Characterizations of Lorentzian manifolds

Abstract

The focus of this paper is to characterize the Lorentzian manifolds equipped with a semi-symmetric non-metric ρ-connection [briefly, $(M,\tilde{\nabla })$ ]. The conditions for a Lorentzian manifold to be a generalized Robertson–Walker spacetime are established and vice versa. We prove that an n-dimensional compact $(M,\tilde{\nabla })$ is geodesically complete. We also study the properties of almost Ricci solitons and gradient almost Ricci solitons on Lorentzian manifolds and Yang pure space, respectively. Finally, we study the properties of semisymmetric $(M,\tilde{\nabla })$ , and it is proven that $(M,\tilde{\nabla })$ is semisymmetric if and only if it is a Robertson–Walker spacetime.