  We consider foliations of the whole three dimensional hyperbolic space H^3 by
oriented geodesics. Let L be the space of all the oriented geodesics of H^3,
which is a four dimensional manifold carrying two canonical pseudo-Riemannian
metrics of signature (2,2). We characterize, in terms of these geometries of L,
the subsets M in L that determine foliations of H^3. We describe in a similar
way some distinguished types of geodesic foliations of H^3, regarding to which
extent they are in some sense trivial in some directions: On the one hand,
foliations whose leaves do not lie in a totally geodesic surface, not even at
the infinitesimal level. On the other hand, those for which the forward and
backward Gauss maps are local diffeomorphisms.
  The subject of this article is within the framework of foliations by
congruent submanifolds, and follows the spirit of the paper by Gluck and Warner
where they understand the infinite dimensional manifold of all the great circle
foliations of the three sphere.
