Three-dimensional Lorentz manifolds admitting a parallel null vector field

On the Ricci operator of locally homogeneous Lorentzian 3-manifolds

“…In [4], the first author completely classified three-dimensional Lorentzian symmetric spaces, showing that besides Lorentzian space forms and the product between a real line and a pseudo-Riemannian surface of constant Gaussian curvature, the only possible case is the one of a Lorentzian manifold admitting a parallel null vector field. Locally symmetric Lorentzian 3-spaces having a parallel null vector field were described in [8]. They admit local coordinates ( ) such that, with respect to the local frame field {(…”

“…If (M = G ) is a three-dimensional Lorentzian Lie group having Lie algebra g 5 , then by (8) we get that the components of the Ricci operator Q are …”

Semi-symmetric Lorentzian Three-manifolds Admitting a Parallel Degenerate Line Field

“…Lorentzian three-manifolds admitting a parallel degenerate line field have been studied in the fundamental paper [9]. These manifolds are described in terms of a suitable system of local coordinates (t, x, y) and form a large class, depending on an arbitrary three-variables function f (t, x, y).…”

Pseudo-Riemannian Jacobi–videv Manifolds

“…One has the following family of examples which are Jacobi-Videv and not Einstein. Manifolds in this family have been studied previously in different contexts, see for example [9,10,11,12,13]; we also refer to [14,15] Definition 1.1. Let k ≥ 1, let ℓ ≥ 1, and m = 2k + ℓ.…”