On Curvature properties of Nariai Spacetimes

The charged Nariai spacetimes are the exact solutions of Einstein-Maxwell field equations with positive cosmological constant and such a spacetime is the direct topological product of a $2$-dimentional de-Sitter spacetime with a round $2$-sphere of constant radius. The present paper deals with the investigation of curvature restricted geometric structures admitted by the charged Nariai spacetime metric and it is shown that such a spacetime is locally symmetric, $2$-quasi-Einstein and $Ein(2)$-space. Moreover it realizes the pseudosymmetric type conditions such as Ricci generalized projectively pseudosymmetric and pseudosymmetric due to Weyl conformal curvature tensor. It is interesting to note that the energy momentum tensor can be expressed explicitly with the help of some $1$-forms. We also evaluate the curvature properties of charged Nariai type topological product metric, and it is found that such metric is semisymmetric but not locally symmetric and under certain restrictions belongs to the family of generalized class of recurrent manifolds. Finally we end up our findings with some concluding remarks.