Let $S$ be a smooth closed (i.e. compact without boundary) surface. A geodesic lamination on $S$ is a nonempty closed subset of $S$ which is a disjoint union of geodesics. Suppose $\alpha$ is a geodesic lamination on $S$ such that for every $p\in S$ there is a geodesic of $\alpha$ containing $p$.
Can we construct a smooth vector field in $S$ using $\alpha$?
I was trying to use the tangents of the geodesics but I am not sure how to give the orientation on them.
P.S.: If necessary assume that $S$ is a hyperbolic surface.
Thanks in advance.