# Structure of second-order symmetric Lorentzian manifolds

Oihane F. Blanco; Miguel Sánchez; José M. Senovilla

Journal of the European Mathematical Society (2013)

- Volume: 015, Issue: 2, page 595-634
- ISSN: 1435-9855

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topBlanco, 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 -

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