Structure of second-order symmetric Lorentzian manifolds

  • Miguel Sánchez

    Universidad de Granada, Spain
  • Oihane F. Blanco

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

    Universidad del Pais Vasco, Bilbao, Spain


Second-order symmetric Lorentzian spaces, that is to say, Lorentzian manifolds with vanishing second derivative R0\nabla \nabla R\equiv 0 of the curvature tensor RR, are characterized by several geometric properties, and explicitly presented. Locally, they are a product M=M1×M2M=M_1\times M_2 where each factor is uniquely determined as follows: M2M_2 is a Riemannian symmetric space and M1M_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., R0\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 M1×M2M_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, 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