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

learn more… | top users | synonyms

5
votes
0answers
65 views

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 ...
17
votes
3answers
450 views

Gromov-Hausdorff limits of 2-dimensional Riemannian surfaces

Let $\{M_i\}$ be a sequence of 2-dimensional orientable closed surfaces of genus $g$ with smooth Riemannian metrics with the Gauss curvature at least $-1$ and diameter at most $D$. By the Gromov ...
7
votes
0answers
117 views

Isometry group of low dimensional Alexandrov space

It is known by the work of Galaz-García and Guijarro, that the dimension of the isometry group of an $n$-dimensional Alexandrov space (of curvature bounded below) is bounded above by ...
11
votes
1answer
227 views

Hausdorff convergence of submanifolds in Riemannian manifolds

Let $(M^n,g)$ be a smooth compact Riemannian manifold. It is well known that any sequence $\{X_i\}$ of compact subsets of $M$ has a subsequence which converges in the Hausdorff metric to a compact ...
4
votes
1answer
129 views

Does Alexandrov space satisfy a reverse doubling condition?

Let $X$ be an $n-$dim Alexandrov space with curvature $\geq k$. Does $X$ satisfy a reverse doubling condition? That is, does there exist a constant $C>1$, s.t., for any $x\in X$, ...
1
vote
0answers
58 views

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 ...
7
votes
1answer
112 views

When a Riemannian manifold with boundary is an Alexandrov space?

Let $M$ be a smooth Riemannian manifold (without boundary). Let $X\subset M$ be a smooth compact submanifold with boundary, $\dim X=\dim M$. Under what conditions $X$, equipped with the induced ...
4
votes
1answer
147 views

Classification of 2-dimensional Alexandrov spaces

Is it possible to classify explicitly compact 2-dimensional Alexandrov spaces with curvature bounded below (either with or without boundary)? If yes, a reference would be helpful. EDIT: If the ...
5
votes
1answer
109 views

Geometry of convex subsets in Alexandrov space/ Riemannian manifold

Let $X^n$ be an $n$-dimensional complete Alexandrov space with curvature bounded below (or a smooth Riemannian manifold, possibly with boundary). Let $U\subset X$ be an open dense subset with the ...
7
votes
1answer
75 views

Convergence of functions on Alexandrov spaces

Consider a sequence of $n-$dim Alexandrov spaces with curvature $\geq$ -1 $\{(M_i,p_i)\}$ Gromov-Hausdroff converging to an $n-$dim Alexandrov space $(M,p)$. Let $f:M\mapsto \mathbb R$ be a Lipschitz ...
2
votes
1answer
67 views

A property of concave functions on Alexandrov spaces

EDIT: Let $X$ be an $n$-dimensional Alexandrov space with curvature bounded below. Let $f_1,\dots, f_n\colon X\to \mathbb{R}$ be $\lambda$-concave functions. Assume that at a fixed point $p$ there ...
2
votes
1answer
80 views

Coordinate chart of concave functions near a regular point in Alexandrov spaces

Let $M$ be an Alexandrov space with curvature $\geqslant -1$. Then we have the following theorem which is often used to perturb a regular point to points we want. Let $g_0$ be a ...
4
votes
1answer
87 views

Gromov-Hausdroff convergence for Alexandrov spaces

Let $\{X_n\}_{n=1}^\infty$ be a sequence of compact Alexandrov spaces (with curvature $\geq k$) converging to (in the sense of Gromov-Hausdroff convergence) an Alexandrov spaces $X$, and ...
4
votes
1answer
71 views

Quantitative version of the splitting theorem

The classical splitting theorem (Toponogov, Milka) says that if a smooth complete Riemannian manifold (more generally, Alexandrov space) $M^n$ of non-negative sectional curvature contains a line (i.e. ...
3
votes
1answer
63 views

Is the boundary of Alexandrov space again an Alexandrov space?

Let $X$ be a finite dimensional (possibly compact) Alexandrov space with curvature $\geq K$. Is it true that its boundary is again Alexandrov space with curvature bounded from below? If yes, is the ...
2
votes
1answer
75 views

A property of geodesic triangles in Alexandrov spaces

Let $X$ be an $n$-dimensional Alexandrov space with curvature at least -1. Assume that at every point it has an $(n,\delta)$-strainer of length $\mu$, where $\delta$ and $\mu$ are independent of a ...
2
votes
1answer
121 views

Set of regular points in an Alexandrov space with curvature bounded below

Let $X^n$ be an $n$-dimensional Alexandrov space with curvature bounded below. A point $x\in X$ is called regular if the space of directions $\Sigma_x$ is isometric to the standard sphere $S^{n-1}$. ...
1
vote
0answers
58 views

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 ...
1
vote
0answers
79 views

A compact Alexandrov space with curvature bounded below has curvature bouneded above? [closed]

For a compact Riemannian manifold, Since the curvature tensor is continuous, we know that the sectional curvature is bounded, i.e. bounded above and below. Now let $M$ be a compact Alexandrov space ...
2
votes
1answer
90 views

Is a cocompact CAT(0) periodic?

Let $X$ be a CAT(0) space and $G$ its group of isometries. Then $X$ is said to be cocompact, if there exists a compact set $K\subset X$ with $X=G.K$. The space $X$ is called periodic, if there exists ...
4
votes
0answers
127 views

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 ...
1
vote
0answers
34 views

Concave functions on a cone over Alexandrov space

Let $\Sigma$ be an Alexandrov space of curvature $\geq 1$ without boundary. Let $X$ be the cone over $\Sigma$. $X$ is well known to be non-negatively curved. Let $f\colon X\to \mathbb{R}$ be a ...
1
vote
1answer
66 views

Two geodesics with angle $\pi$ in Alexandrov space

Let two geodesic segments in an Alexandrov space with curvature bounded from below start at the same point and the angle between them equals $\pi$. It is possible that these segments are not the two ...
1
vote
1answer
124 views

Gradient of distance function at cut points on Alexandrov spaces

Let $M$ be an $n$-dim Alexandrov space with curvature bounded below $sec \geqslant k$, possibly non-compact. We assume that $M$ has no boundary for simplicity. For a compact subset $K \subset M$, the ...
3
votes
1answer
107 views

Gauss-Bonnet formula for 2-dimensional Alexandrov spaces

EDIT: Let $S$ be a closed orientable 2-dimensional surface equipped with a metric with curvature $\geq \kappa$ in the sense of Alexandrov. Questions 1. Can one define a measure $K$ on $S$ (thought ...
6
votes
2answers
126 views

Geodesics on convex hypersufaces

Let $M^n$ be the boundary of a convex compact set in $\mathbb{R}^{n+1}$ with non-empty interior. Question 1. Is $M$ geodesically complete, i.e. is it true that every geodesic (= locally shortest ...
6
votes
2answers
166 views

Alexandrov spaces which are not limits of Riemannian manifolds

Are there important/ interesting/ natural examples of compact Alexandrov spaces with curvature bounded from below which are not Gromov-Hausdorff limits of smooth compact Riemannian manifolds with ...
4
votes
1answer
182 views

when are local quasigeodesics global in CAT(0)

It is well-known (and easily shown) that a local quasi-geodesic (for some value of "local") in a $\delta$-hyperbolic space is global (one can compute the constants, as well, from local data). This is ...
7
votes
1answer
235 views

Is the center of gravity in a CAT(0) space contained in the convex hull?

In reading Greg Kuperberg's partial answer to this question Convex hull in CAT(0) , I started wondering if the center of gravity is always contained in the closed convex hull. More precisely, given ...
4
votes
0answers
52 views

Convex hulls of quasi-convex sets in proper CAT(0) spaces

Let $A$ be a quasi-convex set in some proper CAT(0) space, $X$, and let $\mbox{Hull}(A)$ be the intersection of all convex sets containing A. Can we conclude that $\mbox{Hull}(A)$ is in some bounded ...
9
votes
0answers
98 views

What are the extremal CAT(0) metrics?

(Split off from Does every CAT(0) space embed in a product of trees? ) Fix an integer $k \ge 2$, and let $MC0_k \subset \mathbb{R}^{\binom{k}{2}}$ be the set of possible squared-distances between $k$ ...
12
votes
2answers
374 views

Does every CAT(0) space embed in a measurable integral of $\mathbb{R}$-trees?

Question 1. Does every CAT(0) space embed isometrically inside an integral of $\mathbb{R}$-trees? Here an integral of $\mathbb{R}$ trees means the set of functions from a measure space $\mathcal{F}$ ...
1
vote
1answer
81 views

Alexandrov spaces of constant curvature

Let $X$ be locally compact Alexandrov space whose curvature satisfies both inequalities $\geq K$ and $\leq K$. What can be said about such a space? Is it locally isometric to the standard Riemannian ...
0
votes
1answer
88 views

Existence of shortest paths in complete Alexandrov spaces

Let $X$ be complete finite dimensional Alexandrov space with curvature bounded from below. Is it true that any two points can be connected by a shortest path? If this is not true in general, it it ...
2
votes
1answer
131 views

Shortest paths in Alexandrov spaces

Let $X$ be an Alexandrov space with curvature bounded from below (if necessary, $X$ might be assumed to be finite dimensional or even compact). Question 1. Is it true that every point of $X$ has a ...
4
votes
1answer
177 views

Codimension of the set of topologically singular points of an Alexandrov space.

I am reading Burago, Burago and Ivanov's book A course in metric geometry. In chapter 10 the mention that Alexandrov spaces of curvature bounded below have a stratification into topological manifolds. ...
2
votes
1answer
113 views

Why finite dimensional MCS-space implies $\mathbb{R}^m\times cone$ locally?

Frequently, I see a statment of the result in reference that a finite dimensional Alexandrov spaces is locally a product $\mathbb{R}^m\times cone$. It seems that this result comes from Perelman's ...
3
votes
1answer
82 views

Whether the manifold part of an Alexandrov space is connected?

The title is my question. Alexandrov space here means finite dimensional Alexandrov space with curvature bounded below ,denoted by CBB. Let $\gamma$ be a simple curve in a $n$ dimensional CBB $M$ ...
1
vote
1answer
132 views

Convex subcomplexes of CAT(0) cubical complexes

Is the following statement true? If so, can anyone provide a reference? Let $X$ be a CAT(0) cubical complex, and let $Y$ be a connected subcomplex of $X$. Then the following are equivalent: ...
1
vote
0answers
115 views

A question about a manifold in an $n$-dimensional Alexandrov space with curvature bounded below [duplicate]

Suppose $M$ is an $n$-dimensional Alexandrov space with curvature bounded below(maybe with boundary), subspace $A\subset M$ is an $n$-dimensional manifold without boundary. Then whether every point in ...
3
votes
0answers
112 views

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 ...
6
votes
0answers
152 views

CAT(0) groups that does not act on CAT(0) cubical complex

CAT(0) groups are groups that act on a CAT(0) space properly and cocompactly. If a group acts on a CAT(0) cubical complex properly and cocompactly, then of course it is a CAT(0) Group. I am wondering ...
5
votes
2answers
254 views

Are shortest halving curves simple closed geodesics?

Let $S$ be a smooth convex surface in $\mathbb{R}^3$ (although my question may as well be asked for the surface of a polyhedron). Say that $\gamma$ is a shortest halving curve if (a) it partitions the ...
2
votes
0answers
102 views

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 ...
2
votes
0answers
89 views

isoperimetric problems on Alexandrov spaces

For an Alexandrov space M with curvature bounded from below, the isoperimetric profile $v \to I_M(v)$ defined for every $v\in (0,V(M))$ (the volume of M might be infinite), is given by $$ ...
1
vote
1answer
73 views

The set of strained points in an Alexandrov space is open [closed]

I'm reading Burago, Burago and Ivanov's book, and I'm on the section about Strainers. The authors say that it is obvious that the set of $(m,\varepsilon)$-strained points for any fixed natural number ...
2
votes
0answers
177 views

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 ...
7
votes
1answer
317 views

Length inequalities in trees and CAT(0) spaces

I have a family of possibly related questions. Let me start with an elementary one: Question 1. Fix an integer $n$. For which collections of real numbers $a_{ij}$, $i, j = 1, \dots, n$, is it true ...
11
votes
1answer
361 views

Tverberg's theorem in CAT(0) spaces

Does Tverberg's theorem hold for CAT(0) spaces of covering dimension $d<\infty$: Is it true that for any $d$-dimensional $CAT(0)$-space $X$ and a subset $E\subset X$ of cardinality $(d + 1)(r - ...
13
votes
1answer
445 views

Mapping class group and CAT(0) spaces

I hope the questions are not too vague. Is the mapping class group of an orientable punctured surface $CAT(0)$ ? Is any of the remarkable simplicial complexes (curve complex, arc complex...) built ...