Compact Lorentzian holonomy

Abstract

We consider (compact or noncompact) Lorentzian manifolds whose holonomy group has compact closure. This property is equivalent to admitting a parallel timelike vector field. We give some applications and derive some properties of the space of all such metrics on a given manifold.2010 Mathematical Subject Classification: 53C50; 53C29.

Compact affine manifolds with precompact holonomy are geodesically complete

“…These semi-Riemannian results are independent of Theorem 1.1; in fact, Gutiérrez and Müller [5] have proven recently that, for a Lorentzian metric g, the compactness of Hol(g) implies the existence of a timelike parallel vector field in a finite covering. This conclusion (combined with the cited one in [12]) also gives an alternative proof of Theorem 1.1 in the particular case that ∇ comes from a Lorentzian metric.…”