SIGMA 6 (2010), 005, 8 pages      arXiv:1001.1994      https://doi.org/10.3842/SIGMA.2010.005

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

Giovanni Calvaruso ^a and Eduardo García-Río ^b
a) Dipartimento di Matematica "E. De Giorgi", Università del Salento, Lecce, Italy
^b) Faculty of Mathematics, University of Santiago de Compostela, 15782 Santiago de Compostela, Spain

Received October 01, 2009, in final form January 07, 2010; Published online January 12, 2010

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.

Key words: Lorentzian manifolds; skew-symmetric curvature operator; Jacobi, Szabó and skew-symmetric curvature operators; commuting curvature operators; IP manifolds; C-spaces and P-spaces.

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