2nd-order symmetric Lorentzian manifolds I: characterization and general results

TL;DR

This paper characterizes and classifies n-dimensional Lorentzian manifolds with vanishing second covariant derivative of the Riemann tensor, revealing they are either locally symmetric or possess a covariantly constant null vector field.

Contribution

It provides a complete classification of 2-symmetric Lorentzian manifolds, identifying their structure and relation to Brinkmann's class, and discusses related issues and open questions.

Findings

2-symmetric Lorentzian manifolds are either locally symmetric or have a covariantly constant null vector field.

The subclass with null vector fields corresponds to a specific family within Brinkmann's class.

The paper discusses applications and presents new open questions in the field.

Abstract

The n-dimensional Lorentzian manifolds with vanishing second covariant derivative of the Riemann tensor (2-symmetric spacetimes) are characterized and classified. The main result is that either they are locally symmetric or they have a covariantly constant null vector field, in this case defining a subfamily of Brinkmann's class in n dimensions. Related issues and applications are considered, and new open questions presented.