Characterization of warped product manifolds through the -curvature tensor with applications to relativity

We first investigate how the flatness and the symmetry of the $$ -tensor impact both the base manifold and the fiber manifold of a warped product manifold. In both cases, we determine the form of the $$ -tensor on both the fiber and the base manifolds. Additionally, we establish that the fiber manifold is of constant curvature, while the base manifold is Einstein. It is also proved that a $$ -curvature flat GRW space-time is perfect fluid and static. Furthermore, the space-time either represents dark matter era or the state equation has a specific form.