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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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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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29 views

Why is k-framed subset in Alexandrov space an MCS space?

A compact subset $P$ in an Alexandrov space $X$ is called k-framed if $P$ can be covered by a finite collection of open sets $U_{\alpha}$ such that each $U_{\alpha}$ is a product neighborhood of rank $...
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48 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 $$ \...
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68 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 ...
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221 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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58 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 ...
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85 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 ...
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1answer
60 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 $\...
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82 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.
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81 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)>...
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26 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 ...
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116 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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420 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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212 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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109 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 $\varepsilon \in (0,\varepsilon_0)$...
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181 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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64 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 $\{X_i^n\}_{i\in \mathbb{N}}$ (...
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53 views

Convexity of set of normal directions in a CAT(0)-space

Let $X$ be a $\mathrm{CAT}(0)$ space, $p\in X$ and $v\in T_pX$. Let $N\subset T_pX$ be the set of tagent vectors making an angle greater than or equal to $\pi/2$ with $v$. Is it true that the set $\...
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358 views

Homeomorphism/ homotopy types of non-negatively curved manifolds

A (special case of a) theorem of Gromov says for any $n\in \mathbb{N}$ there exists a constant $C(n)$ such that for any smooth connected closed $n$-dimensional Riemannian manifold with non-negative ...
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1answer
133 views

Fundamental group of Alexandrov space.

Is it true that the fundamental group of a compact finite dimensional Alexandrov space with curvature bounded below is finitely generated?
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103 views

“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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123 views

Smooth approximation of locally CAT(-1) metrics

It is a well known fact that locally CAT(-1) metrics on surfaces can be approximated by hyperbolic polyhedral metrics with cone singularities: roughly speaking you pick a geodesic triangulation of the ...
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77 views

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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265 views

On triangle comparison in Riemannian manifolds with upper sectional curvature bound

I have a question on Riemannian geometry or CAT(k) geometry, which might be simple for experts. Suppose $M$ is a complete smooth Riemannian manifold with sectional curvature bounded from above by $k&...
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311 views

Bishop-Gromov volume comparison on manifolds with negligible negative Ricci curvature

Let us consider a complete Riemannian manifold $M$ of dimension $n$ with $Ric \geq 0$. Then the Bishop-Gromov volume comparison theorem says that for any $p \in M$, the function $$ \frac{\text{Vol}(B(...
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90 views

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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73 views

Convergence of semi-concave functions on Alexandrov spaces

I will use the terminology of this paper by A. Petrunin: https://arxiv.org/pdf/1304.0292.pdf Let a sequence of $n$-dimensional Alexandrov spaces $\{X_i\}$ of curvature at least $-1$ converges to an ...
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42 views

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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62 views

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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1answer
166 views

Multidimensional gluing theorem for Riemannian manifolds

I would like to understand whether the following multidimensional (partial) generalization of the A.D. Alexandrov gluing theorem is true and, if yes, whether there is a reference. (The original ...
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89 views

Is any geodesic in the tangent cone of an Alexandrov space a limit geodesic?

If $X$ is an Alexandrov space of curvature bounded below by a real number $k$, is it true that any geodesic in the tangent cone $T_pX$ can be realized as a limit of geodesics when we view $T_pX$ as ...
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127 views

Estimate of number of boundary components of a compact Riemannian 2-surface

Let $X$ be a compact smooth 2-dimensional Riemannian manifold with boundary. Assume that the Gauss curvature of $X$ is at least $-1$ and the diameter is at most $D$. Assume that near the boundary the ...
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1answer
176 views

Estimate of area of 2-dimensional surface

Let $X$ be a compact smooth 2-dimensional Riemannian manifold with boundary. Assume that the Gauss curvature of $X$ is at least $\kappa$, the diameter is at most $D$, and the second fundamental form ...
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1answer
252 views

Hausdorff vs Gromov-Hausdorff convergence of convex hypersurfaces

Let $\{K_i\}$ be a sequence of convex compact $n$-dimensional subsets in a Euclidean space $\mathbb{R}^n$. Assume $\{K_i\}$ converges in the Hausdorff metric to a convex compact set $K$ which is also $...
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1answer
197 views

Angle estimate in Alexandrov spaces

I am not sure that this is a research level question. Remark 10.9.4 in the book "A course in metric geometry" by Burago, Burago, Ivanov claims the following. Let $X$ be a finite dimensional ...
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148 views

Rademacher type theorem for Alexandrov spaces

The classical Rademacher theorem says that any Lipschitz function on a doman in $\mathbb{R}^n$ has the first derivative almost everywhere. I am wondering if this result can be generalized as follows. ...
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313 views

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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227 views

Can Alexandrov surfaces of CAT(0) type be approximated by CAT(0) polyhedra?

The theory of such surfaces goes back to the book by Alexandrov and Zalgaller (1967 English translation) and from a more analytic viewpoint, work by Reshetnyak where everything is translated into ...
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301 views

Harmonic maps are light

Assume $f\colon \mathbb{D}\to\mathbb{R}^2$ is a harmonic map and $x\notin f(\partial\mathbb{D})$. Is it true that $f^{-1}\{x\}$ is totally disconnected? I hope that the answer is yes. But actually I ...
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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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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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1answer
221 views

Isoperimetric inequality in CAT(0) surfaces

I am looking for a version of the (2-dimensional) isoperimetric inequality for globally CAT(0) (in particular simply-connected) surfaces. I am particularly interested in characterizing the disk of ...
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1answer
96 views

Area of a sphere in Alexandrov spaces

Let $(X,d)$ be an $n (\geq2)$ dim Alexandrov space with curvature $\geq k$. $B(x,r)$ is an open ball in $X$. Let $M_{k,n}$ be the $n$ dim space form of constant curvature $k$. $B_k(r)$ is an open ball ...
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143 views

Can we realize the smooth metric of an Alexandrov space with nonnegative curvature by a Riemannian structure?

We know that a smooth Riemannian manifold with nonnegative curvature is an Alexandrov space (with induced metric) of nonnegative curvature. What about the converse? That is, given a smooth metric d ...
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Lower estimate on sectional curvature of the boundary

Let $M^n$ be an $n$-dimensional smooth compact Riemannian manifold with boundary. Assume that the sectional curvature of $M$ is at least $\kappa$, the diameter is at most $D$, and the second ...
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1answer
218 views

Generalization of Radon's theorem

A Cat(0) metric space $(X,d)$ of constant and finite local dimension is approximately flat if there exists a dense subset $U\subset X$ such that every $x\in U$ has a flat neighborhood (i.e. isometric ...
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566 views

Applications of Alexandrov spaces to Riemannian geometry

I am an expert neither in Riemannian geometry nor in Alexandrov spaces. I am wondering what are the applications of Alexandrov spaces to more classical Riemannian geometry. For example one can show ...
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205 views

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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2answers
186 views

Gluing Alexandrov spaces along parts of boundary

I'm familiar with the Petrunin gluing theorem that states that gluing two Alexandrov spaces $M_1,M_2\in Alex(k)$ along their boundaries via an isometry $:\partial M_1\rightarrow \partial M_2$ results ...
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2answers
328 views

Limit space of a sequence of Riemannian manifolds with uniformly bounded below Ricci curvature

Let $\{M^n_i\}_{i=1}^\infty$ be a sequence of closed smooth Riemannian $n$-dimensional manifolds with uniformly bounded below Ricci curvature and uniformly bounded above diameter. The Gromov ...

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