Four-dimensional pseudo-Riemannian g.o. spaces and manifolds

A g.o. manifold is a homogeneous pseudo-Riemannian manifold whose geodesics are all homogeneous, that is, they are orbits of a one-parameter group of isometries. A g.o. space is a realization of a homogeneous pseudo-Riemannian manifold (M, g) as a coset space M = G/H, such that all the geodesics are homogeneous. We prove that apart from the already classified non-reductive examples (Calvaruso et al., 2015), any four-dimensional pseudo-Riemannian g.o. manifold is naturally reductive. To obtain this result, we shall also provide a complete description up to isometries of four-dimensional pseudo-Riemannian g.o. spaces, and show explicit realizations of the four-dimensional pseudo-Riemannian naturally reductive spaces classified in Batat et al. (2015). (C) 2018 Elsevier B.V. All rights reserved.