This article introduces a new curvature tensor, the S-curvature tensor, which is seen as a comprehensive extension of various curvature tensors. It is demonstrated that a semi-Riemannian manifold with traceless S-curvature tensor is Einstein. It is proved that a S-curvature flat semi-Riemannian manifold is of constant sectional curvature. Moreover, we show that a perfect fluid S-curvature flat space-time represents dark matter era. It is shown that a perfect fluid spacetime with ₕ Sʰ₉₊₋ = 0 is expansion-free and shear-free and its flow is geodesic, but not necessary vorticity-free. We show that a pseudo S-symmetric manifold is reduced to pseudo symmetric manifold if and only if the scalar curvature is constant. Finally, a concrete example of pseudo S-symmetric manifolds is introduced.
Syied et al. (Wed,) studied this question.