  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.
