Let $X^3$ be a simply connected piecewise linear(PL-) manifold with non-positive curvature. I am curious about that can we know about the singular set of $X^3$ explicitly. For example, what is the singular set of the hyperbolic space $H^3$? Is it depend on triangulation of $H^3$?

In my best knowledge, the definition of singular set(Cao and Escobar's paper, Definition 1.10) is as follows.

Let $\tau$ be a given triangulation of $X^3$ and if $k$-dimensional simplex $\sigma^k \subset X^3$ is said to be singular if $Link(\sigma^k, X^3)$ is not isometric to the unit sphere $S^{n-k-1}(1)$.