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. (10th August 2021)

• Record Type: Journal Article
• Authors: Guan, Jianyun; Liu, Haiming
• Other Names: Vetro Calogero Academic Editor.
• Is Part Of: Journal of function spaces. Volume 2021(2021)
• Publication Date: 2021-08-10
• Subjects: Function spaces -- Periodicals; 515.7305
• DOI: 10.1155/2021/1431082
• Languages: English
• ISSNs: 2314-8896

Abstract

Abstract : The group of rigid motions of the Minkowski plane with a general left-invariant metric is denoted by E 1, 1, g λ 1, λ 2, where λ 1 ≥ λ 2 > 0 . It provides a natural 2 -parametric deformation family of the Riemannian homogeneous manifold 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 λ 1, λ 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.