A class of Lorentzian manifolds with indecomposable holonomy groups

TL;DR

This paper studies a special class of Lorentzian manifolds with indecomposable holonomy groups, constructing examples with specific geometric properties and analyzing their holonomy and completeness.

Contribution

It introduces a new class of Lorentzian manifolds with indecomposable holonomy groups, including explicit constructions and analysis of their geometric structures.

Findings

Constructed Lorentzian metrics with indecomposable, reducible holonomy groups.

Provided examples with Hermitian screen holonomy.

Developed complete pp-wave solutions.

Abstract

We consider a class of $S^{1}$-bundles whose total space admits a nowhere vanishing recurrent lightlike vector field with respect to a Lorentzian metric. This metric can be modified such that its restricted holonomy group is indecomposable and reducible. We apply Hodge theory to construct examples with Hermitian screen holonomy. Finally, we construct complete pp-waves.