The objective of the present paper is to study 4-dimensional almost pseudo Ricci symmetric perfect fluid spacetimes $(APRS)_4$. We show that a Robertson-Walker spacetime $(APRS)_4$ and vice versa under certain condition imposed on its scale factor. Some popular toy models $F(R)$-gravity are also studied current setting various energy conditions investigated.

In this paper, we classify almost cosymplectic 3-manifolds with pseudo-parallel characteristic Jacobi operator. The only simply connected and complete non-cosymplectic 3-manifold pseudo parallel operator is the Minkowski motion group.