Sub-Lorentzian Geometry of Curves and Surfaces in a Lorentzian Lie Group

Abstract

We consider the sub-Lorentzian geometry of curves and surfaces in the Lie group $E\left(1,1\right).$ Firstly, as an application of Riemannian approximants scheme, we give the definition of Lorentzian approximants scheme for $E\left(1,1\right)$ which is a sequence of Lorentzian manifolds denoted by $E_{{\lambda }_1,{\lambda }_2}^L$ . By using the Koszul formula, we calculate the expressions of Levi-Civita connection and curvature tensor in the Lorentzian approximants of $E_{{\lambda }_1,{\lambda }_2}^L$ in terms of the basis $\left\{E_1,E_2,E_3\right\}.$ These expressions will be used to define the notions of the intrinsic curvature for curves, the intrinsic geodesic curvature of curves on surfaces, and the intrinsic Gaussian curvature of surfaces away from characteristic points. Furthermore, we derive the expressions of those curvatures and prove two generalized Gauss-Bonnet theorems in $E_{{\lambda }_1,{\lambda }_2}^L$ .