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

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 2n(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 Ivanov– Petrova manifolds, curvature-Ricci commuting (indeed, semi-symmetric) and P-spaces, and that ε-spaces are Ivanov–Petrova and curvature-curvature commuting manifolds.