# 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:

1. $$\CAT(\kappa)$$ space with $$\kappa>0$$,
2. $$\CAT(0)$$ space,
3. $$\CAT(\kappa)$$ space with $$\kappa<0$$.

For definitions/background see below...

## What I Known

CAT(0) Case:

This question is related to the following posts: (MSE - Answered) When $$X$$ admits a metric making it into a $$\CAT(0)$$ space: The summary of that answer is:

• If $$X$$ is a contractable topological manifold with boundary then its interior (as a manifold) admits a complete $$CAT(0)$$ metric see this reference. The linked article also states that if $$X$$ is PL-manifold and tame then such a metric exists.

CAT(1) Case: This open (and old) MO post: (MO) When $$X$$ admits a metric making into a $$\CAT(1)$$ space gives some positive answers when $$X$$ is polyhedral (but the linked article is in Russian...)

## Background

In what follows, we use $$M_{\kappa}$$ to denote the simply connected Riemannian manifold of dimension $$2$$ with constant curvature $$\kappa$$. For example, $$M_0$$ is the Euclidean plane, $$M_1$$ is the surface of the unit sphere, and $$M_{-1}$$ is the hyperbolic plane.
Let $$D_{\kappa}:=\operatorname{diam}(M_{\kappa}) = \begin{cases} \pi/\sqrt{\kappa}&:\, \kappa >0\\ \infty &:\, \kappa\leq 0 \end{cases}$$

Definition: Comparison Triangle. Let $$(X,d)$$ be a geodesic space, $$x_1,x_2,x_3\in X$$ and let $$T$$ be a triangle in $$X$$ whose edges are geodesics connecting $$x_1$$ to $$x_2$$, $$x_2$$ to $$x_3$$, and $$x_1$$ to $$x_3$$. A comparison triangle $$T^{\star}$$ in $$M_{\kappa}$$ for $$T$$ is a triangle in $$M_{\kappa}$$ with vertices $$y_1,y_2,y_3$$ such that $$d(x_i,x_j)=d_{M_{\kappa}}(y_i,y_j)$$ for $$i,j=1,2,3$$.

Definition $$\operatorname{CAT}(\kappa)$$-space A geodesic metric space $$(X,d)$$ is if $$\operatorname{CAT}(\kappa)$$ if for each geodesic triangle $$T$$ in $$(X,d)$$ with perimeter at-most $$2D_{\kappa}$$ there is a comparison trianble $$T^{\star}$$ in $$M_{\kappa}$$ with sideos of the same length as those of $$T$$ and such that the distances between points in $$T$$ are no more than those in $$T^{\star}$$.

Further discussion can be found here or in Martin R. Bridson Andre Haefliger's book: Metric Spaces of non-positive curvature.

• Is this question useful without a link to the definition of CAT($\kappa$) ? Dec 5, 2021 at 20:29
• @GeraldEdgar I added some definitions, backgrounds and links; I hope this helps my question reach a wider audience. Dec 5, 2021 at 22:12
• The way it is asked, it is just a subquestion of my old MO question. A better question would have been with two parts: for CAT(-1) and for CAT(0) metrics. Dec 6, 2021 at 11:22
• @Carl_Petterson Every CAT(-1) space is a CAT(1) space: This is an easy consequence of the definition and comparison of hyperbolic and spherical triangles. Should be in Burago-Burago-Ivanov book. I will check. Dec 6, 2021 at 16:48
• @Carl_Petterson: No. Dec 22, 2021 at 16:54