All Questions
Tagged with mg.metric-geometry alexandrov-geometry
43 questions with no upvoted or accepted answers
15
votes
0
answers
753
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 curvature ...
- 11.1k
10
votes
0
answers
387
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 ...
- 4,123
8
votes
0
answers
276
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 ...
- 45k
7
votes
0
answers
477
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 ...
- 21.8k
6
votes
0
answers
134
views
Nearby convex set in a nearby space
Let $K$ be a convex set in a CAT(0) space $X$. Suppose $X'$ is a CAT(0) space that is very close to $X$.
Is there a convex set $K'\subset X'$ that is close to $K\subset X$?
Two spaces $X$ and $X'$ ...
- 45k
6
votes
0
answers
386
views
Is there a Bishop-Gromov inequality for manifolds with boundary?
EDIT. Let $M^n$ be a smooth compact Riemannian manifold with smooth boundary.
Assume in addition that near the boundary $M$ is locally geodesically convex.
Assume that the Ricci curvature satisfies $...
- 21.8k
5
votes
0
answers
161
views
Yamaguchi submersion theorem
Let me remind first a theorem of Yamaguchi (1996).
Given $n\in \mathbb{N}, \mu_0>0$. Then there exist $\delta_n>0$ and $\epsilon_n(\mu_0)>0$ with the following property. Let $X$ be an $n$-...
- 21.8k
5
votes
0
answers
195
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 non-...
- 2,371
5
votes
0
answers
145
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 $\...
- 401
4
votes
1
answer
96
views
Sequence of 2-cylinders converging to a segment in the Gromov-Hausdorff metric
Let $\{C_i\}_{i=1}^\infty$ be a sequence of (compact) 2-dimensional cylinders with smooth Riemannian metrics with Gauss curvature at least $-1$ and geodesically convex boundary (equivalently, the ...
- 21.8k
4
votes
0
answers
196
views
An analogue of the Milnor-Švarc lemma for Busemann boundaries
The Milnor-Švarc lemma, is, without doubt, regarded as one of the most important statements in geometric group theory. (Edit) One of the corollaries of this lemma says that if a hyperbolic group $G$ ...
- 193
4
votes
0
answers
108
views
"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 ...
- 21.8k
4
votes
0
answers
100
views
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 ...
4
votes
0
answers
51
views
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 ...
- 21.8k
3
votes
0
answers
75
views
non-negative curvature condition for polyhedral manifolds
A polyhedral manifold P, i.e, a topological manifold with a triangulation where each simplex is isometric to a simplex in Euclidean space (other constant curvature spaces are allowed), is said to have ...
- 85
3
votes
0
answers
179
views
An inequality in Perelman's paper "Elements of Morse theory on Aleksandrov spaces"
I am trying to understand Perelman's paper "Elements of Morse theory on Aleksandrov spaces", St. Petersburg Math. J. 5 (1994), no. 1, 205–213. A version in Russian is here.
Let $\Sigma^n$ be ...
- 21.8k
3
votes
0
answers
179
views
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 ...
- 21.8k
3
votes
0
answers
87
views
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 ...
- 21.8k
3
votes
0
answers
195
views
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 ...
- 21.8k
3
votes
0
answers
99
views
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 ...
- 21.8k
3
votes
0
answers
322
views
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 ...
- 31
3
votes
0
answers
196
views
Why is a negatively curved cone surface locally CAT(-1)?
Recently, I was reading a paper about the rigidity of negatively curved cone surfaces written by S. Hersonsky and F. Paulin. The authors said that a negatively curved cone surface is locally CAT(-1). ...
- 31
2
votes
0
answers
81
views
Nested convex hulls in Hadamard manifold
Let $F$ be a finite set in a Hadamard manifold $H$, and $W\supset F$ is its neighborhood.
Is it true that the closure of the convex hull of $F$ lies in the interior of the convex hull of $W$?
...
- 45k
2
votes
0
answers
62
views
Ball Covering Property in Non-negatively Curved Spaces [Reference Request]
$\DeclareMathOperator{\vol}{Vol}$
Suppose we are working inside a Riemannian $n$-manifold $M$ of non-negative Ricci curvature. In his PhD thesis (see pp.8–9), Zhong-dong Liu presents an incredibly ...
- 458
2
votes
0
answers
209
views
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 $\...
- 21.8k
2
votes
0
answers
85
views
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; ...
- 447
2
votes
0
answers
141
views
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 ...
- 21.8k
2
votes
0
answers
87
views
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 ...
- 21.8k
2
votes
0
answers
152
views
Do Alexandrov spaces with non-empty boundary satisfy $RCD^*$ condition?
Let $M$ be an $n-$ dim compact Alexandrov space with curvature $\geq k$ with non-empty boundary $\partial M$.
Recently, a notion of generalized lower Ricci curvature bound on metric measure spaces ...
- 169
2
votes
0
answers
135
views
homotopy type of metric spheres
Let $(X,d)$ be a metric space, $p\in X$ and $U_p$ a neighborhood of $p$ such that there exists a bi-Lipschitz map
$F:U_p\to \mathbb{R}^n$
(we regard $\mathbb{R}^n$ with the usual Euclidean metric). ...
- 85
1
vote
0
answers
67
views
Quasi-geodesics in Alexandrov spaces
I am trying to understand the notion of quasi-geodesic in Alexandrov space with curvature bounded below following the Perelman-Petrunin paper. I have two questions:
Is it true that the shortest ...
- 21.8k
1
vote
0
answers
31
views
Cut locus of linear isometric action quotients
Given a compact group $G\leq \operatorname{O}(d)$ of linear isometries on $\mathbb R^d$, equip its quotient $\mathbb R^d/G$ with the canonical orbital metric.
I am curious about the following. Is ...
- 71
1
vote
0
answers
33
views
Collapse of Moebius bands with bounded below Gauss curvature and convex boundary
Let $\{M_i\}_{i=1}^\infty$ be a sequence of (compact) Moebius bands with Riemannian metrics with Gauss curvature at least $-1$ and such that the boundaries are geodesically convex (equivalently, the ...
- 21.8k
1
vote
0
answers
164
views
Reference to equivariant Gromov-Hausdorff convergence
I am looking for a reference to the following notions and facts below which, I think, I can prove, but which might be known to experts.
Let us fix a finite group $G$. Consider the class of all compact ...
- 21.8k
1
vote
0
answers
46
views
Essential singular set of an Alexandrov space
Let $X$ be a locally compact Alexandrov space with curvature bounded below. Suppose $C$ is a closed subset that consists of the essential singular points, where a point $p$ is called an essential ...
- 891
1
vote
0
answers
79
views
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 ...
- 14.3k
1
vote
0
answers
43
views
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 ...
1
vote
0
answers
79
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 $E\...
- 169
1
vote
0
answers
129
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 $\Gamma_2$ $$
\Gamma_2 [f,g;\phi]:=\frac12 \int_X (\Gamma (f,g)L\phi -(\Gamma(f,Lg)+\Gamma(...
- 689
1
vote
0
answers
85
views
curvature of subset of Alexandrov spaces
If M is a Riemannian manifold with $Ric \ge - \left( {n - 1} \right)$, $$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 \...
- 689
0
votes
0
answers
33
views
Request for resources on directional derivative of the Riemannian distance function, and Berger's lemma about geodesics realizing the diameter
I've been recently interested in directional derivatives of the Riemannian distance function, and I came across this question, and its answer by Sergei Ivanov, where he stated an important result: (I ...
- 1,467
0
votes
0
answers
42
views
Reference request: in Alexandrov geometry gradient flows preserve extremal subsets
It is mentioned in literature that in Alexandrov geometry gradient flows of semi-concave functions preserve each extremal subset.
I am looking for a proof of this fact.
- 21.8k
0
votes
0
answers
214
views
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 ...
- 1