Title: Geodesics revisited

Abstract: Geodesics generalize the notion of straight lines to curved spaces. They are curves on a (semi)-Riemannian manifold which solve the geodesic equation, a second order ordinary differential equation. In Riemannian geometry existence and uniqueness of geodesics thus follows from the standard theory for ordinary differential equations (Picard-Lindeloef Theorem). However, this requires the Riemannian metric to be of regularity C^{1,1} or more. If the Riemannian metric has just slightly less regularity the existence of geodesics is still valid but otherwise all hell breaks loose. We will discuss some instructive but little known examples of 2-dimensional Riemannian manifolds that amongst others violate local uniqueness of geodesics and the local length-minimizing property. Some (conjectured) fixes to restore the usual behavior of geodesics also in lower regularity will be discussed at the end.