A unit speed geodesic $\gamma:I\to M$ on a Riemannian manifold $M$ can be lifted to a curve $\sigma=(\gamma,\gamma')$ on $SM$, the unit sphere bundle (the unit tangent bundle) of $M$. Let us say that the geodesic $\gamma$ is dense on the tangent bundle if the trace $\sigma(I)$ of $\sigma$ is dense on $SM$. The existence of a dense geodesic in this sense can be regarded as a form of ergodicity (or mixing) of the geodesic flow.
Are there simply connected Riemannian manifolds for which a geodesic is dense on the tangent bundle? Does the answer depend on dimension?
I know that such geodesics exist on all compact Anosov manifolds. This includes negatively curved closed surfaces, but they have genus $\geq2$ and are not simply connected.
I failed to find an example or proof of non-existence.