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 E1,1,gλ1,λ2, where λ1≥λ2>0. It provides a natural 2-parametric deformation family of the Riemannian homogeneous manifold Sol3=E1,1,g1,1 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 C2-smooth surface in E1,1,gLλ1,λ2 away from characteristic points and signed geodesic curvature for the Euclidean C2-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.