**1**

vote

**0**answers

75 views

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

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 ...

**2**

votes

**0**answers

39 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 ...

**5**

votes

**2**answers

213 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

**0**answers

91 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

**0**answers

63 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
$$
...

**0**

votes

**1**answer

54 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

**0**answers

118 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 ...

**6**

votes

**1**answer

185 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

**1**answer

312 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 - ...

**5**

votes

**0**answers

111 views

### Fourier analysis for the discrete cube in CAT(0) spaces?

Is there a meaningful Fourier analysis of mappings from the discrete cube into CAT(0) spaces?
Examples for what I have in mind:
Fix a CAT(0) space $X$, a mapping $f:\{-1,1\}^n \to X$, and ...

**14**

votes

**3**answers

367 views

### Embedding expanders in CAT(0) spaces

It is well-known that expanders are hard to embed into Hilbert (or $\ell^p$) spaces - any embedding of an expander with $n$ vertices has distortion $\Omega(\log n)$.
Can anyone provide a reference ...

**1**

vote

**0**answers

81 views

### Alexandrov spaces satisty $BE(K,N)$ and $BE(K,\infty)$?

Assume the Dirichlet form $\varepsilon$ adimits a Carre du champ $\Gamma$ and introduce the multilinear form $\Gamma_2$ $$
\Gamma_2 [f,g;\phi]:=\frac12 \int_X (\Gamma (f,g)L\phi ...

**1**

vote

**0**answers

58 views

### curvature of subset of Alexandrov spaces

If M is a Riemannian manifold with $Ric \ge - \left( {n - 1} \right)$, $$ds_M^2 = d{t^2} + \exp \left( {2t} \right)ds_N^2$$ N is a submanifold of M. Then by Gauss-equation, we can prove $Ric\left( N ...

**3**

votes

**0**answers

97 views

### Generalized flag complex?

Assume we glue an $n$-dimensional simplicial complex $K$
from copies of an $n$-simplex $\Delta$ with fixed spherical metric.
We may think that $\Delta$ has colored vertices
and we glue so that the ...

**-1**

votes

**1**answer

181 views

**0**

votes

**0**answers

81 views

### Connected sum in Alexandrov spaces

Is it possible to take connected sums of Alexandrov spaces? More explicitly, can one put a metric that turns the connected sum into an Alexandrov space? Does it matter if the curvature bound is from ...

**8**

votes

**2**answers

206 views

### Distortion of tree embedding in Alexandrov spaces

It is a well-known theorem first proved by Bourgain that any map $\varphi:T_n\to H$ from the binary tree of height $n$ to a Hilbert space has distortion at least $C \sqrt{\ln n}$ where $C$ is a ...

**1**

vote

**1**answer

96 views

### examples of space of direction at a point in an infinite dim Alexandrov space compact

The space of direction at a point in an infinite dim Alexandrov space can be compact?Please give examples or prove it's wrong.

**4**

votes

**1**answer

167 views

### Does convex set in Alexandrov space has positive reach?

Let $M$ be a metric space, $A$ a subset of $M$. The reach (defined by Federer) of $A$ in $M$ is the largest $r_0\ge 0$ such that if $x\in M$ and the $d(x, A)< r_0$, then $A$ contains a unique point ...

**0**

votes

**0**answers

103 views

### Must a hyperbolic cone over Riemannian manifold be manifold?

M is a hyperbolic cone over an n-1 dim Riemannian manifold N with $Ric(N) \ge - \left( {n - 2} \right)$ ie
$M = R \times {}_{\cosh \left( t \right)}N$,Surely N is an Alexandrov space,must M be a ...

**0**

votes

**0**answers

68 views

### examples of Alexandrov space with sec>=-1 and first eigenvalue =(n-1)^2/4

could someone give some examples :nonRiemannian manifold Alexandrov space with sec>=-1 and the first eigenvalue equal to (n-1)^2/4

**7**

votes

**1**answer

303 views

### Is the tangent cone of a totally convex subset again totally convex?

To need not worry about the possibly broadest context let:
$X$ be an Alexandrov's space with lower curvature bound and $C$ be a totally convex subset, i.e. for any $x,y \in C$ and any geodesic ...

**0**

votes

**1**answer

292 views

### about parabolic cone

I want to prove some Alexandrov space M is parabolic cone X x R.Since Alex has no Riemannian metric,so how to do?Is there any (triangle) formula about the relation of distance of two points in M and ...

**0**

votes

**0**answers

99 views

### Is level set of Busemann function on Alexandrov space again Alexandrov space?

M is an Alexandrov space with curv>=-1,containing a line(ray).Is level set of Busemann function on M again Alexandrov space?If not,can you give a counterexample?

**-2**

votes

**2**answers

232 views

### examples of totally geodesic subset

Could you give examples of totally geodesic subset of codim>1 in positively curved Alexandrov space?

**1**

vote

**1**answer

506 views

### a result of soul theorem,right?

X is an n-dim positively curved manifold and Y is a totally geodesic submanifold of codimension 1.Then cutting along Y we get n-dim positively curved manifolds without boundary,by soul theorem these ...

**13**

votes

**1**answer

339 views

### Is $\partial X$ a sphere for $X$ a complete CAT$(0)$ space?

Let $X$ be a complete CAT$(0)$ metric space, and $\partial X$ its boundary.
One way to define $\partial X$ is as the equivalence class of geodesic rays
$\gamma(t), \gamma'(t)$
that remain within a ...

**0**

votes

**1**answer

200 views

### positively curved Alexandrov space

I heard a conjecture "3-dim positively curved Alexandrov space is of the form S^3/J.(I cannot make sure my statement is accurate).
What is the classification of n-dim positively curved Alexandrov ...

**4**

votes

**1**answer

270 views

### Extend the Wilking Connectiveity Theorem to Alexandrov spaces

In the conference "on Manifolds with Non-negative Sectional Curvature" held in 2007,
Problem 6 is:
Extend the Wilking Connectivity Theorem to Alexandrov spaces, i.e. if $X$ is a positively curved ...

**5**

votes

**1**answer

271 views

### 3-dim positively curved Alexandrov space

What is the classification of 3-dim positively curved Alexandrov space?
And if a 3-dim positively curved Alexandrov space has a totally (quasi)geodesic subset,then the classification?

**7**

votes

**2**answers

358 views

### Source for: Geodesics in CAT(0) spaces

I am seeking a good introductory reference that could lead to an understanding of
the properties of geodesics in
complete CAT(0) metric spaces.
I am especially interested in learning the differences ...

**10**

votes

**1**answer

259 views

### Connecting Lemma in the Alexandrov's existence theorem.

At the moment I am polishing my lecture notes which in particular cover Alexandrov's existence theorem.
Denote by $\mathbf{P}_k$ the space of isometry classes of polyhedral
metrics on the $\mathbb ...

**3**

votes

**2**answers

324 views

### Alexandrov geometry techniques for Finsler manifolds.

Hi, first I would like to apologize for my English. It's not my native language and I feel my grasp of it is limited.
I've been reading Burago's book on metric geometry and I've that it mentions ...

**9**

votes

**0**answers

269 views

### Is it overkill to invoke Kirszbraun theorem to prove the following fact ?

Given a small enough convex triangle $(abc)$ in a (smooth or Alexandrov) surface $(X,d)$ of curvature greater than $-1$, let $(\overline{abc})$ be its comparison triangle in $\mathbb{H}^2$. Then there ...

**7**

votes

**1**answer

477 views

### Smoothability of compact Alexandrov surfaces with curvature bounded from below.

Let $(X,d)$ be compact metric space of curvature greater than $-1$ (in the sense of comparison triangles), assume that its Hausdorff dimension is $2$. Then a result of Perelman says that $X$ is a ...

**5**

votes

**1**answer

245 views

### Flat sector in a proper cocompact CAT(0) space

Let $X$ be a complete CAT(0) space with a proper and cocompact group action by isometries, and suppose there are $\xi, \xi' \in \partial X$ with $\angle (\xi, \xi') < \pi$. Using proposition 9.5 ...

**12**

votes

**2**answers

621 views

### infinite dimensional CAT(0) groups

Usually a CAT(0) group is defined to be a group acting properly isometrically and cocompactly on a CAT(0) space, but I would like to consider only those groups that act properly, isometrically and ...

**4**

votes

**1**answer

342 views

### Examples of CAT(0)-groups

My question is the following:
Let M be a simply connected Riemannian manifold whose sectional curvatures
are all nonpositive and let G be a group. Suppose that G acts in M properly discontinuous and
...

**11**

votes

**1**answer

466 views

### Is $SL(n,\mathbb{Z})$ a CAT(0) group?

Is it possible to find a CAT(0) space on which the matrix group $SL(n,\mathbb{Z})$ acts properly discontinuously and cocompactly? Note: when the cocompactness is dropped , it is possible.

**2**

votes

**3**answers

318 views

### Is this the CAT(0) metric on an affine building?

Let $R$ be a discrete valuation ring qith quotient field $Q$ and let $t\in R$ be a generator of the unique maximal ideal in $R$. Let $V$ be a finite-dimensional $Q$-vector space. Then one can consider ...

**7**

votes

**1**answer

526 views

### Metric spheres in CAT(0) manifolds

Let $X$ be a topological manifold of dimension $n$, equipped with a compatible CAT(0) metric.
Are sufficiently small metric spheres in $X$ homeomorphic to metric spheres in Euclidean space ...

**10**

votes

**3**answers

854 views

### (1-Lipschitz) + (length-preserving) = isometry

I am looking for an elementary way to prove the following theorem.
Theorem. Let $\alpha$ and $\beta$ be two simple convex closed curves in $\mathbb R^2$.
Assume
$$\mathop{\rm length} ...

**9**

votes

**1**answer

514 views

### Metrically singular Alexandrov space.

Perelman's stability theorem shows in particular that a finite dimensional compact Alexandrov space $(X,d)$ such that $X$ is not a topological manifold cannot be approximated in the Gromov-Hausdorf ...

**12**

votes

**1**answer

667 views

### Minimizing geodesic on a convex surface

Let $\Sigma$ be a smooth convex surface in Euclidean 3-space
and $\gamma$ be a unit speed minimizing geodesic in $\Sigma$.
Assume that for some $a < b < c$, we have
...

**17**

votes

**4**answers

792 views

### When is a extension of $\mathbb{Z}$ by a free group a CAT(0) group?

The question has an easy answer, if one replaces free by free abelian: Then the resulting group is always solvable and a solvable subgroup of a CAT(0) group is virtually abelian.
If the resulting was ...

**8**

votes

**1**answer

1k views

### Rigidity of triangle comparison in Alexandrov spaces

For $CAT(\kappa)$ spaces $X$ we have following rigidity result: if equality holds in any of the comparison distances between a triangle $\Delta$ in $X$ and the corresponding comparison triangle ...

**14**

votes

**4**answers

1k views

### Braid groups acting on CAT(0)-complexes

Does the braid group $B_n, n\ge 3$, act properly by isometries on a CAT(0) cube complex?
Update 1. During a recent talk of Nigel Higson in Pennstate Dmitri Burago asked whether the braid groups are ...

**12**

votes

**0**answers

570 views

### Are all these groups CAT(0) groups?

Given a geodesic metric space $X$ together with a choice of midpoints
$m:X\times X\rightarrow X$ (i.e. $d(m(x,y),x)=d(m(x,y),y)=d(x,y)/2$).
Assume furthermore, that the following nonpositive ...

**3**

votes

**1**answer

186 views

### Stability of midpoints in CAT(0) spaces

Given a CAT(0) space $X$ and a compact, convex subset $A$ of $X$. One can define its midpoint $m(A)$ as the point, at which the following function attains its minimum.
$f:A\rightarrow ...

**6**

votes

**1**answer

308 views

### Contracting a geodesic on a space of curvature less than 1

I would like to ask for a reference to the following statement (hopefully correct):
Let $M$ be a manifold of sectional curvature at most $1$ and let $\gamma$ be a closed geodesic.
Suppose that ...