Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 6 (2010), 005, 8 pages      arXiv:1001.1994      http://dx.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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