Consider a Riemannian manifold $\Sigma$ of dimension two homeomorphic to a torus. When is there a non-closed geodesic on $\Sigma$ which does not intersect itself — are there reasonable necessary or sufficient conditions for this?

(This is a reference request. My guess is that this question does not have an amazing answer, but I am curious if it was considered in the literature. By the way, a flat torus obviously has this property, but I am interested in a more general setting.)