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

TL;DR

This paper investigates the algebraic properties of curvature tensors in Lorentzian manifolds with large isometry groups, revealing specific properties of Egorov and $\varepsilon$-spaces related to Ivanov-Petrova and commuting conditions.

Contribution

It establishes new algebraic characterizations of Egorov and $\varepsilon$-spaces within Lorentzian manifolds with large isometry groups.

Findings

Egorov spaces are Ivanov-Petrova, semi-symmetric, and $\\mathcal P$-spaces.

$\varepsilon$-spaces are Ivanov-Petrova and curvature-curvature commuting.

Curvature tensors of these spaces satisfy several algebraic properties.

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 $\varepsilon$-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 Ivanov-Petrova manifolds, curvature-Ricci commuting (indeed, semi-symmetric) and $\mathcal P$-spaces, and that $\varepsilon$-spaces are Ivanov-Petrova and curvature-curvature commuting manifolds.