On the Full Holonomy of Lorentzian Manifolds with Parallel Weyl Tensor

TL;DR

This paper determines the full holonomy group of certain compact Lorentzian manifolds with parallel Weyl tensor, revealing their geometric structure and isometry group composition.

Contribution

It provides a complete computation of the holonomy group for non-conformally flat, non-symmetric Lorentzian manifolds with parallel Weyl tensor, including their isometry group structure.

Findings

Manifolds are geodesically complete.

The isometry group is a semidirect product of a subgroup of SO(n) with the Heisenberg group.

Holonomy group characterization for these manifolds.

Abstract

We compute the full holonomy group of compact Lorentzian manifolds with parallel Weyl tensor, which are neither conformally flat nor locally symmetric, for the case where the fundamental group is contained in a distinguished subgroup G of the isometry group of the universal cover. To prove this, we show that every such compact Lorentzian manifold has to be geodesically complete. Moreover we characterize the identity component of the isometry group for this universal cover and show that G is up to a discrete factor contained in the latter. Concretely, we prove that the identity component of the isometry group is isomorphic to a semidirect product of a subgroup of SO(n) with the Heisenberg group.