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Assume that $(M, g)$ is a connected Riemannian manifold which is either open or is compact with zero Euler characteristic.

Is there a non vanishing vector field $X$ on $M$ such that all trajectories of $X$ are geodesics, after a possible reparametrization?

The question is somehow a converse question to the following question:

Limit cycles as closed geodesics(in negatively or positively curved space)

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If that would be the case, then $M$ would have a foliation by geodesics. The article

Zeghib, A., On continuous geodesic foliations of hyperbolic manifolds, Invent. Math. 114, No.1, 193-206 (1993). ZBL0789.57019.

shows that for a closed hyperbolic $3$-manifold there is even no continuous foliation.

A simpler example (with a non-compact manifold) is a metric on $\mathbb{R}^2$ obtained by smoothing the vertex of a cone with the angle $< \pi$ at the vertex. There, geodesics far away from the vertex are self-intersecting.

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  • $\begingroup$ Thank you for your very interesting answer and the paper by Abdel Ghani Zeghib. $\endgroup$
    – Ali Taghavi
    Commented May 10, 2017 at 11:23

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