On generalized Robertson--Walker spacetimes satisfying some curvaturecondition

We give necessary and sufficient conditions for warped product manifolds (M, g), of dimension \ 4, with 1-dimensional base, and in particular, for generalized Robertson--Walker spacetimes, to satisfy some generalized Einstein metric condition. Namely, the difference tensor R. C - C. R, formed from the curvature tensor R and the Weyl conformal curvature tensor C, is expressed by the Tachibana tensor Q (S, R) formed from the Ricci tensor S and R. We also construct suitable examples of such manifolds. They are quasi-Einstein, i. e. at every point of M rank (S - a g) \ 1, for some a \ R, or non-quasi-Einstein.