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 ε-spaces exhaust the class of n-dimensional Lorentzian manifolds admitting a group of isometries of dimension at least ½n(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.

Keywords

• Lorentzian manifolds
• skew-symmetric curvature operator
• Jacobi
• Szabó and skew-symmetric curvature operators
• commuting curvature operators
• IP manifolds
• C-spaces and P-spaces