Conformal deformations of a specific class of lorentzian manifolds with non-irreducible holonomy representation

Abstract

Concerning holonomy theory or in the context of the existence of parallel spinors, Lorentzian manifolds with indecomposable, but non-irreducible holonomy representation have considerable significance. In this paper, we have comprehensively concentrated on conformal deformations of a particular class of four dimensional Lorentzian manifolds with indecomposable, non-irreducible holonomy representation which admit a recurrent light-like vector field. This type of Lorentzian manifolds are denoted by pr-waves and their holonomy algebra is contained in the parabolic algebra $ \big(\mathbb{R}\oplus \mbox{so(2)}\big)\ltimes \mathbb{R}^2 $. Moreover, it is mainly illustrated that for an arbitrary conformal diffeomorphism by inducing some specific structural conditions a pr-wave manifold behaves totally analogous to Einstein manifolds. Particularly, it is demonstrated that in some special circumstances the structure of a pr-wave manifold is precisely the same as a manifold equipped with a warped product metric.