# Questions tagged [non-positive-curvature]

The non-positive-curvature tag has no usage guidance.

34
questions

6
votes

2
answers

396
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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 $...

1
vote

0
answers

65
views

### 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)...

1
vote

0
answers

67
views

### Parallel Jacobi fields in a Hadamard manifold

Let $M$ be a Hadamard manifold and let $c: \mathbb{R}\rightarrow M$ be a geodesic. A Jacobi field $Y$ along $c$ is called parallel if $Y'(t) = 0$ for every $t\in \mathbb{R}$. If we assume that $M$ is ...

4
votes

0
answers

67
views

### Representing homotopy classes of Kähler manifolds by harmonic maps

Let $(M,g_M)$ be a compact Kähler manifold with negative bisectional curvature. Let $\alpha : (S,g_S) \to M$ be a continuous map from a compact Riemannian manifold $(S,g_S)$.
Is $\alpha$ homotopic to ...

2
votes

1
answer

127
views

### Upper bound on volume growth of area minimizers

Let $M^n$ be a complete simply connected Riemannian manifold with $\operatorname{sec}_M \leq 0$ (i.e. a Hadamard manifold) and assume that there is a constant $a \geq 0$ such that $\operatorname{sec}...

1
vote

0
answers

59
views

### Are Carnot groups ever CAT(𝜅) spaces?

Let $G$ be a free Carnot group of homogeneous dimension $d$, equipped with the Carnot–Carathéodory metric. Is $(G,d)$ ever $\operatorname{CAT}(\kappa)$ for some $\kappa\in \mathbb{R}$?

4
votes

1
answer

136
views

### Are geometric actions on CAT(0) spaces with isolated flats minimal on the boundary?

Suppose $X$ is a $CAT(0)$ space with isolated flats, $\partial X$ its visual boundary and $G$ acts properly discontinuously amd cocompactly on $X$. Must the $G$ action on $\partial X$ have a dense ...

3
votes

0
answers

151
views

### When do Polish spaces admit complete metric making them $\mathrm{CAT}(\kappa)$?

Question
$\DeclareMathOperator\CAT{CAT}$Let $X$ be a Polish space. When are there known conditions under which $X$'s topology can be metrized by a metric $d$ such that $(X,d)$ is a:
$\CAT(\kappa)$ ...

9
votes

0
answers

430
views

### Volume of a geodesic ball in $\operatorname{SL}(n) / {\operatorname{SO}(n)}$?

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}$Crossposted on MSE: https://math.stackexchange.com/questions/4261809/volume-of-a-geodesic-ball-in-sln-son
Question: What is the volume of a ...

1
vote

1
answer

98
views

### $L^p$-barycenters via continuous selectors

Let $\mathbb{P}$ be a Borel probability measure on $\mathbb{R}^n$ with $\int_{x \in \mathbb{R}^n} \|x\|^p\mathbb{P}(dx)<\infty$. For which $p\in [1,\infty)$ (other than $p=2$) is the following ...

0
votes

0
answers

81
views

### Gromov–Hausdorff closure of non-positively curved graphs

Setup:
Let $\Gamma$ be the set of non-positively curved weighted connected graphs, with finitely many points, which are isometrically embedded in $\mathbb{R}^n$; for some $n\in \mathbb{N}$;$n\geq 2$. ...

2
votes

0
answers

56
views

### Is there always a purely real representative for a metrized solvable Lie group?

Alekseevski proves for Heintze groups (a special class of solvable Lie groups) that any such group admits a (left-invariant) metric which is isometric to a purely real Heintze group (again equipped ...

4
votes

1
answer

502
views

### Sectional curvature of the manifold of symmetric positive definite matrices

I am interested in the sectional curvatures of the manifold of symmetric positive definite $n \times n$ matrices with the affine metric and more precisely in a tight lower bound. It's fairly well ...

10
votes

0
answers

208
views

### Does a rank 1 CAT(0) space with a proper cocompact group action contain a zero width axis?

A geodesic in a proper CAT(0) space is said to be rank 1 if it does not bound a flat half-plane and zero-width if it does not bound a flat strip of any width.
Let $X$ be a geodesically complete CAT(0) ...

3
votes

1
answer

123
views

### 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 ...

0
votes

0
answers

47
views

### how to normalize curvature to be between -1 and -1/4

On paper "Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one", lemma 8.1, when the author tried to construct finite energy map from Quaternion hyperbolic ...

-1
votes

1
answer

114
views

### Horospherical distance in CAT($-1$) space

In $\mathbb{H}^n$, equipped with its hyperbolic metric of constant curvature $-1$, if we have two points $p,q$ on a common horosphere $\partial S$, then $$d_{\mathbb{H}}(p,q) = 2\sinh^{-1} (d_{\...

4
votes

0
answers

74
views

### Proximal isometries in CAT($-1$) metric space

Let $X$ be a rank $1$ symmetric space of non-compact type and $G$ its isometry group. $G$ is a semisimple linear algebraic Lie group of non-compact type with trivial center. Let $\rho$ be a ...

6
votes

1
answer

216
views

### Is the completion of a CAT(0) open ball a closed ball?

It is well-known that the completion of a metric space which is homeomorphic to a ball can be very wild; in fact, I think, every compact manifold is the closure of an open ball!
But CAT(0) spaces are ...

4
votes

1
answer

579
views

### The midpoint geodesic

Let $(M,g)$ be a complete simply connected Riemannian manifold with non-positive curvature. Because of the Hopf-Rinow theorem, any two points are connected by a geodesic segment.
Pick three distinct ...

9
votes

0
answers

460
views

### When does a CAT(0) group contain a rank one isometry

Let $G$ be a CAT(0) group which acts on the CAT(0) space $X$ properly and cocompactly via isometry. Let $g \in G$ be a hyperbolic isometry of $X$. Then $g$ is called $\textbf{rank one}$ if no axis of $...

3
votes

1
answer

113
views

### Billera Tree Space

I am studying the tree space of Billera and I do not really understand why it is an Hadamard Space. I have already read L. Billera, S. Holmes, K. Vogtmann, Geometry of the space of phylogenetic trees, ...

5
votes

1
answer

128
views

### Convex embedding with a positivity condition

We have a $n$-dimensional hypersurface $\Sigma$ embedded in the Euclidean $(n+1)$-space $\mathbb{R}^{n+1}$. We know that $\Sigma$ is compact without boundary, convex (not necessarly strictly convex), ...

7
votes

1
answer

279
views

### Convexity of Isoperimetric Domains

I am interested in what is known about the convexity of isoperimetric domains in compact Cartan-Hadamard manifolds (Riemannian manifolds that are complete and simply-connected and have non-positive ...

1
vote

1
answer

158
views

### Understanding the definition of an F-connected simplicial complex

I'm reading the classic paper "Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one" by Gromov-Schoen. In Section 6, they define the notion of F-connectedness ...

10
votes

1
answer

458
views

### Is the center of gravity in a CAT(0) space contained in the convex hull?

In reading Greg Kuperberg's partial answer to this question Convex hull in CAT(0) ,
I started wondering if the center of gravity is always contained in the closed convex hull.
More precisely, given $...

6
votes

0
answers

142
views

### Convex hulls of quasi-convex sets in proper CAT(0) spaces

Let $A$ be a quasi-convex set in some proper CAT(0) space, $X$, and let $\mbox{Hull}(A)$ be the intersection of all convex sets containing A. Can we conclude that $\mbox{Hull}(A)$ is in some bounded ...

12
votes

1
answer

308
views

### What are the extremal CAT(0) metrics?

(Split off from Does every CAT(0) space embed in a measurable integral of $\mathbb{R}$-trees? )
Fix an integer $k \ge 2$, and let
$MC0_k \subset \mathbb{R}^{\binom{k}{2}}$ be the set of possible ...

14
votes

3
answers

731
views

### Does every CAT(0) space embed in a measurable integral of $\mathbb{R}$-trees?

Question 1. Does every CAT(0) space embed isometrically inside an integral of $\mathbb{R}$-trees?
Here an integral of $\mathbb{R}$ trees means the set of functions from a measure space $\mathcal{F}$ ...

4
votes

1
answer

158
views

### Geodesic comparison in Hadamard space

Let $X$ be a hadamard space and $\gamma_1, \gamma_2 \colon \mathbb{R}\rightarrow X$ be two geodesics. Part 2 of Coroallary 2.5 in
http://www.iam.uni-bonn.de/fileadmin/WT/Inhalt/people/Karl-...

5
votes

0
answers

142
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 $\...

15
votes

3
answers

694
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 (...

6
votes

0
answers

259
views

### Negative curvature in the middle of $R^{3}$

What's a simple example of metric on $R^{3}$ which has negative scalar curvature inside of a (limited) set N, and is equal to the standard (Euclidean) metric outside?
Basically, I am asking for a ...

4
votes

1
answer

526
views

### Volume of a geodesic simplex on a manifold of non-positive curvature.

Let $M$ be a simply connected manifold that admits a metric of non-positive curvature. For example take $\mathbb{R}^k\times \mathbb{H}^n$. Take $m+1$ points $x_0$, $x_1, \ldots$ $x_m$, $m>k+1$ and ...