAT(0) spaces
Does there exist a survey on the early developments of CAT(k) spaces, with the first motivations and the first problems considered? I looked at Bridson and Haefliger's book On metric spaces of non-...
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Fourier analysis for the discrete cube in CAT(0) spaces?
Is there a meaningful Fourier analysis of mappings from the discrete cube into CAT(0) spaces?
Examples for what I have in mind:
Fix a CAT(0) space $X$, a mapping $f:\{-1,1\}^n \to X$, and $\...
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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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"Uniqueness" of the Yamaguchi submersions of Riemannian manifolds
The Yamaguchi submersion theorem says the following. Let $\{M_i\}$ be a sequence of $n$-dimensional smooth connected closed Riemannian manifolds of diameter at most $D$ and sectional curvature at ...
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Is there a fiber bundle for Alexandrov spaces collapsing to a manifold?
Let $\Psi(i)\to 0$ as $i\to \infty$.
Let $A_i$ be a sequence of n dimensional Alexandrov spaces with curvature $\geqslant k$, that Gromov Hausdorff converge to an m dimensional closed Riemannian ...
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Lower estimate on length of boundary of 2d Riemannian surface
Fix constants $\kappa\in \mathbb{R}, D>0,A>0$. Does there exist a constant $C>0$ depending on $\kappa, D, A$ only such that for any compact 2-dimensional Riemannian surface (or more generally ...
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Surfaces with curvature $\leq K$ are of bounded integral curvature
One characteristic of a CBA($K$) surface (a topological surface with an intrinsic metric of curvature $\leq K$ in the sense of Alexandrov) is that $\delta_K(T) \leq 0$, where $\delta_K(T)$
is the ...
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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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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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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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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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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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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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Volume of boundary of Alexandrov space.
Let $X$ be an $n$-dimensional compact Alexandrov space with curvature bounded below which has non-empty boundary. Is it true that the boundary has Hausdorff dimension $n-1$? If yes, does it have ...
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Curvature $\geq-1$ but not $\geq1$
(Edited again)
In the following, for brevity, I will say that $$X\ \ \mathrm{has}\ \ \kappa_{\mathrm{max}}=k$$ if $X$ is a compact ($n$-dimensional with $n\geq2$, with empty boundary) Aleksandrov ...
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Why is a negatively curved cone surface locally CAT(-1)?
Recently, I was reading a paper about the rigidity of negatively curved cone surfaces written by S. Hersonsky and F. Paulin. The authors said that a negatively curved cone surface is locally CAT(-1). ...
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Convex functions with non-singular hessian measure are continuously differentiable?
It is known that every convex function $f: \Omega\to \mathbb{R}$, $\Omega$ convex subset of $\mathbb{R}^n$, has a weak derivative of bounded variation $Df\in BV_{loc}(\mathbb{R}^n)$ (e.g. Evans and ...
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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 $W$?
...
2
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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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46
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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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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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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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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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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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Boundary of 2-dimensional Alexandrov space
Let $X$ be a compact 2-dimensional Alexandrov space with curvature at least $\kappa$. My question is somewhat vague.
What is known about the boundary of $X$?
For example:
1) Is the boundary ...
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Do Alexandrov spaces with non-empty boundary satisfy $RCD^*$ condition?
Let $M$ be an $n-$ dim compact Alexandrov space with curvature $\geq k$ with non-empty boundary $\partial M$.
Recently, a notion of generalized lower Ricci curvature bound on metric measure spaces ...
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130
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homotopy type of metric spheres
Let $(X,d)$ be a metric space, $p\in X$ and $U_p$ a neighborhood of $p$ such that there exists a bi-Lipschitz map
$F:U_p\to \mathbb{R}^n$
(we regard $\mathbb{R}^n$ with the usual Euclidean metric). ...
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Cusp points in Alexandrov spaces
Given a space of bounded integral curvature (by which I mean a topological surface with an intrinsic metric, such that the sum of excesses of any finite collection of non-overlapping simple triangles ...
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114
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Connected sum in Alexandrov spaces
Is it possible to take connected sums of Alexandrov spaces? More explicitly, can one put a metric that turns the connected sum into an Alexandrov space? Does it matter if the curvature bound is from ...
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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 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 $3$-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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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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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 ...
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Lipschitzian extension of mapping between Alexandrov spaces
Let $X$ be an $n-$dim (compact, if needed) Alexandrov space with curvature $\geq -k$, with $k\geq0$, and let $Y$ be an Alexandrov space with curvature $\leq0$ globally. Given any bounded nonempty $E\...
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145
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Doubling theorem for Alexandrov spaces
Is there a user friendly exposition of the notion of boundary of an Alexandrov space with curvature bounded from below and of the Doubling theorem?
The only reference I am aware of is the original ...
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Alexandrov spaces satisty $BE(K,N)$ and $BE(K,\infty)$?
Assume the Dirichlet form $\varepsilon$ adimits a Carre du champ $\Gamma$ and introduce the multilinear form $\Gamma_2$ $$
\Gamma_2 [f,g;\phi]:=\frac12 \int_X (\Gamma (f,g)L\phi -(\Gamma(f,Lg)+\Gamma(...
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curvature of subset of Alexandrov spaces
If M is a Riemannian manifold with $Ric \ge - \left( {n - 1} \right)$, $$ds_M^2 = d{t^2} + \exp \left( {2t} \right)ds_N^2$$ N is a submanifold of M. Then by Gauss-equation, we can prove $Ric\left( N \...
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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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123
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Must a hyperbolic cone over Riemannian manifold be manifold?
M is a hyperbolic cone over an n-1 dim Riemannian manifold N with $Ric(N) \ge - \left( {n - 2} \right)$ ie
$M = R \times {}_{\cosh \left( t \right)}N$,Surely N is an Alexandrov space,must M be a ...