  In this paper we give a complete local parametric classification of the
hypersurfaces with dimension at least three of a space form that carry a
totally geodesic foliation of codimension one. A classification under the
assumption that the leaves of the foliation are complete was given in
\cite{drt} for Euclidean hypersurfaces. We prove that there exists exactly one
further class of local examples in Euclidean space, all of which have rank two.
We also extend the classification under the global assumption of completeness
of the leaves for hypersurfaces of the sphere and show that there exist plenty
of examples in hyperbolic space.
