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

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

2010-01-12

arxiv.org/abs/1001.1994v1

SIGMA 6 (2010), 005, 8 pages

Mathematics

Differential Geometry

Scientific paper

10.3842/SIGMA.2010.005

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.

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