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 (Q1981830)

Summary: The group of rigid motions of the Minkowski plane with a general left-invariant metric is denoted by \((E(1,1), g(\lambda_1, \lambda_2))\), where \(\lambda_1\geq \lambda_2>0\). It provides a natural \(2\)-parametric deformation family of the Riemannian homogeneous manifold \(\mathrm{Sol}_3=(E(1, 1), g(1, 1))\) 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 \(C^2\)-smooth surface in \((E(1, 1), g_L (\lambda_1, \lambda_2))\) 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.