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On Lorentzian manifolds with highest first Betti number

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.