Lorentzian approximations for a Lorentzian $\alpha$-Sasakian manifold and Gauss-Bonnet theorems

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.