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 1,1 . 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 1,1 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 1,1 with the second left-invariant Lorentzian metric g 2 .

Gauss–Bonnet Theorem Related to the Semi-Symmetric Metric Connection in the Heisenberg Group

“…Proof. Based on similar discussions in [12][13][14][15][16][17][18], we assume that all points satisfy ω( γi (t)) ̸ = 0 and d dt (ω( γi (t))) ̸ = 0 on the curve γ i . Since our proof of Proposition 6 is based on the approximation argument relying on the Lebesgue Dominated Convergence Theorem, the finite sets are negligible.…”

“…According to the relevant studies described above, there is little research on the geometric properties related to semi-symmetric connections in the Heisenberg group. The research on the Gauss-Bonnet theorems related to different connections on between Lie groups can be found at the following references ( [12][13][14][15][16][17][18]). Under the influence of the above work, this paper attempts to research geometric properties related to the semi-symmetric connection in the Heisenberg group by employing the method of the Riemannian approximation scheme.…”