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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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 ...
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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 ...
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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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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 ...
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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 $\...
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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 $\Gamma_2$ $$
\Gamma_2 [f,g;\phi]:=\frac12 \int_X (\Gamma (f,g)L\phi -(\Gamma(f,Lg)+\Gamma(...
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curvature of subset of Alexandrov spaces
If M is a Riemannian manifold with $Ric \ge - \left( {n - 1} \right)$, $$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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Uniquely geodesic and CAT(0) spaces?
Improvement after J-M Schlenker's comment below :
This post has been divided into two parts, the second part is here.
Question : Is a finite dimensional metric space, uniquely geodesic if and only ...
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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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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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Alexandrov angles in Riemannian manifolds
Dear all, I am teaching a course in Riemannian geometry, and I would like to prove some comparison theorems in the next lessons, building on the well-known theory of Jacobi fields, and of Rauch ...
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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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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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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 spaces with sec>=-1 and first eigenvalue (n-1)^2/4
Could someone give examples of non-Riemannian manifolds that are Alexandrov spaces with $\mathrm{sec}\geq-1$ and the first eigenvalue equal to $(n-1)^2/4$?
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Walls of CAT(0) cube complex sufficiently far apart implies intersection of stabilizers finite
I was reading through Agol's paper on the Virtual Haken Conjecture and I came across a claim whose proof I am after. It seems to boil down to the following claim about the hyperplanes and their ...
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Is the tangent cone of a totally convex subset again totally convex?
$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 $\gamma$ (that is a locally shortest path) connecting $x$ and $y$ ...
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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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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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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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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 ...
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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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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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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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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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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 S^...
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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 2-...
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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 (3)...
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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 ...
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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 $\mathbb{E}^...
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(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} \alpha=\...
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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 ...
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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
$$\gamma'(a)=\gamma'(b)=\...
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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 curvature ...
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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 \mathbb{R}\qquad ...
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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 $\...
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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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