Triangle inequality for geodesic distance on groups

Question

Consider $G$ a compact Lie group. I am trying to understand the properties of the geodesic distances on $G$. In particular, if we denote by $|x|$ a geodesic distance induced by the bi-invariant Riemannian metric on $G$ from $x\in G$ to the identity, is there an analogue of the triangle inequality? By that I mean $$|y^{-1}x|\leq C(|x|+|y|)$$ for some $C>0$ and all $x,y\in G$. Also, is it true that $|x^{-1}|=|x|$ for all $x\in G$? These seem like elementary questions, but I can't find any book or reference that addresses them.

Answer 1

Saying "geodesic distance" is not sufficient to characterize the distance function on a Lie group with, say, a bi-invariant metric (note that the latter may not be unique), because the geodesic joining a pair of points may not be unique, and furthermore it may not be minimizing. The standard approach is to define the geodesic distance as the least length of a path joining the two points. Then of course the triangle inequality is automatically satisfied, by composing the two minimizing paths. Note that a minimizing path is necessarily a geodesic (by a variational argument).