Characterizations of a Lorentzian manifold with a semi-symmetric metric connection

In this article, we characterize a Lorentzian manifold $\mathcal{M}$ with a semi-symmetric metric connection. At first, we consider a semi-symmetric metric connection whose curvature tensor vanishes and establish that if the associated vector field is a unit time-like torse-forming vector field, then $\mathcal{M}$ becomes a perfect fluid spacetime. Moreover, we prove that if $\mathcal{M}$ admits a semi-symmetric metric connection whose Ricci tensor is symmetric and torsion tensor is recurrent, then $\mathcal{M}$ represents a generalized Robertson—Walker spacetime. Also, we show that if the associated vector field of a semi-symmetric metric connection, whose curvature tensor vanishes, is an $f-\operatorname{Ric}$ vector field, then the manifold is a space of constant curvature. Therefore, the spacetime reduces to the Minkowski spacetime whenever the scalar curvature $R=0$; de Sitter spacetime whenever $R>0$; anti de Sitter spacetime whenever $R<0$. Moreover, if the associated vector field is a torqued vector field, then the manifold becomes a perfect fluid spacetime. Finally, we apply this connection to investigate the Ricci solitons.