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

Jianyun Guan,Haiming Liu,Calogero Vetro

doi:10.1155/2021/1431082

Jianyun Guan, Haiming Liu + Show 1 more

Open Access

https://doi.org/10.1155/2021/1431082

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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.

Highlights

In [1], Proposition 2.6 stated that any left-invariant metric on the group of rigid motions of the Minkowski plane Eð1, 1Þ is isometric to one of the metric gðλ1, λ2, λ3Þ with λ1 ≥ λ2 > 0 and λ3 = 1/λ1λ2
In [2], the metric gðλ1, λ2, λ3Þ was denoted by gðλ1, λ2Þ = gðλ1, λ2, 1/λ1λ2Þ as we take in this paper, and the authors classified parallel surfaces in the groups of rigid motions of the Euclidean plane and the Minkowski plane
We consider ð Eð1, 1Þ, gðλ1, λ2ÞÞ which is the group of rigid motions of the Minkowski plane with the general left-invariant metric gðλ1, λ2Þ: This group is very interesting and important for the reason that it provides a natural 2-parametric deformation family of one of the Riemannian homogeneous manifold Sol3 = ðEð1, 1Þ, gð1, 1ÞÞ which is the model space to solve-geometry in the eight model geometries of Thurston

Summary

Introduction

We try to solve this problem for the group of rigid motions of the Minkowski plane with the general left-invariant metric gðλ1, λ2Þ: We compute the sub-Riemannian limits of the Gaussian curvature for a Euclidean C2-smooth surface in ðEð1, 1Þ, gLðλ1, λ2ÞÞ away from characteristic points and signed geodesic curvature for the Euclidean C2-smooth curves on surfaces. We compute the sub-Riemannian limit of the curvature of curves in ðEð1, 1Þ, gLðλ1, λ2ÞÞ. 1⁄2X1, X3 = λ12X2: To compute the curvatures of curves and surfaces in the motion group of the Minkowski plane with respect to gLðλ1 , λ2Þ, we use the Levi-Civita connection ∇L on ðEð1, 1Þ, gLð λ1, λ2ÞÞ.

Results

Conclusion