**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:

- $\CAT(\kappa)$ space with $\kappa>0$,
- $\CAT(0)$ space,
- $\CAT(\kappa)$ 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.

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