Symmetry and pseudosymmetry properties with Ricci soliton of the Reissner-Nordström-de Sitter spacetime

The primary objective of the article is to investigate the symmetry and pseudosymmetry properties of the Reissner-Nordström-de Sitter (briefly, RNdS) spacetime. The secondary aim of the paper is to explore the notion of Ricci solitons in RNdS spacetimes. The study is important due to the conceding of almost Ricci soliton and almost Ricci Yamabe soliton of the RNdS spacetime. The analysis shows that this spacetime satisfies multiple types of symmetric and pseudosymmetric conditions. It is interesting to note that RNdS spacetime revealed pseudosymmetry, Weyl conformal pseudosymmetry, Weyl projective pseudosymmetry, conharmonic pseudosymmetry, and concircular pseudosymmetry. Also the RNdS spacetime is pseudosymmetric due to conformal, projective, conharmonic curvature tensors. Furthermore, in the RNdS spacetime, the second order commutator tensor R · R is linearly dependent on Q(Ric, R) and Q(g, C), and the tensor R · C does not commute with the tensor C · R. The investigation also shows that RNdS spacetime is a Roter type. It is demonstrated that the RNdS spacetime is 2-quasi Einstein and an Ein(2) space with recurrent conformal 2-forms. Riemann compatibility preserved under geodesic mapping and it has a great geometric insight in determining the nature of Pontryagian forms. We derive the general form of the compatible tensors of the RNdS spacetime. The energy-momentum tensor of the RNdS spacetime is shown to be pseudosymmetric, and also the energy momentum tensor is pseudosymmetric due to conformal, conharmonic, concircular and projective curvature tensor. We note that geodesic maps do not necessarily preserve Weyl compatibility. However such kind of compatibility remains invariant under conformal maps. The presence of a Weyl-compatible vector ensures that the Weyl tensor assumes an algebraically special form, and this condition is both necessary and sufficient for the vanishing of its magnetic component. The energy momentum tensor is compatible with projective, conharmonic, concircular, Riemann and conformal curvature. It is shown that the energy momentum tensor of the RNdS spacetime is Weyl compatible and hence the Weyl tensor of a such a spacetime is algebraically special and hence its magnetic component vanishes. Consequently the Weyl tensor in RNdS spacetime is purely electric. We study a generalized notion of curvature inheritance and find that, with respect to the non-Killing vector fields ∂/∂r and ∂/∂θ, the RNdS spacetime does not satisfy such inheritance condition. However, the RNdS spacetime is shown to admit an almost Ricci soliton and an almost η-Ricci Yamabe soliton with respect to the non-Killing vector field ∂/∂r, but with respect to the non-Killing vector field ∂/∂θ the spacetime does not admit such notions. Finally, we present a comparison between the RNdS and Vaidya-Bonner-de Sitter (VBdS) spacetimes.