Lorentzian homogeneous structures with indecomposable holonomy

TL;DR

This paper characterizes Lorentzian homogeneous spaces with indecomposable, non-irreducible holonomy, showing they are locally isometric to plane waves, and explores related properties of Lorentzian connections with parallel torsion.

Contribution

It generalizes existing classifications by linking algebraic conditions on the isotropy group to geometric structures like plane waves in Lorentzian homogeneous spaces.

Findings

Lorentzian homogeneous space with certain holonomy is a plane wave.

Connections with parallel torsion are characterized.

Results extend to 2-symmetric Lorentzian connections.

Abstract

For a Lorentzian homogeneous space, we study how algebraic conditions on the isotropy group affect the geometry and curvature of the homogeneous space. More specifically, we prove that a Lorentzian locally homogeneous space is locally isometric to a plane wave if it admits an Ambrose--Singer connection with indecomposable, non-irreducible holonomy. This generalises several existing results that require a certain algebraic type of the torsion of the Ambrose--Singer connection and moreover is in analogy to the fact that a Lorentzian homogeneous space with irreducible isotropy has constant sectional curvature. In addition, we prove results about Lorentzian connections with parallel torsion and for 2-symmetric connections.