On Lorentzian connections with parallel skew torsion

TL;DR

This paper classifies Lorentzian metric connections with parallel skew torsion, exploring their holonomy, torsion, and curvature, and constructs examples and classifications of naturally reductive homogeneous spaces in low dimensions.

Contribution

It extends the understanding of Lorentzian connections with parallel skew torsion, building on Riemannian results and providing classifications and construction methods.

Findings

Complete classification of Lorentzian naturally reductive homogeneous spaces in low dimensions.

Construction methods for Lorentzian spaces from Riemannian counterparts.

Analysis of holonomy algebras, torsion, and curvature for these connections.

Abstract

The paper is devoted to metric connections with parallel skew-symmetric torsion in Lorentzian signature. This is motivated by recent progress in the Riemannian signature and by possible applications to supergravity theories. We provide a complete information about holonomy algebras, torsion and curvature of the considered connections up to the corresponding objects from the Riemannian signature. Various examples are constructed. It is shown how to construct all simply connected Lorentzian naturally reductive homogeneous spaces of arbitrary dimension from Riemannian naturally reductive homogeneous spaces. This leads to complete classification of Lorentzian naturally reductive homogeneous spaces in low dimensions.