Structure of second-order symmetric Lorentzian manifolds

Second-order symmetric Lorentzian spaces, that is, Lorentzian manifolds with vanishing second derivative del del R equivalent to 0 of the curvature tensor R, are characterized by several geometric properties, and explicitly presented. Locally, they are a product M = M-1 x 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., del R not equal 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 x 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 on spaces with a parallel lightlike vector field, the so-called Brinkmann spaces.