# Structure of second-order symmetric Lorentzian manifolds

### Miguel Sánchez Caja

Universidad de Granada, Spain### Oihane F. Blanco

Universidad de Granada, Spain### José M. M. Senovilla

Universidad del Pais Vasco, Bilbao, Spain

## 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.

## Cite this article

Miguel Sánchez Caja, Oihane F. Blanco, José M. M. Senovilla, Structure of second-order symmetric Lorentzian manifolds. J. Eur. Math. Soc. 15 (2013), no. 2, pp. 595–634

DOI 10.4171/JEMS/368