In this paper, we define the Lorentzian approximations of a $3$-dimensional Lorentzian $\alpha$-Sasakian manifold. Moreover, we define the notions of the intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on Lorentzian surfaces and spacelike surfaces and the intrinsic Gaussian curvature of Lorentzian surfaces and spacelike surfaces away from characteristic points. Furthermore, we derive the expressions of those curvatures and prove Gauss-Bonnet theorems for the Lorentzian surfaces and spacelike surfaces in the Lorentzian $\alpha$-Sasakian manifold.

In this paper, we defined affine Yano–Ricci solitons and perturbed Yano–Ricci solitons on three-dimensional Lorentzian Lie groups and obtain their complete classification. The major results indicated that all groups $G_3,\cdots ,G_7$ admit both affine Yano–Ricci solitons and their perturbed companions, whereas $G_1$ supports neither. As a corollary, we found that the Lorentzian Lie groups $G_3,G_5$, and $G_6$ are affine Einstein manifolds with respect to the Yano connection.