Geodesic
Structure of second-order symmetric Lorentzian manifolds
Journal of the European Mathematical Society, Tome 15 (2013) no. 2, pp. 595-634

Voir la notice de l'article provenant de la source EMS Press

Second-order symmetric Lorentzian spaces, that is to say, Lorentzian manifolds with vanishing second derivative ∇∇R≡0 of the curvature tensor R, are characterized by several geometric properties, and explicitly presented. Locally, they are a product M=M1​×M2​ where each factor is uniquely determined as follows: M2​ is a Riemannian symmetric space and M1​ 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., ∇R=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​×M2​. 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.
DOI : 10.4171/jems/368
Classification : 53-XX, 58-XX, 00-XX
Keywords: Second-order symmetric spaces, curvature conditions, Brinkmann spaces, Lorentzian symmetric spaces, plane waves, holonomy of Lorentzian manifolds 
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     title = {Structure of second-order symmetric {Lorentzian} manifolds},
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Oihane F. Blanco; José M. M. Senovilla; Miguel Sánchez. Structure of second-order symmetric Lorentzian manifolds. Journal of the European Mathematical Society, Tome 15 (2013) no. 2, pp. 595-634. doi: 10.4171/jems/368

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