Structure of second-order symmetric Lorentzian manifolds

TL;DR

This paper classifies second-order symmetric Lorentzian manifolds, showing they are locally products of symmetric Riemannian spaces and specific Lorentzian spaces, with a detailed analysis of their geometric and PDE properties.

Contribution

It provides a complete local and global classification of second-order symmetric Lorentzian manifolds, including explicit descriptions and technical PDE analysis.

Findings

Locally, these manifolds are products of symmetric Riemannian spaces and special Lorentzian spaces.

The curvature tensor is characterized by a local affine function in the proper case.

Complete second-order symmetric spaces are globally products of specific factors.

Abstract

Second-order symmetric Lorentzian spaces, that is to say, Lorentzian manifolds with vanishing second derivative of the curvature tensor R, are characterized by several geometric properties, and explicitly presented. Locally, they are a product M=M_1 x 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., with non-zero first covariant derivative of R at some point), the curvature tensor turns out to be described by some local affine function which characterizes a globally defined parallel lightlike line. 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 x 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.