# 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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### 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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### 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 $X$.
...

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### 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 ...

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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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### 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 ...

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### 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 $(...

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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 ...

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### 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 ...

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### 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 $\...

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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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### 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 ...