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.



Simply connected manifolds with dense geodesics on the tangent bundle
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](http://www.ams.org/mathscinet-getitem?mr=1486942). 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](https://mathoverflow.net/questions/127530/surfaces-filled-densely-by-a-geodesic) 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](https://mathoverflow.net/a/127552/5701) to that question references a 2004 paper by Donnay and Pugh that builds on the Burns-Donnay construction from the paper above.


