  We consider Lorentzian manifolds with parallel light-like vector field V.
Being parallel and light-like, the orthogonal complement of V induces a
codimension one foliation. Assuming compactness of the leaves and non-negative
Ricci curvature on the leaves it is known that the first Betti number is
bounded by the dimension of the manifold or the leaves if the manifold is
compact or non-compact, respectively. We prove in the case of the maximality of
the first Betti number that every such Lorentzian manifold is - up to finite
cover - diffeomorphic to the torus (in the compact case) or the product of the
real line with a torus (in the non-compact case) and has very degenerate
curvature, i.e. the curvature tensor induced on the leaves is light-like.
