Abstract
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.
