The Sub-Riemannian Limit of Curvatures for Curves and Surfaces and a Gauss-Bonnet Theorem in the Group of Rigid Motions of Minkowski Plane with General Left-Invariant Metric

Abstract

The group of rigid motions of the Minkowski plane with a general left-invariant metric is denoted by $\left(E\left(1,1\right),g\left({\lambda }_1,{\lambda }_2\right)\right)$ , where ${\lambda }_1\ge {\lambda }_2>0$ . It provides a natural $2$ -parametric deformation family of the Riemannian homogeneous manifold ${\text{Sol}}_3=\left(E\left(1,1\right),g\left(1,1\right)\right)$ which is the model space to solve geometry in the eight model geometries of Thurston. In this paper, we compute the sub-Riemannian limits of the Gaussian curvature for a Euclidean $C^2$ -smooth surface in $\left(E\left(1,1\right),g_L\left({\lambda }_1,{\lambda }_2\right)\right)$ away from characteristic points and signed geodesic curvature for the Euclidean $C^2$ -smooth curves on surfaces. Based on these results, we get a Gauss-Bonnet theorem in the group of rigid motions of the Minkowski plane with a general left-invariant metric.