Homogeneous structures on three-dimensional Lorentzian manifolds

Four-Dimensional Pseudo-Riemannian Generalized Symmetric Spaces Which are Algebraic Ricci Solitons

“…Now, for instance, a three-dimensional homogeneous Lorentzian manifold is necessarily reductive. This was proved in [7] and it also follows independently from the classification obtained by Calvaruso in [2]. Furthermore, the existence of non-reductive four-dimensional pseudo-Riemannian homogeneous manifolds was proven in [7].…”

“…Now, consider a non-symmetric simply-connected four-dimensional generalized symmetric space (M = G/H, g) of type A and signature (2,2). Then, taking into account the results of [4] and [5], the Lie algebra g of the Lie group G may be decomposed into the vector space direct sum g = h ⊕ m where h denotes the Lie algebra of H and m is a vector subspace of g.…”

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].…”

“…The aim of this paper is to answer the question above. As the first author proved in [4], a locally homogeneous Lorentzian 3-manifold is either locally symmetric or locally isometric to some three-dimensional Lie group equipped with a leftinvariant Lorentzian metric. Because of this result, the problem essentially reduces to determining precisely which Segre types occur for all three-dimensional Lorentzian Lie groups and all Lorentzian symmetric 3-spaces.…”

Examples of signature (2, 2) manifolds with commuting curvature operators

“…Lorentzian Walker metrics have been studied extensively in the physics literature since they constitute the background metric of the pp-wave models ( [2,25,26,30] to list a few of the many possible references; the literature is a vast one); a pp-wave spacetime admits a covariantly constant null vector field U and therefore it is trivially recurrent (i.e., ∇U = ω ⊗ U for some oneform ω). Lorentzian Walker metrics present many specific features both from the physical and geometric viewpoints [12,14,27,32]. We also refer to related work of Hall [21] and of Hall and da Costa [22] for generalized Lorentzian Walker manifolds (i.e.…”