Generalized weakly Ricci-symmetric (CS)4-spacetimes

(CS)4-spacetimes with Einstein field equations under some curvature restriction named generalized weakly Ricci-symmetry have been studied. We have proved that if the characteristic vector field ξ of a generalized weakly Ricci-symmetric (CS)4-spacetime obeying Einstein equation is a Killing vector field, then such a spacetime admits (i) curvature collineation, (ii) conformal collineation, (iii) conharmonic collineation, (iv) concircular collineation, (v) projective collineation, (vi) m -projective collineation. It is further proved that each of conformally flat, conharmonically flat, concircularly flat, projectively flat and m-projectively flat generalized weakly Ricci-symmetric (CS)4 -spacetime is infinitesimally spatially isotropic relative to the unit timelike vector field ξ.