# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### “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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...