Assume that $S$ is a finite 2-dimensional simplicial complex equipped with a metric $d$ such that each triangle is isometric to a plane triangle (so $(S,d)$ is a polyhedral space).
Is it possible to subdivide the triangulation of $S$ so that it will have acute triangles only?
It is well known that any triangle admits a triangulation with only acute triangels; one can see the proof in the following picture. But I do not see a way to fit such triangulations together.
Such a triangulation would be useful in the proof of Zalgaller's theorem: any n-dimensional polyhedral space admits a length-preserving piecewise linear map to $\mathbb R^n$. Well, it would help only if $n=2$, for larger $n$, we should say that a simplex is acute if it contains its circumcenter.