Let $X$ be a Riemann surface of genus $g\ge 2$ and $q$ a non-vanishing holomorphic quadratic differential on $X$. Denote by $\mathbb{H}$ the universal cover of $X$ and by $\widetilde q$ the pullback of $q$ to $\mathbb{H}$. The space $\mathbb{H}$ endowed by the flat singular metric $|\widetilde q|$ is a Cat(0) metric space.


For every $x,y,z\in (\mathbb{H},|\widetilde q|)$ denote by $\Delta$ the corresponding geodesic triangle (i.e. the subset of $(\mathbb{H},|\widetilde q|)$ composed by $x,y,z$ and the three geodesics connecting them). It seems to me that it makes sense to talk about the interior of $\Delta$: it is the bounded region of $(\mathbb{H},|\widetilde q|)$ delimited by $\Delta$ (and of course if can by empty). Denote by $\Delta'$ the union of $\Delta$ with its interior.


Is there a nice characterization of such $\Delta'$? In particular, is it true that they are always isometric to polygons with possibly one dimensional components (it seems to me that there can not be zeroes of $\widetilde q$ in the interior of $\Delta$)?



Geodesic triangles in particular Cat(0) space
Triangles do not have singular points in their interior ⟺ curvature of any singular point is at least π. 


In this case each triangle admits a length-preserving immersion into the plane.
(The interior admits an locally isometric immersion since it is locally flat, then you can extend it to the closure of the interior in the solid triangle and then add line segments for each vertex with overlapping adjacent sides.)


The total negative turn of sides can not exceed π.
It is sufficient to conclude that the length-preserving immersion is actually an embedding.


*Does it answer you question?*


