# Journal of the European Mathematical Society

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**Volume 15, Issue 2, 2013, pp. 595–634**

**DOI: 10.4171/JEMS/368**

Structure of second-order symmetric Lorentzian manifolds

Oihane F. Blanco^{[1]}, Miguel Sánchez

^{[2]}and José M. M. Senovilla

^{[3]}(1) Universidad de Granada, Spain

(2) Universidad de Granada, Spain

(3) Universidad del Pais Vasco, Bilbao, Spain

*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.

*Keywords: *Second-order symmetric spaces, curvature conditions, Brinkmann spaces, Lorentzian symmetric spaces, plane waves, holonomy of Lorentzian manifolds

Blanco Oihane, Sánchez Miguel, Senovilla José: Structure of second-order symmetric Lorentzian manifolds. *J. Eur. Math. Soc.* 15 (2013), 595-634. doi: 10.4171/JEMS/368