A Study on Killing Vector Fields in Four-Dimensional Spaces

Abstract

In the present study, some properties of Killing vector fields are investigated on $4-$dimensional manifolds in case of the signature of the metric tensor $g$ is either Lorentz or positive definite or neutral. First of all, the notation and the main object of the study are introduced on these manifolds. Later on, some special subalgebras are examined for the members of the Killing algebra when the Killing vector field vanishes at a point of the manifold admitting any of these metric signatures. The constraints of this examination to the Weyl conformal curvature tensor and the Ricci tensor are then studied and some results are obtained. Finally, some examples related to these results are given for all metric signatures.