Simply connected manifolds with dense geodesics on the tangent bundle 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.
 A: Burns and Donnay proved that every surface (including a sphere) admits a Riemannian metric that makes the geodesic flow ergodic with respect to Liouville measure, and hence topologically transitive (there is some $v\in SM$ whose orbit under the geodesic flow is dense in $SM$, in other words, the corresponding geodesic is dense in $SM$ in the sense you describe in the question):  see
"Embedded surfaces with ergodic geodesic flows", Internat. J. Bifur. Chaos Appl. Sci. Engrg. 7 (1997), no. 7, 1509–1527.  The abstract to that paper reads as follows:

Following ideas of Osserman, Ballmann and Katok, we construct smooth
  surfaces with ergodic, and indeed Bernoulli, geodesic flow that are
  isometrically embedded in $\mathbb{R}^3$. These surfaces can have
  arbitrary genus and can be made analytic.

As Misha's comment points out, this is related to an earlier question of Joseph O'Rourke, although that one seems to be about density on the surface, rather than in the unit tangent bundle.  Misha's answer to that question references a 2004 paper by Donnay and Pugh that builds on the Burns-Donnay construction from the paper above.
