# Questions tagged [non-positive-curvature]

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### 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: $\CAT(\kappa)$ ...
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### 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 ...
87 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 ...
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### 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 $n\in \mathbb{N}$;$n\geq 2$. ...
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### Convex separation in CAT(0) spaces

Are there any convex separation theorems for CAT(0) spaces? Particularly, can one separate two disjoint convex sets by a horosphere? In Riemannian manifolds, I have seen some results on convex ...
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### 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 ...
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### 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 ...
136 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) ...
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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 ...
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### 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 ...
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### 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_{\...
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### 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 ...
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### 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 ...
469 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 ...
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### 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 ...
213 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 ...
629 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 $\mathcal{F}$ ...
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-...