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

[摘要] 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 ½ 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 Ivanov-Petrova manifolds, curvature-Ricci commuting (indeed, semi-symmetric) and P -spaces, and that ε-spaces are Ivanov-Petrova and curvature-curvature commuting manifolds.

[关键词] Lorentzian manifolds;skew-symmetric curvature operator;Jacobi;Szab´o and skew-symmetric curvature operators;commuting curvature operators;IP manifolds;Cspaces and P-spaces