Weakening Holonomy

Institute for Mathematical Physics Weakening Holonomy

Screen bundles of Lorentzian manifolds and some generalisations of pp-waves

A pp-wave is a Lorentzian manifold with a parallel light-like vector field satisfying a certain curvature condition. We introduce generalisations of pp-waves, on one hand by allowing the vector field to be recurrent and on the other hand by weakening the curvature condition. These generalisations are related to the screen holonomy of the Lorentzian manifold. While pp-waves have a trivial screen...

The Spin Holonomy Group in General Relativity

It has recently been shown by Goldberg et al that the holonomy group of the chiral spin-connection is preserved under time evolution in vacuum general relativity. Here, the underlying reason for the timeindependence of the holonomy group is traced to the self-duality of the curvature 2-form for an Einstein space. This observation reveals that the holonomy group is time-independent not only in v...

Finsler manifolds with non-Riemannian holonomy

The aim of this paper is to show that holonomy properties of Finsler manifolds can be very different from those of Riemannian manifolds. We prove that the holonomy group of a positive definite non-Riemannian Finsler manifold of non-zero constant curvature with dimension > 2 cannot be a compact Lie group. Hence this holonomy group does not occur as the holonomy group of any Riemannian manifold. ...

Projective Geometry II: Holonomy Classification

The aim of this paper and its prequel is to introduce and classify the holonomy algebras of the projective Tractor connection. This is achieved through the construction of a ‘projective cone’, a Ricci-flat manifold one dimension higher whose affine holonomy is equal to the Tractor holonomy of the underlying manifold. This paper uses the result to enable the construction of manifolds with each p...