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?*

**Addendum.** As Misha remarks in his answer, it is easy to construct to metrics on any closed surface that do not carry a transitive geodesic. However, the metrics I'm interested in have the additional property that *the lifted metric on the universal cover has no trapped geodesics.* In other words, no geodesic stays forever in a compact subset of the open unit disc.