Global decomposition of a Lorentzian manifold as a Generalized Robertson–Walker space

Abstract

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 nontrivial complete Lorentzian manifolds with more than one GRW decomposition. Moreover, we show that the Friedmann Cosmological Models admit an unique GRW decomposition, even locally.