Let $M$ be a smooth closed positively curved surface in Euclidean space $R^3$, $T$ be a geodesic triangulation of $M$, and $E$ be the edge graph of the convex hull of vertices of $T$. 

It is easy to see that $T$ may not always be isomorphic to $E$, because $E$ is determined by the vertices of $T$, while a set of points in $M$ may form vertices of different triangulations of $M$. For instance, one might take any pairs of adjacent acute triangles in $T$ and replace their common edge with a geodesic connecting the vertices of these triangles not on that edge.

Are there some conditions which one may impose on $T$, as a subset of $M$ (e.g., involving edge lengths, angles, or curvatures of $M$) in order to ensure that $T$ is isomorphic to $E$?