Structure of second-order symmetric Lorentzian manifolds

Blanco, Oihane F., Sánchez, Miguel, and Senovilla, José M.. "Structure of second-order symmetric Lorentzian manifolds." Journal of the European Mathematical Society 015.2 (2013): 595-634. <http://eudml.org/doc/277477>.

@article{Blanco2013,
abstract = {$\textit \{Second-order symmetric Lorentzian spaces\}$, 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., $\nabla R\ne 0$ 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.},
author = {Blanco, Oihane F., Sánchez, Miguel, Senovilla, José M.},
journal = {Journal of the European Mathematical Society},
keywords = {second-order symmetric spaces; curvature conditions; Brinkmann spaces; Lorentzian symmetric spaces; plane waves; holonomy of Lorentzian manifolds; second-order symmetric spaces; curvature conditions; Brinkmann spaces; Lorentzian symmetric spaces; plane waves; holonomy of Lorentzian manifolds},
language = {eng},
number = {2},
pages = {595-634},
publisher = {European Mathematical Society Publishing House},
title = {Structure of second-order symmetric Lorentzian manifolds},
url = {http://eudml.org/doc/277477},
volume = {015},
year = {2013},
}

TY - JOUR
AU - Blanco, Oihane F.
AU - Sánchez, Miguel
AU - Senovilla, José M.
TI - Structure of second-order symmetric Lorentzian manifolds
JO - Journal of the European Mathematical Society
PY - 2013
PB - European Mathematical Society Publishing House
VL - 015
IS - 2
SP - 595
EP - 634
AB - $\textit {Second-order symmetric Lorentzian spaces}$, 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., $\nabla R\ne 0$ 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.
LA - eng
KW - second-order symmetric spaces; curvature conditions; Brinkmann spaces; Lorentzian symmetric spaces; plane waves; holonomy of Lorentzian manifolds; second-order symmetric spaces; curvature conditions; Brinkmann spaces; Lorentzian symmetric spaces; plane waves; holonomy of Lorentzian manifolds
UR - http://eudml.org/doc/277477
ER -