All Questions
Tagged with mg.metric-geometry alexandrov-geometry
171 questions
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32
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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 ...
- 1,465
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
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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
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0
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42
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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
4
votes
1
answer
96
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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
1
vote
0
answers
33
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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
1
answer
127
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Connectedness of fibers of almost Riemannian submersions
EDIT: Let $M,N$ be compact connected smooth Riemannian manifolds. Let us assume that $N$ is closed, while $M$ might have a geodesically convex boundary.
Given $f\colon M\to N$ be an $\varepsilon$-...
- 21.8k
2
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answers
81
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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
8
votes
2
answers
489
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Amalgamated product acting on CAT(0) cube complex
I was reading the following result from the book Metric spaces of non-positive curvature by Bridson and Haefliger.
Result:
Let $F_0,F_1$ and $H$ be groups acting properly
by isometries on complete $...
- 349
4
votes
1
answer
131
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Does the Alexandrov angle define convex functions along geodesics in CAT(0) spaces?
Let $X$ be a CAT(0) space and suppose $a,b,c,d\in X$ satisfy
$$
\max\{\angle_a(b,c),\angle_a(b,d)\}<\frac{\pi}{2}.
$$
Let $\gamma:[0,\ell]\to X$ be the geodesic with $\gamma(0)=c$ and $\gamma(\ell)...
- 743
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answers
75
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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
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&...
- 85
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1
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97
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When is the angular metric on the space of directions intrinsic?
Suppose we have a point $p$ in an Alexandrov space $X$ of curvature bounded below and let $\Sigma_pX$ denote the space of directions of $X$ at $p$. What conditions on $X$ are necessary to ensure that $...
- 458
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2
answers
159
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Tangent cone of a proper CAT(0) is a proper CAT(0) space
Let $(X,d)$ be a proper CAT$(0)$ space. Let $x\in X$ and let $T_x X$ be the tangent cone of $X$ at $x$ equipped with its usual distance denoted $d_x$. It is a known fact that $(T_x X, d_x)$ is a ...
- 91
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179
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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
4
votes
1
answer
314
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Question on G. Perelman's paper "Elements of Morse theory on Aleksandrov spaces"
I am reading Perelman's paper "Elements of Morse theory on Aleksandrov spaces", St. Petersburg Math. J. 5 (1994), no. 1, 205–213. Here is a Russian version (I cold not find the English one)....
- 21.8k
1
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0
answers
238
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Example of CAT($k$) space [closed]
Good time of day. I repeat the question from MSE (https://math.stackexchange.com/questions/4464888/question-about-example-of-catk-space) because no response has been received.Question is the following:...
- 103
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2
answers
164
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Isometric classification of 1-dimensional Alexandrov spaces
It is well known and easy to see (modulo standard basic facts) that any compact 1-dimensional Alexandrov space with curvature bounded from below is isometric either to a circle or to a segment.
I am ...
- 21.8k
2
votes
1
answer
114
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Convergence of extremal subsets in Alexandrov spaces
Let $\{X_i^n\}$ be a sequence of $n$-dimensional Alexandrov spaces with curvature uniformly bounded from below which converges in the Gromov-Hausdorff sense to a compact $n$-dimensional Alexandrov ...
- 21.8k
6
votes
1
answer
148
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Isometric imbedding of a 2-disk into Euclidean 3-space
Let us call a cap the intersection of the boundary of 3-dimensional convex compact set $K$ in $\mathbb{R}^3$ with a half-space bounded by a plane $H$ such that the orthogonal projection to $H$ of this ...
- 21.8k
3
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1
answer
118
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Is a cap an Alexandrov space?
Let us call a cap the intersection of the boundary of 3-dimensional convex compact set $K$ in $\mathbb{R}^3$ with a half-space bounded by a plane $H$ such that the orthogonal projection to $H$ of this ...
- 21.8k
1
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0
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164
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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
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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
4
votes
1
answer
99
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Does codimension-1 collapsing with bounded curvature have boundary?
Let $(M^n,g_i)$ be a sequence of smooth complete Riemannian manifold with $|sec_{g_i}| \le 1$. Suppose $(M_i^n,g_i)$ converges to a limit space $(X^{n-1},d)$ in the Gromov-Hausdorff sense, where the ...
- 1,441
7
votes
1
answer
246
views
Rigidity for convex surfaces in elliptic/hyperbolic space
From Alexandrov's work we know that any metric on the sphere with lower curvature bound $\kappa$ (in the sense of Alexandrov) can be realized as a closed convex surface (i.e. boundary of a compact ...
- 115
7
votes
2
answers
136
views
Completion of an Alexandrov space
Let $X$ be an incomplete Alexandrov space with sec $\ge -1$ in the sense that for any point in $X$ there exists a small neighborhood in which the four-points criterion is satisfied.
Suppose $X$ is ...
- 2,535
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answers
134
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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
4
votes
1
answer
239
views
Alexandrov spaces of zero curvature
Let $X$ be an $n$-dimensional Alexandrov space with curvature satisfying both $\ge 0$ and $\le 0$. Can we prove that any tangent cone of $X$ must be isometric to $\mathbb R^{k} \times C(S^{n-k-1}/\...
- 2,535
10
votes
1
answer
377
views
Translation lengths in CAT(0) spaces
Let $a,b$ be two loxodromic isometries of a CAT(0) space. Assume that, for every $n \geq 1$, $a^nb$ is also loxodromic. Is it possible for the translation length of $a^nb$ to be bounded independently ...
- 8,401
4
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196
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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
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
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0
answers
179
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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
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
8
votes
1
answer
225
views
Geodesic line with endpoints in interior of Riemannian manifold or Alexandrov space
Let $X$ be a finite dimensional Alexandrov space with curvature bounded below and non-empty boundary. Let $\gamma$ be a shortest geodesic path in $X$ whose endpoints belong to the interior of $X$.
...
- 21.8k
3
votes
1
answer
159
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Tangent cone of metric graph
I am starting to study some lecture notes about metric geometry and I would appreciate it if someone could some questions regarding the notion of the tangent cone.
Consider 3 half lines joined by ...
- 91
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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 ...
- 3,399
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
2
answers
364
views
Unbounded curvature implies infinite diameter on complete metric spaces
I recently asked this question Unbounded sectional curvature implies infinite diameter?.
I would like now to ask something similar, but in another context.
Suppose you have a complete metric space $(...
- 1,774
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
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
0
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0
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214
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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
6
votes
1
answer
218
views
Continuity of volume of boundary of Riemannian manifolds in the Gromov-Hausdorff sense
Let $\{X_i^n\}$ be a sequence of smooth compact Riemannian $n$-dimensional manifolds with boundary. Assume that this sequence has uniformy bounded below sectional curvature, and each $X_i$ is ...
- 21.8k
2
votes
1
answer
188
views
Measure of the boundary of Alexandrov space
Let $X$ be a compact $n$-dimensional Alexandrov space with curvature bounded below. Let $\partial X$ denote its boundary in the sense of the theory of Alexandrov spaces.
Is it true that if $\...
- 21.8k
3
votes
1
answer
195
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Dimension of Alexandrov space which is homeomorphic to a manifold
Let $M^n$ be a smooth manifold of dimension $n$. Let $M$ given a metric with curvature bounded below in the sense of Alexandrov which induces the original topology of $M$.
It is true that the ...
- 21.8k
0
votes
1
answer
117
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Neighborhood of $(n,\delta)$-strained point in Alexandrov space homeomorphic to $\mathbb{R}^n$, how big is $\delta$?
Let $M$ be an $n$-dimensional Alexandrov space with curvature $\geqslant k$. A point $p\in M$ is said to be an $(n,\delta)$ strained point if there are $n$ pairs of points $a_i, b_i$ such that
$$
\...
3
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0
answers
195
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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
8
votes
1
answer
605
views
Why 2-tori with Gauss curvature $\geq -1$ cannot collapse to segment?
Let $\{(\mathbb{T}^2,g_i)\}_{i=1}^\infty$ be a sequence of 2-dimensional tori with smooth Riemannian metrics with Gauss curvature at least $-1$. It was explained in the final answer to the post Gromov-...
- 21.8k
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
3
votes
1
answer
163
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A.D. Alexandrov imbedding theorem for metrics with symmetry
A well known theorem due to A.D. Alexandrov says that any metric on the 2-sphere $S^2$ with curvature at least -1 (in the sense of Alexandrov) can be isometrically realized either as convex surface in ...
- 21.8k
5
votes
1
answer
251
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If M times circle admits a locally CAT(0)-metric, then M also carries a locally CAT(0)-metric?
A locally CAT(0) metric on length space means that every point in it has a geodesically convex neighborhood such that every triangle in it is slimmer than the comparison triangle in the Euclidean ...
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