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 $\mathrm{CAT}[\kappa]$).

Filter by
Sorted by
Tagged with
6
votes
1answer
99 views

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 ...
3
votes
1answer
71 views

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 ...
0
votes
0answers
94 views

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 ...
2
votes
0answers
29 views

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 ...
1
vote
0answers
35 views

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 \...
1
vote
0answers
34 views

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 ...
2
votes
0answers
35 views

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,\...
4
votes
1answer
81 views

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 ...
6
votes
0answers
130 views

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 ...
6
votes
2answers
86 views

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 ...
1
vote
0answers
74 views

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 ...
3
votes
0answers
56 views

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,...
6
votes
0answers
103 views

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
votes
1answer
151 views

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}/\...
0
votes
0answers
45 views

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 ...
11
votes
1answer
280 views

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
votes
1answer
76 views

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 ...
2
votes
2answers
120 views

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
58 views

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 ...
2
votes
0answers
44 views

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 ...
3
votes
0answers
143 views

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 ...
2
votes
0answers
27 views

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 ...
2
votes
0answers
72 views

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 ...
2
votes
0answers
107 views

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
167 views

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
65 views

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 ...
2
votes
0answers
78 views

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
201 views

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 $(...
3
votes
0answers
64 views

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 ...
1
vote
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 ...
0
votes
0answers
109 views

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
125 views

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 $\...
3
votes
0answers
109 views

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 ...
0
votes
1answer
76 views

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 $$ \...
2
votes
0answers
100 views

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 ...
8
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-...
2
votes
0answers
71 views

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 ...
3
votes
1answer
136 views

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 ...
1
vote
1answer
89 views

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 $\...
1
vote
1answer
120 views

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.
2
votes
1answer
82 views

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)>...
1
vote
0answers
28 views

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 ...
21
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 ...
8
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 $...
6
votes
1answer
116 views

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 $\...
0
votes
1answer
68 views

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 $\{X_i^n\}_{i\in \mathbb{N}}$ (...