  This paper is devoted to the study of Gibbs u-states for the geodesic flow
tangent to a foliation with negatively curved leaves. On the one hand we give
sufficient conditions for the existence of transverse invariant measures. In
particular we prove that when this foliated geodesic flow preserves a Gibbs
su-state, i.e. a measure with Lebesgue disintegration both in the stable and
unstable horospheres, then it has to be obtained by combining a transverse
invariant measure and the Liouville measure on the leaves. On the other hand we
study in detail the projections of Gibbs u-states along the unit spheres
tangent to the foliation. We show that they have Lebesgue disintegration in the
leaves and that the local densities possess an integral representation analogue
to the Poisson representation of harmonic functions.
