# 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. – Misha 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. – Joonas Ilmavirta 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 $SM$.) – Joonas Ilmavirta Oct 5 '16 at 5:37
• Link to my earlier question: Surfaces filled densely by a geodesic. – Joseph O'Rourke Oct 5 '16 at 14:31

## 1 Answer

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.

• 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. – Joonas Ilmavirta 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. – Vaughn Climenhaga Oct 5 '16 at 12:48