I would appreciate if you consider the following two questions on $1$ dimensional foliations whose leaves are geodesic.
1)Assume that $M$ is a Riemannian manifold which is either an open manifold or is a compact manifold with zero Euler characteristic. Does $M$ admit a foliation by geodesics?
2)Assume that $M$ is a Riemannian surface which admit at least one foliation by geodesics. Does there necessarily exist a foliation of $M$ by geodesics which satisfy the "Isocline Locale property"?
The Isocline local property is defined as follows:
For every $x\in M$ there is locally a geodesic $\alpha $ which is transverse to the foliation and it intersect all leaves with the same angle.