A well-known result of Hedlund and Morse states that if a Riemannian metric on a closed surface of genus $g > 1$ has no conjugate points, then it carries transitive geodesics (i.e., geodesics whose velocity vectors are dense in the unit tangent bundle).

Are there Riemannian metrics on closed surfaces of genus $g > 1$ that do not carry a transitive geodesic and if so what is the weakest condition known under which the existence of transitive geodesics has been proved?