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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 ...
Learning math's user avatar
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 ...
Lucas L.'s user avatar
1 vote
1 answer
98 views

Intersection of conical neighbourhoods on a polyhedral space

Let $P$ be a non-negatively curved (in the Alexandrov sense) polyhedral space (of dimension 3, say), $p,q\in P$ be vertices, and let $e$ be an edge connecting $p$ and $q$. Assume $e$ has cone angle $0&...
Lucas L.'s user avatar
8 votes
1 answer
282 views

Length and curvature for closed curves in negatively curved spaces

In the Euclidean plane, for a closed smooth curve of length $\ell$ whose curvature is bounded above by $\epsilon$ we have the inequality $$ \ell \ge 2\pi \epsilon^{-1} $$ which follows from the fact ...
Jean Raimbault's user avatar
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 ...
aglearner's user avatar
  • 14.3k
2 votes
1 answer
93 views

In a manifold, $\angle xpy>\frac{\pi}{2}$, for $q$ on $px$ or $py$, $B_q(r)$ homeomorphic to $B_p(r)$?

Let M be an n-dimensional Riemannian manifold without boundary, with sectional curvature $\geqslant -1$. For a point $p\in M$, suppose there exist $l, \delta>0$, $x,y \in M$ with $d(p,x),d(p,y)>...
mathmetricgeometry's user avatar
8 votes
1 answer
412 views

Homeomorphism/ homotopy types of non-negatively curved manifolds

A (special case of a) theorem of Gromov says for any $n\in \mathbb{N}$ there exists a constant $C(n)$ such that for any smooth connected closed $n$-dimensional Riemannian manifold with non-negative ...
asv's user avatar
  • 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 ...
mathmetricgeometry's user avatar
3 votes
1 answer
704 views

Bishop-Gromov volume comparison on manifolds with negligible negative Ricci curvature

Let us consider a complete Riemannian manifold $M$ of dimension $n$ with $Ric \geq 0$. Then the Bishop-Gromov volume comparison theorem says that for any $p \in M$, the function $$ \frac{\text{Vol}(B(...
user116108's user avatar
8 votes
1 answer
400 views

Multidimensional gluing theorem for Riemannian manifolds

I would like to understand whether the following multidimensional (partial) generalization of the A.D. Alexandrov gluing theorem is true and, if yes, whether there is a reference. (The original ...
asv's user avatar
  • 21.8k
7 votes
1 answer
162 views

Estimate of number of boundary components of a compact Riemannian 2-surface

Let $X$ be a compact smooth 2-dimensional Riemannian manifold with boundary. Assume that the Gauss curvature of $X$ is at least $-1$ and the diameter is at most $D$. Assume that near the boundary the ...
asv's user avatar
  • 21.8k
7 votes
1 answer
231 views

Estimate of area of 2-dimensional surface

Let $X$ be a compact smooth 2-dimensional Riemannian manifold with boundary. Assume that the Gauss curvature of $X$ is at least $\kappa$, the diameter is at most $D$, and the second fundamental form ...
asv's user avatar
  • 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 ...
user84068's user avatar
  • 169
3 votes
1 answer
109 views

Area of a sphere in Alexandrov spaces

Let $(X,d)$ be an $n (\geq2)$ dim Alexandrov space with curvature $\geq k$. $B(x,r)$ is an open ball in $X$. Let $M_{k,n}$ be the $n$ dim space form of constant curvature $k$. $B_k(r)$ is an open ball ...
user84068's user avatar
  • 169
7 votes
1 answer
259 views

Can we realize the smooth metric of an Alexandrov space with nonnegative curvature by a Riemannian structure?

We know that a smooth Riemannian manifold with nonnegative curvature is an Alexandrov space (with induced metric) of nonnegative curvature. What about the converse? That is, given a smooth metric d ...
Jialong Deng's user avatar
  • 1,799
5 votes
2 answers
546 views

Limit space of a sequence of Riemannian manifolds with uniformly bounded below Ricci curvature

Let $\{M^n_i\}_{i=1}^\infty$ be a sequence of closed smooth Riemannian $n$-dimensional manifolds with uniformly bounded below Ricci curvature and uniformly bounded above diameter. The Gromov ...
asv's user avatar
  • 21.8k
8 votes
1 answer
315 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 $\frac{n(n+1)}{2}$...
Melquíades Ochoa's user avatar
5 votes
1 answer
346 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 $x\in X$, $0<r<\...
Shman's user avatar
  • 51
7 votes
1 answer
121 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 ...
David Born's user avatar
5 votes
1 answer
163 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 $f_n:X_n\...
Someone1987's user avatar
1 vote
0 answers
181 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 ...
mafan's user avatar
  • 471
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). ...
Huiping Pan's user avatar
2 votes
1 answer
662 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 ...
mafan's user avatar
  • 471
4 votes
1 answer
272 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 ...
Igor Rivin's user avatar
  • 96.4k
2 votes
1 answer
232 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 ...
asv's user avatar
  • 21.8k
5 votes
1 answer
196 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 $$ I_M(v)=...
user53063's user avatar
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(...
jiangsaiyin's user avatar
4 votes
1 answer
305 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 ...
Ralph's user avatar
  • 283
9 votes
1 answer
531 views

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 $y$ ...
Wolfgang Spindeler's user avatar
-2 votes
2 answers
298 views

examples of totally geodesic subset

Could you give examples of totally geodesic subset of codim>1 in positively curved Alexandrov space?
jiangsaiyin's user avatar
2 votes
1 answer
807 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 ...
jiangsaiyin's user avatar
17 votes
1 answer
526 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 ...
Joseph O'Rourke's user avatar
4 votes
1 answer
385 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 ...
jiangsaiyin's user avatar
10 votes
1 answer
935 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 2-...
Thomas Richard's user avatar
10 votes
1 answer
933 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 ...
Thomas Richard's user avatar
9 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 $\...
Luc's user avatar
  • 265
6 votes
2 answers
365 views

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 ...
Peter Samuelson's user avatar