In this paper, we characterize weakly Ricci-symmetric (shortly, (WRS)4) spacetimes and their solutions in [Formula: see text]-gravity. It is demonstrated that a (WRS)4 spacetime represents a stiff matter fluid. In addition, we obtain that a conformally flat (WRS)4 spacetime is a space of quasi-constant sectional curvature. Moreover, we establish that a Ricci symmetric (WRS)4 spacetime represents a static spacetime. Finally, we investigate the effect of (WRS)4 spacetime solutions in [Formula: see text]-gravity.

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In this study, we analyze generalized quasi-Einstein spacetimes endowed with Gray’s decomposition, as well as generalized Robertson–Walker spacetimes. It is shown that the Ricci tensor of a generalized quasi-Einstein spacetime assumes the form of a perfect fluid in all Gray’s subspaces under certain restrictions. Finally, it is established that a generalized quasi-Einstein generalized Robertson–Walker spacetime is a perfect fluid spacetime.

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