Algebraic Properties of Curvature Operators in Lorentzian Manifolds with Large Isometry Groups

Abstract

Abstract. Together with spaces of constant sectional curvature and products of a real line with a manifold of constant curvature, the socalled Egorov spaces and ε-spaces exhaust the class of n-dimensional Lorentzian manifolds admitting a group of isometries of dimension at least 1 2 n(n − 1) + 1, for almost all values of n [Patrangenaru V., Geom. Dedicata 102 (2003), 25-33]. We shall prove that the curvature tensor of these spaces satisfy several interesting algebraic properties. In particular, we will show that Egorov spaces are IvanovPetrova manifolds, curvature-Ricci commuting (indeed, semi-symmetric) and P-spaces, and that ε-spaces are Ivanov-Petrova and curvature-curvature commuting manifolds.

Ricci solitons on Lorentzian manifolds with large isometry groups

“…Hence observe that ∂ v is a parallel null vector field and thus that Egorov spaces are Walker metrics [6], [3]. By an explicit calculation on the geodesic equations, it has been shown in [1] that Egorov spaces are geodesically complete.…”

“…In opposition to ε-spaces, Egorov spaces are not homogeneous in general. However the Ricci tensor is recurrent and so is the curvature tensor since they are locally conformally flat (see [1], [6]). …”

“…Cahen-Wallach symmetric spaces are locally conformally flat if and only if κ 1 = · · · = κ n , in which case the resulting manifold is an ε-space (we refer to [4] and [5] for more information on the geometry of Cahen-Wallach spaces and to [1] and [6] for ε-spaces).…”

“…The geometry of Egorov spaces and ε-spaces has been investigated in the literature (see for example [1], [5], [6], [7], [14] and references therein) where it is shown that both are Walker manifolds. It is worth emphasizing here that although ε-spaces are locally symmetric, Egorov spaces are not homogeneous in general.…”