TITLE: Two questions on "foliation by geodesics"
QUESTION [5 upvotes]: 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.

REPLY [7 votes]: 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.