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
type: Item
instance of: scholarly article
title: 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
zbMATH Open document ID: 1477.53051
DOI: 10.1155/2021/1431082
author: Jianyun Guan
author: Haiming Liu
published in: Journal of Function Spaces
publication date: 2021-09-07T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 53C17
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 53A35
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 53A04
zbMATH DE Number: 7391734
zbMATH Keywords: geometries of Thurston
zbMATH Keywords: Gauss-Bonnet theorem
Wikidata QID: Q114070130
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2021/1431082
OpenAlex ID: W3189508400
cites work: Classifying 3 and 4 dimensional homogeneous Riemannian manifolds by Cartan triples
cites work: Parallel surfaces in the motion groups \(E(1,1)\) and \(E(2)\)
cites work: A complete classification of parallel surfaces in three-dimensional homogeneous spaces
cites work: Intrinsic curvature of curves and surfaces and a Gauss-Bonnet theorem in the Heisenberg group
cites work: Correction to: ``Intrinsic curvature of curves and surfaces and a Gauss-Bonnet theorem in the Heisenberg group''
cites work: Gauss-Bonnet theorems in the affine group and the group of rigid motions of the Minkowski plane
cites work: Gauss-Bonnet theorems in the BCV spaces and the twisted Heisenberg group
cites work: The sub-Riemannian limit of curvatures for curves and surfaces and a Gauss-Bonnet theorem in the rototranslation group
cites work: Gauss-Bonnet theorem in lorentzian Sasakian space forms
cites work: An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem
description: scientific article; zbMATH DE number 7391734