Einstein-like metrics on three-dimensional homogeneous Lorentzian manifolds

Abstract

We completely classify three-dimensional homogeneous Lorentzian manifolds, equipped with Einstein-like metrics. Similarly to the Riemannian case (E. Abbena et al., Simon Stevin Quart J Pure Appl Math 66:173-182, 1992), if (M, g) is a three-dimensional homogeneous Lorentzian manifold, the Ricci tensor of (M, g) being cyclic-parallel (respectively, a Codazzi tensor) is related to natural reductivity (respectively, symmetry) of (M, g). However, some exceptional examples arise.

Homogeneity of Lorentzian three-manifolds with recurrent curvature

“…Further observe that all non-conformally flat left-invariant metrics on threedimensional Lie groups (see, for example the discussion in [8], [19]) with nilpotent Ricci operators are locally isometric to the manifold N b .…”

On the Ricci operator of locally homogeneous Lorentzian 3-manifolds

“…The curvature of all three-dimensional Lorentzian Lie groups was described in [5]. In particular, for all Lie algebras g 1 − g 7 , the Ricci components ( ) with respect to the pseudo-orthonormal frame field for which (4)- (10) …”

“…are the only possibly non-vanishing components of the Levi Civita connection (we refer to [5,6] for more details). Notice that structure constants α β γ uniquely determine A B C and conversely, by the Koszul formula [14] it follows Proof.…”

Curvature Properties of Four-Dimensional Generalized Symmetric Spaces

“…In Lorentzian settings, Einstein-like metrics have been studied in threedimensional Lorentzian manifolds admitting a parallel null vector field [6] and in three-dimensional homogeneous Lorentzian three-manifolds [4].…”

“…The curvature of all three-dimensional homogeneous Lorentzian manifolds was studied by the first author in [4], the corresponding study in the Riemannian framework having been made by Milnor in [15] and by Abbena, Garbiero and Vanhecke in [1]. Here, after describing the curvature of generalized pseudo-Riemannian spaces in dimension four, as an immediate application we classify Einstein-like metrics on generalized symmetric spaces.…”