Given a null hypersurface $L$ of a Lorentzian manifold, we construct a Riemannian metric $\widetilde{g}$ on it from a fixed transverse vector field $\zeta$. We study the relationship between the ambient Lorentzian manifold, the Riemannian manifold $(L,\widetilde{g})$ and the vector field $\zeta$. As an application, we prove some new results on null hypersurfaces, as well as known ones, using Riemannian techniques.Comment: 26 page

Generalized Robertson-Walker (GRW) spaces constitute a quite important family in Lorentzian geometry, and it is an interesting question to know whether a Lorentzian manifold can be decomposed in such a way. It is well known that the existence of a suitable vector field guaranties the local decomposition of the manifold. In this paper, we give conditions on the curvature which ensure a global decomposition and apply them to several situations where local decomposition appears naturally. We also study the uniqueness question, obtaining that the de Sitter spaces are the only non trivial complete Lorentzian manifolds with more than one GRW decomposition. Moreover, we show that the Friedmann Cosmological Models admit an unique GRW decomposition, even locally.

The results of the paper [1] remain true as stated, but there is a mistake in the proof of Proposition 2.3, namely the assertion that the direct sum decomposition (2.1) is an orthogonal decomposition. This fact has also been used in the proofs of Proposition 2.4 and Theorem 2.6. The gaps should be amended as follows. Proof of Proposition 2.3. Let V ∈ X(T M) be given byĝ(V v , X) = g(c(X), K πv) + g(v, ∇ dπ(X) K), for every X ∈ T v T M. Thus we haveĝ(V v , A v) = 1, at any v ∈ C K (M), and so D v = Span{V v , A v } is a Lorentzian vector subspace of T v T M (recall thatĝ(A, A) = 0 on C K M). One easily checks that X ∈ T v C K M if and only ifĝ(A v , X) =ĝ(V v , X) = 0. Therefore, (C K M,ĝ) is a Lorentzian manifold. The proof for spacelike fibres remains valid. Finally, [T v (C K M) p ] ⊥ = {X ∈ T v T M : c(X) = −g(v, ∇ dπ(X) K) v}, and so π restricted to C K M is a semi-Riemannian submersion.

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Abstract. We give a Lorentzian metric on the null congruence associated with a timelike conformal vector field. A Liouville type theorem is proved and a boundedness for the volume of the null congruence, analogous to a well-known Berger-Green theorem in the Riemannian case, will be derived by studying conjugate points along null geodesics. As a consequence, several classification results on certain compact Lorentzian manifolds without conjugate points on its null geodesics are obtained. Finally, several properties of null geodesics of a natural Lorentzian metric on each odd-dimensional sphere have been found.