# 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.

• I think Joseph O'Rourke already asked this some time ago and the answer was positive, such metrics exist on 2-spheres. Oct 4 '16 at 22:51
• I am not sure but I also think there are examples on the sphere. I'd appreciate it if you could clarify what you exactly mean by "trace of $\sigma$" and the connection with ergodicity.
– vap
Oct 4 '16 at 23:06
• @vap, I clarified the question a bit. Oct 5 '16 at 5:34
• @Misha, Joseph's question is related and useful, but it only studies embedded surfaces instead of all manifolds. Also, he only requires density on the base $M$, not on the sphere bundle $SM$. (For example, the flat torus has geodesics dense on the base, but none dense on $SM$.) Oct 5 '16 at 5:37
• Link to my earlier question: Surfaces filled densely by a geodesic. Oct 5 '16 at 14:31

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.
• Thanks! By a dense geodesic do you mean one that is dense on $SM$ or only on $M$? The second does not imply the first (example: flat torus). I think this is a sufficient answer for embedded surfaces, and I hope there are similar results on more general manifolds. Oct 5 '16 at 5:42
• Right, I should have been more explicit - I mean the stronger property, density on $SM$. I edited the answer to reflect this. Oct 5 '16 at 12:48