Stable geodesics on a K3 surface Abstract: This thesis studies stable closed geodesics on a Ricci flat Kummer K3 surface. The motivation for counting stable closed geodesics is explained and put into the context of comparison geometry. Building on estimates due to R. Kobayashi restrictions on stable closed geodesics on arbitrary Kummer K3s are derived. Then in a complementary effort isometry and special Lagrangian arguments are put forward to construct concrete stable closed geodesics on a highly symmetric Kummer K3. This gives a partial affirmative answer to a conjecture put forward by P. Gao and M. Douglas namely that any Ricci flat compact Calabi Yau manifold will always carry stable closed geodesics