arXiv subjects
Publications and source records attributed to Benjamin Olea.
Given a null hypersurface $L$ of a Lorentzian manifold, we construct a Riemannian metric $\widetilde{g}$ on it from a fixed transverse vector field $ζ$. We study the relationship between the ambient Lorentzian manifold, the Riemannian manifold $(L,\widetilde{g})$ and the vector field $ζ$. As an application, we prove some new results on null hypersurfaces, as well as known ones, using Riemannian techniques.
Given a Lorentzian manifold $(M,g_L)$ and a timelike unitary vector field $E$, we can construct the Riemannian metric $g_R=g_L+2ω\otimesω$, being $ω$ the metrically equivalent one form to $E$. We relate the curvature of both metrics, especially in the case of $E$ being Killing or closed, and we use the relations obtained to give some results about $(M,g_L)$.
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.
New splitting theorems in a semi-Riemannian manifold which admits an irrotational vector field (not necessarily a gradient) with some suitable properties are obtained. According to the extras hypothesis assumed on the vector field, we can get twisted, warped or direct decompositions. Some applications to Lorentzian manifold are shown and also $\mathbf{S}^{1}\times L$ type decomposition is treated.