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



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> Is there a non vanishing vector field $X$ on $M$ such that all trajectories of $X$ are geodesics, after a possible reparametrization?
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The question is somehow a converse question to the following question:


[Limit cycles as closed geodesics(in negatively or positively curved space)](https://mathoverflow.net/questions/160945/limit-cycles-as-closed-geodesicsin-negatively-or-positively-curved-space)



A non vanishing vector field compatible to a Riemannian metric
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**](http://dx.doi.org/10.1007/BF01232666), Invent. Math. 114, No.1, 193-206 (1993). [ZBL0789.57019](https://zbmath.org/?q=an:0789.57019).


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.


