# A non vanishing vector field compatible to a Riemannian metric

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)

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