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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.

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    $\begingroup$ Is this question useful without a link to the definition of CAT($\kappa$) ? $\endgroup$
    – Gerald Edgar
    Commented Dec 5, 2021 at 20:29
  • $\begingroup$ @GeraldEdgar I added some definitions, backgrounds and links; I hope this helps my question reach a wider audience. $\endgroup$
    – Carlos_Petterson
    Commented Dec 5, 2021 at 22:12
  • $\begingroup$ 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. $\endgroup$
    – Moishe Kohan
    Commented Dec 6, 2021 at 11:22
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    $\begingroup$ @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. $\endgroup$
    – Moishe Kohan
    Commented Dec 6, 2021 at 16:48
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    $\begingroup$ @Carl_Petterson: No. $\endgroup$
    – Moishe Kohan
    Commented Dec 22, 2021 at 16:54

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