Lorentzian Approximations and Gauss–Bonnet Theorem for E 1,1 with the Second Lorentzian Metric

Abstract

In this paper, we consider the Lorentzian approximations of rigid motions of the Minkowski plane $E_L^2\left(\mathrm{1,1}\right)$ . By using the method of Lorentzian approximations, we define the notions of the intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on Lorentzian surface, and the intrinsic Gaussian curvature of Lorentzian surface in $E\left(\mathrm{1,1}\right)$ with the second Lorentzian metric away from characteristic points. Furthermore, we derive the expressions of those curvatures and prove Gauss–Bonnet theorem for the Lorentzian surface in $E\left(\mathrm{1,1}\right)$ with the second left-invariant Lorentzian metric $g_2$ .