Gauss—Bonnet Theorems in the Lorentzian Heisenberg Group and the Lorentzian Group of Rigid Motions of the Minkowski Plane

Sining Wei,Yong Wang

doi:10.3390/sym13020173

Abstract

The aim of this paper was to obtain Gauss–Bonnet theorems on the Lorentzian Heisenberg group and the Lorentzian group of rigid motions of the Minkowski plane. At the same time, the sub-Lorentzian limits of Gaussian curvature for surfaces which are C2-smooth in the Lorentzian Heisenberg group away from characteristic points and signed geodesic curvature for curves which are C2-smooth on surfaces are studied. Using a similar method, we also studied the corresponding contents on Lorentzian group of rigid motions of the Minkowski plane.

Highlights

The Gauss–Bonnet theorem and the definition of Gaussian curvature for surfaces which are non-horizontal in sub-Riemannian Heisenberg space H1 have been introduced by Diniz-Veloso in [1]
We will use similar methods to get the Gauss–Bonnet theorem in the Lorentzian Heisenberg group and the Lorentzian group of rigid motions of the Minkowski plane
The main results of this paper are Gauss–Bonnet type theorems for the spacelike surfaces and Lorentzian surfaces in Lorentzian Heisenberg group and the Lorentzian group of rigid motions of the Minkowski plane

Summary

Introduction

The Gauss–Bonnet theorem and the definition of Gaussian curvature for surfaces which are non-horizontal in sub-Riemannian Heisenberg space H1 have been introduced by Diniz-Veloso in [1]. Let H1L be Lorentzian Heisenberg group, γ : I → R3 = H1L be a C2-smooth regular curve, where I is a open interval in R and γ(t) = γ1(t), γ2(t), γ3(t).

So we have p

We have

Findings

Conclusions