This note proves the geodesic completeness of any compact man-ifold endowed with a linear connection such that the closure of its holonomy group is compact.

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. In this paper, all manifolds are assumed to be smooth, connected and without boundary. 2 By the holonomy of (M, g) we mean the group Holx(M, g) := {Pγ | γ loop in x} ⊂ O(TxM, gx) of parallel displacements along loops closed in x ∈ M.arXiv:1311.6723v3 [math.DG] 24 Oct 2014 3 We define the first Betti number of any manifold M to be the rank of H 1 (M, R). 4 Essentially his assumption on the light-like vector field V was weaker. Namely he just assumed V to be recurrent, i.e. such that ∇ g V = ω ⊗ V , with ω ∈ Ω 1 (M) and ker ω = V ⊥ . 5 Note that by [Con74], all leaves are diffeomorphic and either dense or closed. Indeed, a timeorientable indecomposable, non-irreducible Lorentzian manifold is obviously transversally parallelizable, see e.g. [Lär11, Lemma 2.47].

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