# Two questions on “foliation by geodesics”

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.

No to the first question. Let $M$ be a Riemannian $2$-manifold whose universal cover is the hyperbolic plane $H$, and whose fundamental group is not cyclic. For any foliation of $H$ by geodesics (lines), it seems to me that the endpoints of these lines on the boundary circle of $H$ will fill up the whole circle except for two points. These two points will have to be preserved by deck transformations, so any discrete group of isometries of $H$ that preserves the foliation must be cyclic.
• @Thiku: The $M$ in this answer is an open manifold. – Lee Mosher Jun 16 '18 at 17:59
• To be honest I can not understand this answer. For example consider $M=H\setminus \{p,q\}$. Then is not $H$ the universal covering space of $M$? But M admit a foliation by geodesic. The vertical foliation. Am I mistaken? – Ali Taghavi Jun 17 '18 at 10:54
• This $M$ is not complete, as it would have to be if its universal covering space were to be metrically isomorphic to $H$. – Tom Goodwillie Jun 17 '18 at 11:44