I am not well-versed in geometry relating to manifolds so pardon my lack of correct terminology.
Simply put, if I have some geometric surface (that can be found in the real physical world, such as the surface of a mountain range), and given 3 points $A, B, C$ on it.
Say I know the geodesic distance $AB$ and $AC$, now can I conclude that the geodesic distance $BC$ obeys the triangle inequality? (i.e. $AC \le AB + BC $)
Geodesic distance on surface is infimum of lengths of curves. Hence $|A-B| + |B-C|$ is sum of lengths of curves $c_i$ joining $A,\ B$ and $B,\ C$. That is, $c_1\bigcup c_2$ is also curve for $A$ and $C$.