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

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 $\epsilon$-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 $\mathcal{P}$-spaces, and that $\epsilon$-spaces are Ivanov-Petrova and curvature-curvature commuting manifolds.

References in corpus (5)

• Natural Intrinsic Geometrical Symmetries
• Pseudo-Riemannian Jacobi-Videv Manifolds
• Algebraic curvature tensors for indefinite metrics whose skew-symmetric curvature operator has constant Jordan normal form
• Stanilov-Tsankov-Videv Theory
• The classification of simple Jacobi--Ricci commuting algebraic curvature tensors