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

Fatemeh Ahangari

doi:10.3934/naco.2019039

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2019, Volume 9, Issue 4: 401-412. Doi: 10.3934/naco.2019039

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

Fatemeh Ahangari^,

Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, P.O.Box 1993893973, Iran

^* Corresponding author
^* Corresponding author

The reviewing process of the paper is handled by Gafurjan Ibragimov, Siti Hasana Sapar and Siti Nur Iqmal Ibrahim

Received: November 30, 2017

Revised: August 31, 2018

Published: November 2019

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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.

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Mathematics Subject Classification: Primary: 53B30, 53C29, 53C12; Secondary: 53C50.

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