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

Giovanni Calvaruso ; Eduardo García-Río

Symmetry, integrability and geometry: methods and applications, Tome 6 (2010)

Résumé

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 $\frac12 n(n-1)+1$, 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.

Détail

Keywords: Lorentzian manifolds; skew-symmetric curvature operator; Jacobi, Szabó and skew-symmetric curvature operators; commuting curvature operators; IP manifolds; $\mathcal C$-spaces and $\mathcal P$-spaces.

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