A Note on Generalized Robertson-Walker Spacetimes and f(ℛ)-Gravity

First, we prove that a Ricci symmetric spacetime is a perfect fluid spacetime if and only if it is a generalized Robertson–Walker spacetime. Consequently, we establish that under certain constraints on the associated scalar, such a spacetime turns into a static spacetime and is classified under Petrov classification [Formula: see text], [Formula: see text], or [Formula: see text]. Also, we notice that a semi-symmetric as well as pseudo-Ricci symmetric generalized Robertson–Walker spacetime is a perfect fluid spacetime. We examine conformally flat generalized Robertson–Walker spacetime as a solution of [Formula: see text]-gravity theory and describe the physical relevance. For the model [Formula: see text], numerous energy conditions in terms of associated scalars are investigated. This model agrees with the current investigations which states that the Cosmos is in an accelerating phase.