Mixed generalized quasi-Einstein spacetimes in f(R)-gravity

In this study, we investigate mixed generalized quasi-Einstein (MGQE) spacetimes of constant scalar curvature within the framework of f ( R ) -gravity. It is proved that a Ricci semi-symmetric perfect-fluid MGQE spacetime satisfies either p = − ϱ or has its energy density and pressure fixed by f ( R ) and f ′ ( R ) , while a Ricci-recurrent MGQE spacetime is necessarily of pure-radiation type. MGQE generalized Robertson–Walker spacetimes are also examined, and their fiber is shown to be quasi-Einstein. Furthermore, the mixed term is identified with the heat flux, and the corresponding energy conditions are derived. It is also shown that vacuum solutions are Einstein manifolds satisfying R 0 f ′ ( R 0 ) = 2 f ( R 0 ) . Finally, explicit examples are constructed and verified to be exact solutions of the f ( R ) field equations.