I learned that if we are given a $C^0$ Riemannian metric on a smooth manifold $M$, geodesics (i.e. length minimizing curves) are absolutely continuous, and if the metrics is $C^{0,\alpha}$, then the geodesics are even $C^{1, \beta}$ where $\beta = \alpha/(2-\alpha)$.

However, I did not find any results on the existence of geodesics. A priori, it could be possible that no geodesics exist at all, or am I wrong?

**Are there results that there exist "enough" geodesics in some sense, like when two points are close enough, or for almost any two points (suppose maybe that $M$ is compact)?**

A problem in considering the geodesics equation seems to be that it involves the Christoffel symbols which are not continuous. If we have a solution to the geodesic equation, is it length minimizing for two points close enough?

It is known that given a continuous metric $g$, there exists a sequence of smooth Riemannian metrics that converges to $g_n$ such that the corresponding distance functions converge to the one associated to $g$. What happens with the geodesics (i.e. solutions to the geodesics equation) in this case? Do they converge in some sense?