Eur. Phys. J. Plus (2017) 132: 70
https://doi.org/10.1140/epjp/i2017-11375-2

Regular Article

Geometric properties of the Kantowski-Sachs and Bianchi-type Killing algebra in relation to a Klein-Gordon equation

¹ School of Mathematics and Centre for Differential Equations, Continuum Mechanics and Applications, University of the Witwatersrand, Wits 2001, Johannesburg, South Africa
² Faculty of Engineering Sciences GIK Institute of Engineering Sciences and Technology, Topi, Swabi, KPK, Pakistan

^* e-mail: Sameerah.Jamal@wits.ac.za

Received: 1 December 2016
Accepted: 28 January 2017
Published online: 15 February 2017

Abstract

We study the geometric properties of generators for the Klein-Gordon equation in Kantowski-Sachs and certain Bianchi-type spaces. Several versions of the Klein-Gordon equation are derived from its dependence on a potential function. The criteria for different versions of the (1+3) Klein-Gordon equation originates from analyzing three sources, viz. through generators that are identically the Killing algebra, or with the Killing vector fields that are recast into linear combinations and thirdly, real sub-algebras within the conformal algebra. In turn, these equations admit a catalogue of infinitesimal symmetries that are equivalent to the corresponding Killing vector fields in Kantowski-Sachs, Bianchi type III, IX, VIII, VI₀ and VII₀ space-times, with the exception of a linear vector in every case. The sheer number of results are displayed in appropriate tables. Subsequently, in application, we derive some Noetherian conservation laws and identify some exact solutions by quadratures.