Geometry of $\mathsf{W}_{3}$-curvature tensor in $f(\mathsf{R})$-gravity

In this paper, we explore the behaviour of $\mathsf{W}_{3}$-flat curvature tensor in $f(\mathsf{R})$-gravity. We describe $\mathsf{W}_{3}$-flat curvature tensor perfect fluid spacetime solutions of $f(\mathsf{R})$-gravity. It is noticed that after the establishment of $\mathsf{W}_{3}$-flat geometries of spacetime, we have found that the matter field has constant energy density and isotropic pressure. We then make the condition weaker and discuss the effects of $\mathsf{W}_{3}$-conservative spacetime geometries in $f(\mathsf{R}$)-gravity theory. We establish that the spacetime becomes a generalized Robertson-Walker spacetime with a shear, whirl, and acceleration independent perfect fluid with a particular form of expansion scalar presented in terms of scale factor. Moreover, $f(\mathsf{R})$-gravity model for $\mathsf{W}_{3}$-curvature tensor has been constructed to explain the different energy conditions with free parameter. We also study the behaviour of scale factor `a' of the universe with the chosen model.