Compact Lorentzian holonomy

TL;DR

This paper investigates Lorentzian manifolds with holonomy groups having compact closure, establishing their equivalence to the existence of a parallel timelike vector field and exploring the structure of such metrics.

Contribution

It characterizes Lorentzian manifolds with compact closure holonomy groups and analyzes the space of these metrics, providing new insights into their geometric properties.

Findings

Holonomy group with compact closure is equivalent to admitting a parallel timelike vector field

Properties of the space of metrics with this holonomy condition are derived

Results apply to both compact and noncompact Lorentzian manifolds

Abstract

We consider (compact or noncompact) Lorentzian manifolds whose holonomy group has compact closure. Among other results, we obtain that this property is equivalent to admitting a parallel timelike vector field. We also derive some properties of the space of all such metrics on a given manifold.