Characterizations of Lorentzian Manifolds Satisfying Gray’s $$\boldsymbol{\mathcal{C}^{\bot}}$$ Condition

In this study, we characterize a Lorentzian manifold $$(M^{n},g)$$ of dimension $$\geq 3$$ satisfying Gray’s $$\mathcal{C}^{\bot}$$ condition. First, we prove that if a Lorentzian manifold $$M$$ satisfies Gray’s $$\mathcal{C}^{\bot}$$ condition and whose Ricci curvature annihilates the curvature transformation, then in the neighborhood of a point where $$|\nabla\,r_{g}|\neq 0$$ , $$M$$ is a generalized Robertson–Walker (GRW) space-time. Next, it is established that if a quasi-Einstein Lorentzian manifold satisfies Gray’s $$\mathcal{C}^{\bot}$$ condition, then in the neighborhood of a point where $$\beta\neq 0$$ , $$M$$ is a GRW space-time. Furthermore, it is confirmed that any perfect fluid GRW space-time is Bach-flat.