Characterizations of Lorentzian Manifolds Satisfying Gray’s C^ Condition

In this study, we characterize a Lorentzian manifold (Mⁿ,g) of dimension ≥ 3 satisfying Gray’s C^ condition. First, we prove that if a Lorentzian manifold M satisfies Gray’s C^ condition and whose Ricci curvature annihilates the curvature transformation, then in the neighborhood of a point where |∇\,rg|≠ 0 , M is a generalized Robertson–Walker (GRW) space-time. Next, it is established that if a quasi-Einstein Lorentzian manifold satisfies Gray’s C^ condition, then in the neighborhood of a point where β≠ 0 , M is a GRW space-time. Furthermore, it is confirmed that any perfect fluid GRW space-time is Bach-flat.