It is known that hyperbolic spaces admit geodesic foliations (that is, a smooth unit vector field all of whose integral curves are geodesics, see https://arxiv.org/abs/1411.6700).
Suppose a complete open Riemannian manifold $M$ has a totally geodesic foliation $\mathcal F$ by $n$-dimensional hyperbolic spaces. Given a complete geodesic in a leaf of $\mathcal F$, can we extend it to a geodesic foliation of $M$ which is everywhere tangent to the leaves of $\mathcal F$?
This is the question I am interested in. The more general question would be: given a foliation $\mathcal F$ of $M$ and a foliation $\mathcal F_L$ of a leaf $L$ of $\mathcal F$, under which conditions can we extend $\mathcal F_L$ to a foliation $\hat{\mathcal F_L}$ of $M$ so that $\hat{\mathcal F_L}$ is a sub-foliation of $\mathcal F$?
