I'm trying to prove that if $\langle M,g\rangle$ is a riemannian manifold and $G = Isom(M)$ acts properly discontinuous on $M$, then a geodesic $c$ is send to another geodesic by the map $\pi: M \twoheadrightarrow M/G$ (canonical projection), where the metric in $M/G$ is given by $h = \pi_{*}g$. I'm trying to prove the result using only the affine connection, however I have no idea how to proceed.
Thanks in advance.