On the Geometry of Circle Bundles with Special Holonomy

TL;DR

This paper explores the geometric and holonomy properties of Lorentzian manifolds that are circle bundles, providing conditions for completeness, examples of Ricci-flat spaces, and analyzing their holonomy types.

Contribution

It offers new insights into the conditions for completeness and holonomy classifications of Lorentzian circle bundles, including explicit examples.

Findings

Conditions for completeness of Lorentzian circle bundles

Examples of Ricci-flat Lorentzian manifolds with special holonomy

Holonomy classification results for these manifolds

Abstract

We investigate geometric properties of indecomposable but non-irreducible Lorentzian manifolds, which are total spaces of circle bundles. We investigate under which conditions these manifolds are complete and give examples which fulfill the obtained conditions. In particular we investigate the Einstein equation for these spaces yielding examples for complete compact Ricci flat Lorentzian manifolds and manifolds with timelike Killing vector fields. Finally we study their holonomy and obtain in particular complete examples for Lorentzian manifolds with holonomy of so called type 4.