Alexandrov geometry is the study of the geometry of Alexandrov spaces, which are non smooth analogues of Riemannian manifolds with curvature bounded from below.

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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 - ...
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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 ...
12
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3answers
327 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 ...
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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 ...
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45 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 ...
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91 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 ...
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1answer
159 views

CAT(K) and Busemann [closed]

Can a Busemman space be CAT(1)?
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66 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 ...
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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 ...
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38 views

curvature of cone over a cuboid bounded below?

As we know,if M is an Alexandrov space with sec>=1,then the cone over M has sec>=0.What if when M is a cuboid with side length r1,...,rn,dia(M)<=π,then the cuvature of the cone over M bounded below ...
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90 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.
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152 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 ...
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99 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 ...
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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
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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 ...
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286 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 ...
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93 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?
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225 views

examples of totally geodesic subset

Could you give examples of totally geodesic subset of codim>1 in positively curved Alexandrov space?
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490 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 ...
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322 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 ...
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1answer
188 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 ...
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257 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 ...
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241 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?
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337 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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575 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
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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 ...
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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.
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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 ...
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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 ...
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837 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} ...
8
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472 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
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1answer
660 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
6
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1answer
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Details of Perelman's example about soul of Alexandrov space

Reading Perelman's preprint(1991) Alexandrov space II now. Got confused about the last section 6.4, which contains an example which indicate that the statement ".... manifold is diffeomorphic to the ...
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Is there Domain Invariance for Alexandrov spaces?

A colleague asked me this question recently. Every injective continuous map between manifolds of the same (finite) dimension is open - this is Brouwer's Domain Invariance Theorem. Is the same true for ...
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In a locally CAT(k) space, does uniqueness of geodesics imply the lack of conjugate points?

A complete, simply connected Riemannian manifold has no conjugate points if and only if every geodesic is length-minimizing. I just realized that I don't know whether the same is true for a locally ...
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259 views

Is Level set of Regular functions in Alexandrov spaces again an Alex. space?

Let $X^n$ be an Alexandrov space, and $f: X^n\to \mathbb R^k$ a regular map, does the level set necessary be an Alexandrov space? In my mind, the intrinsic metric on the level set is 'comparable' to ...
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Are isometries the only geodesic preserving maps in a CAT(0)-space?

Given any CAT(0) space $X$, we can define a map $s:X\times X\times [0;1]\rightarrow X$, such that $s(x,y,-)$ is the constant speed geodesic from $x$ to $y$ . Any isometry $f$ of $X$ is compatible with ...
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Soul theorem for non-negativly curved open Alexandrov manifolds?

Alexandrov manifold means Alexandrov space which happens to be a manifold, i.e. the space of directions is homeomorphic to shpere. Sorry for introducing this new term. For such a open manifold does ...
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Why is GL(n,C)/U(n) a CAT(0) space?

The title says it all. In one of his answers to the question "Convex hull in CAT(0)" (I don't have the points to post a link, if someone doesn't mind link-ifying this that would be cool), Greg ...