Structure of second-order symmetric Lorentzian manifolds

Abstract

Second-order symmetric Lorentzian spaces , that is to say, Lorentzian manifolds with vanishing second derivative \nabla \nabla R\equiv 0 of the curvature tensor R , are characterized by several geometric properties, and explicitly presented. Locally, they are a product M=M_1\times M_2 where each factor is uniquely determined as follows: M_2 is a Riemannian symmetric space and M_1 is either a constant-curvature Lorentzian space or a definite type of plane wave generalizing the Cahen–Wallach family. In the proper case (i.e., \nabla R \neq 0 at some point), the curvature tensor turns out to be described by some local affine function which characterizes a globally defined {parallel lightlike direction}. As a consequence, the corresponding global classification is obtained, namely: any complete second-order symmetric space admits as universal covering such a product M_1\times M_2 . From the technical point of view, a direct analysis of the second-symmetry partial differential equations is carried out leading to several results of independent interest relative to spaces with a parallel lightlike vector field—the so-called Brinkmann spaces.