# 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

Authors: Jianyun Guan; Haiming Liu
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The group of rigid motions of the Minkowski plane with a general left-invariant metric is denoted byE1,1,gλ1,λ2, where λ1≥λ2>0. It provides a natural 2-parametric deformation family of the Riemannian homogeneous manifold Sol3=E1,1,g1,1 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 C2-smooth surface in E1,1,gLλ1,λ2 away from characteristic points and signed geodesic curvature for the Euclidean C2-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.


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## Body

## 1. Introduction

In [1], Proposition 2.6 stated that any left-invariant metric on the group of rigid motions of the Minkowski plane E1,1 is isometric to one of the metric gλ1,λ2,λ3 with λ1≥λ2>0 and λ3=1/λ1λ2. In [2], the metric gλ1,λ2,λ3 was denoted by gλ1,λ2=gλ1,λ2,1/λ1λ2 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. In [3], they completed the classification of parallel and totally geodesic surfaces in all three-dimensional homogeneous spaces by solving the problem in three-dimensional Lie groups with a left-invariant metric yielding a three-dimensional isometry group. In this paper, we consider E1,1,gλ1,λ2 which is the group of rigid motions of the Minkowski plane with the general left-invariant metric gλ1,λ2. 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=E1,1,g1,1 which is the model space to solve-geometry in the eight model geometries of Thurston.In [4, 5], Balogh et al. used a Riemannian approximation scheme to define a notion of intrinsic Gaussian curvature for a Euclidean C2-smooth surface in the Heisenberg group H1 away from characteristic points, and a notion of intrinsic signed geodesic curvature for the Euclidean C2-smooth curves on surfaces. These results were then used to prove a Heisenberg version of the Gauss-Bonnet theorem. They proposed an interesting question to understand to what extent similar phenomena hold in other sub-Riemannian geometric structures. In [6, 7], Wang and Wei solved this problem for the affine group, the group of rigid motions of the Minkowski plane E1,1,g1,1, the BCV spaces, and the twisted Heisenberg group. Recently, we got the Gauss-Bonnet theorems in the rototranslation group and the Lorentzian Sasakian space forms [8, 9]. In this paper, we try to solve this problem for the group of rigid motions of the Minkowski plane with the general left-invariant metric gλ1,λ2. We compute the sub-Riemannian limits of the Gaussian curvature for a Euclidean C2-smooth surface in E1,1,gLλ1,λ2 away from characteristic points and signed geodesic curvature for the Euclidean C2-smooth curves on surfaces. We get a generalized Gauss-Bonnet theorem in E1,1,gLλ1,λ2.In Section2, we provide a short introduction to E1,1,gLλ1,λ2 and the notions which we will use throughout the paper, such as the Levi-Civita connection in the Riemannian approximants of E1,1,gλ1,λ2. Furthermore, we compute the sub-Riemannian limit of the curvature of curves in E1,1,gLλ1,λ2. In Sections 3 and 4, we compute the sub-Riemannian limits of the geodesic curvature of curves on surfaces and the Riemannian Gaussian curvature of surfaces in E1,1,gLλ1,λ2. In Section 5, we get the Gauss-Bonnet theorem in the group of rigid motions of the Minkowski plane with the general left-invariant metric.


## 2. The Sub-Riemannian Limit of Curvature of Curves inE1,1,gLλ1,λ2

In this section, some basic notions in the motion group of the Minkowski plane will be introduced. LetE1,1 be the motion group of the Minkowski plane:
(1)E1,1=ez0x0e−zy001x,y,z∈ℝ.The Lie algebraℓ1,1 is given explicitly by
(2)ℓ1,1=ω0u0−ωv000u,v,ω∈ℝ.We consider the group of rigid motions of the Minkowski plane with a general left-invariant metric,E1,1,gλ1,λ2. As a model of E1,1,gλ1,λ2, we choose the underlying manifold ℝ3. On ℝ3, we let
(3)X1=λ1λ2∂∂z,X2=1λ12−ez∂∂x+e−z∂∂y,X3=−1λ22ez∂∂x+e−z∂∂y.Then, we have(4)∂∂x=−22e−zλ1X2+λ2X3,∂∂y=22ezλ1X2−λ2X3,∂∂z=1λ1λ2X1,(5)spanX1,X2,X3=TE1,1,gλ1,λ2.LetH=spanX1,X2 be the horizontal distribution on E1,1,gλ1,λ2. Let
(6)ω1=1λ1λ2dz,ω2=λ12−e−zdx+ezdy,ω=−λ22e−zdx+ezdy,be the dual coframe field. Then, H=kerω. The Riemannian approximation scheme used in [4] can in general depend on the choice of the complement to the horizontal distribution. In the context of E1,1,gλ1,λ2, the choice is similar to E1,1,g1,1 in [6]. Let L>0 and define a metric gLλ1,λ2=ω1⊗ω1+ω2⊗ω2+Lω⊗ω, so that X1,X2,X3~≔L−1/2X3 are orthonormal basis on TE1,1,gLλ1,λ2 with respect to gLλ1,λ2. Notice that
(7)gLλ1,λ2=1λ12λ22dz2+λ122−e−zdx+ezdy2+Lλ222e−zdx+ezdy2,and gλ1,λ2=g1λ1,λ2 be the Riemannian metric on E1,1,gλ1,λ2. The approach in this paper is to define sub-Riemannian objects as limits of horizontal objects in E1,1,gLλ1,λ2, where a family of metrics gLλ1,λ2 is essentially obtained as an anisotropic blow-up of the Riemannian metric g1λ1,λ2. At the heart of this approach is the fact that the intrinsic horizontal geometry does not change with L. In general, the metric gLλ1,λ2 does not fit in the family gλ1,λ2 for the case of L≠1. We have
(8)X1,X2=λ22X3,X2,X3=0,X1,X3=λ12X2.To compute the curvatures of curves and surfaces in the motion group of the Minkowski plane with respect togLλ1,λ2, we use the Levi-Civita connection ∇L on E1,1,gLλ1,λ2. A straightforward calculation shows the following proposition.Proposition 1.
LetE1,1,gLλ1,λ2 be the motion group of the Minkowski plane with a general left-invariant metric, relative to the coordinate frame X1, X2, X3~; then, the Levi-Civita connection on E1,1,gLλ1,λ2 is given by
(9)∇XjLXj=0,1≤j≤3,∇X1LX2=λ22L−λ122LX3,∇X2LX1=−λ22L−λ122LX3,∇X1LX3=λ12−λ22L2X2,∇X3LX1=−λ12−λ22L2X2,∇X2LX3=∇X3LX2=λ12+λ22L2X1.Proof.
It follows from a direct application of the Koszul identity, which here simplifies(10)2∇XiLXj,XkL=Xi,Xj,XkL−Xj,Xk,XiL+XkXi,XjL,where i,j,k=1,2,3. By (8) and (9), we have
(11)2∇XjLXj,XkL=Xj,Xj,XkL−Xj,Xk,XjL+Xk,Xj,XjL=−Xj,Xk,XjL+Xk,Xj,XjL=2Xk,Xj,XjL.
Whenj=1, we compute ∇X1LX1,XkL=Xk,X1,X1L. It follows that ∇X1LX1,X1L=0, ∇X1LX1,X2L=X2,X1,X1L=−λ22X3,X1L=0, and ∇X1LX1,X3L=X3,X1,X1L=0. Hence, ∇X1LX1=0. Similarly, ∇X2LX2=0 and ∇X3LX3=0. By the following equation,
(12)2∇X1LX2,XkL=X1,X2,XkL−X2,Xk,X1L+Xk,X1,X2L=λ22X3,XkL−X2,Xk,X1L+Xk,X1,X2L,we get ∇X1LX2=λ22L−λ12/2LX3. Other cases follow by similar computations. ☐☐Definition 2.
Letγ:a,b⟶E1,1,gLλ1,λ2 be a Euclidean C1-smooth curve. We say that γ is regular, if γ̇≠0 for every t∈a,b. Moreover, we say that γt is a horizontal point of γ if
(13)ωγ̇t=−λ22e−γ3dx+eγ3dyγ̇1t∂∂x+γ̇2t∂∂y+γ̇3t∂∂z=−λ222e−γ3γ̇1t+eγ3γ̇2t=0,where γt=γ1t,γ2t,γ3t.Definition 3.
Letγ:a,b⟶E1,1,gLλ1,λ2 be a Euclidean C2-smooth regular curve in the Riemannian manifold E1,1,gLλ1,λ2. The curvature κγL of γ at γt is defined as follows:
(14)κγL≔∇γ̇Lγ̇L2γ̇L4−∇γ̇Lγ̇,γ̇L2γ̇L6.Proposition 4.
Letγ:a,b⟶E1,1,gLλ1,λ2 be a Euclidean C2-smooth regular curve in the Riemannian manifold E1,1,gLλ1,λ2. Then, we have
(15)κγL=1λ1λ2γ¨3t+λ132+λ1λ222L2−e−γ3γ̇1t+eγ3γ̇2tωγ̇t2+λ122γ¨2eγ3+γ̇2γ̇3eγ3−γ¨1e−γ3+γ̇1γ̇3e−γ3−Lωγ̇tγ̇3t2+Lddtωγ̇t−λ1222λ2L−e−γ3γ̇1t+eγ3γ̇2tγ̇3t2·1λ1λ2γ̇3t2+λ122−e−γ3γ̇1t+eγ3γ̇2t2+Lωγ̇t2−2−1λ1λ2γ̇3t1λ1λ2γ¨3t+λ132+λ1λ222L2−e−γ3γ̇1t+eγ3γ̇2tωγ̇t+λ122−e−γ3γ̇1t+eγ3γ̇2tλ122γ¨2eγ3+γ̇2γ̇3eγ3−γ¨1e−γ3+γ̇1γ̇3e−γ3−Lωγ̇tγ̇3t+Lωγ̇tddtωγ̇t−λ1222λ2L−e−γ3γ̇1t+eγ3γ̇2tγ̇3t2·1λ1λ2γ̇3t2+λ122−e−γ3γ̇1t+eγ3γ̇2t2+Lωγ̇t2−31/2.
In particular, ifγt is a horizontal point of γ,
(16)κγL=1λ1λ2γ¨3t2+λ122γ¨2eγ3+γ̇2γ̇3eγ3−γ¨1e−γ3+γ̇1γ̇3e−γ32+Lddtωγ̇t−λ1222λ2L−e−γ3γ̇1t+eγ3γ̇2tγ̇3t2·1λ1λ2γ̇3t2+λ122−e−γ3γ̇1t+eγ3γ̇2t2−2−1λ1λ2γ̇3t1λ1λ2γ¨3t+λ122−e−γ3γ̇1t+eγ3γ̇2tγ¨2eγ3+γ̇2γ̇3eγ3−γ¨1e−γ3+γ̇1γ̇3e−γ32·1λ1λ2γ̇3t2+λ122−e−γ3γ̇1t+eγ3γ̇2t2−31/2.Proof.
By (4), we have
(17)γ̇t=γ̇1t∂∂x+γ̇2t∂∂y+γ̇3t∂∂z=−22e−γ3λ1X2+λ2X3γ̇1t+22eγ3λ1X2−λ2X3γ̇2t+1λ1λ2X1γ̇3t=1λ1λ2γ̇3tX1+−22e−γ3λ1γ̇1t+22eγ3λ1γ̇2tX2+−22e−γ3λ2γ̇1t+22eγ3λ2γ̇2tX3=1λ1λ2γ̇3tX1+λ122−e−γ3γ̇1t+eγ3γ̇2tX2+ωγ̇tX3.
By Proposition1 and (17), we have
(18)∇γ̇LX1=1λ1λ2γ̇3t∇X1LX1+λ122−e−γ3γ̇1t+eγ3γ̇2t∇X2LX1+ωγ̇t∇X3LX1=−λ12−λ22L2ωγ̇tX2+−λ1λ222L−λ1324L−e−γ3γ̇1t+eγ3γ̇2tX3,(19)∇γ̇LX2=1λ1λ2γ̇3t∇X1LX2+λ122−e−γ3γ̇1t+eγ3γ̇2t∇X2LX2+ωγ̇t∇X3LX2=λ12+λ22L2ωγ̇tX1+λ22L−λ122λ1λ2Lγ̇3tX3,(20)∇γ̇LX3=1λ1λ2γ̇3t∇X1LX3+λ122−e−γ3γ̇1t+eγ3γ̇2t∇X2LX3+ωγ̇t∇X3LX3=λ132+λ1λ222L4−e−γ3γ̇1t+eγ3γ̇2tX1+λ12−λ22L2λ1λ2γ̇3tX2.
By (17) and (18), we have
(21)∇γ̇Lγ̇=∇γ̇L1λ1λ2γ̇3tX1+λ122−e−γ3γ̇1t+eγ3γ̇2tX2+ωγ̇tX3=γ̇1λ1λ2γ̇3tX1+1λ1λ2γ̇3t∇γ̇LX1+γ̇λ122−e−γ3γ̇1t+eγ3γ̇2tX2+λ122−e−γ3γ̇1t+eγ3γ̇2t∇γ̇LX2+γ̇ωγ̇tX3+ωγ̇t∇γ̇LX3=1λ1λ2γ¨3t+λ132+λ1λ222L2−e−γ3γ̇1t+eγ3γ̇2tωγ̇tX1+λ122γ¨2eγ3+γ̇2γ̇3eγ3−γ¨1e−γ3+γ̇1γ̇3e−γ3−Lωγ̇tγ̇3tX2+ddtωγ̇t−λ1222λ2L−e−γ3γ̇1t+eγ3γ̇2tγ̇3tX3.
By (14), (17), and (21), we get
(22)∇γ̇Lγ̇L2=1λ1λ2γ¨3t+λ132+λ1λ222L2−e−γ3γ̇1t+eγ3γ̇2tωγ̇t2+λ122γ¨2eγ3+γ̇2γ̇3eγ3−γ¨1e−γ3+γ̇1γ̇3e−γ3−Lωγ̇tγ̇3t2+Lddtωγ̇t−λ1222λ2L−e−γ3γ̇1t+eγ3γ̇2tγ̇3t2,γ̇L4=1λ1λ2γ̇3t2+λ122−e−γ3γ̇1t+eγ3γ̇2t2+Lωγ̇t22,∇γ̇Lγ̇,γ̇L2=1λ1λ2γ̇3t1λ1λ2γ¨3t+λ132+λ1λ222L2−e−γ3γ̇1t+eγ3γ̇2tωγ̇t+λ122−e−γ3γ̇1t+eγ3γ̇2t·λ122γ¨2eγ3+γ̇2γ̇3eγ3−γ¨1e−γ3+γ̇1γ̇3e−γ3−Lωγ̇tγ̇3t+Lωγ̇tddtωγ̇t−λ1222λ2L−e−γ3γ̇1t+eγ3γ̇2tγ̇3t2,γ̇L6=1λ1λ2γ̇3t2+λ122−e−γ3γ̇1t+eγ3γ̇2t2+Lωγ̇t23.
By the definition ofκγL, we get Proposition 4. ☐☐Definition 5.
Letγ:a,b⟶E1,1,gLλ1,λ2 be a Euclidean C2-smooth regular curve in the Riemannian manifold E1,1,gLλ1,λ2, we define the intrinsic curvature κγ∞ of γ at γt to be
(23)κγ∞≔limL⟶∞κγL,if the limit exists.We introduce the following notation: for continuous functionsf1,f2:0,+∞⟶ℝ,
(24)f1L~f2L,asL⟶+∞⇔limL⟶∞f1Lf2L=1.Proposition 6.
Letγ:a,b⟶E1,1,gLλ1,λ2 be a Euclidean C2-smooth regular curve in the Riemannian manifold E1,1,gLλ1,λ2. Then, we have
(25)κγ∞=λ12λ24/2−e−γ3γ̇1t+eγ3γ̇2t2+γ̇3t2ωγ̇t,ifωγ̇t≠0.κγ∞=1λ1λ2γ¨3t2+λ122γ¨2eγ3+γ̇2γ̇3eγ3−γ¨1e−γ3+γ̇1γ̇3e−γ32·1λ1λ2γ̇3t2+λ122−e−γ3γ̇1t+eγ3γ̇2t2−2−1λ1λ2γ̇3t1λ1λ2γ¨3t+λ122−e−γ3γ̇1t+eγ3γ̇2tγ¨2eγ3+γ̇2γ̇3eγ3−γ¨1e−γ3+γ̇1γ̇3e−γ32·1λ1λ2γ̇3t2+λ122−e−γ3γ̇1t+eγ3γ̇2t2−31/2,ifωγ̇t=0,ddtωγ̇t=0,limL⟶∞κγLL=d/dtωγ̇t1/λ1λ2γ̇3t2+λ12/2−e−γ3γ̇1t+eγ3γ̇2t2,ifωγ̇t=0,ddtωγ̇t≠0.Proof.
Whenωγ̇t≠0, we have
(26)∇γ̇Lγ̇,∇γ̇Lγ̇~λ12λ242−e−γ3γ̇1t+eγ3γ̇2t2+γ̇3t2ωγ̇t2L2asL⟶+∞,γ̇L2~Lωγ̇t2,∇γ̇Lγ̇,γ̇L2~OL2asL⟶+∞.
Therefore, we have(27)∇γ̇Lγ̇L2γ̇L4⟶λ12λ24/2−e−γ3γ̇1t+eγ3γ̇2t2+γ̇3t2ωγ̇t2,asL⟶+∞,∇γ̇Lγ̇,γ̇L2γ̇L6⟶0,asL⟶+∞.
Ifωγ̇t≠0, by (14), we have
(28)κγ∞=λ12λ24/2−e−γ3γ̇1t+eγ3γ̇2t2+γ̇3t2ωγ̇t.
By (16) and d/dtωγ̇t=0, we have
(29)κγ∞=1λ1λ2γ¨3t2+λ122γ¨2eγ3+γ̇2γ̇3eγ3−γ¨1e−γ3+γ̇1γ̇3e−γ32·1λ1λ2γ̇3t2+λ122−e−γ3γ̇1t+eγ3γ̇2t2−2−1λ1λ2γ̇3t1λ1λ2γ¨3t+λ122−e−γ3γ̇1t+eγ3γ̇2tγ¨2eγ3+γ̇2γ̇3eγ3−γ¨1e−γ3+γ̇1γ̇3e−γ32·1λ1λ2γ̇3t2+λ122−e−γ3γ̇1t+eγ3γ̇2t2−31/2.
Whenωγ̇t=0 and d/dtωγ̇t≠0, we have
(30)∇γ̇LγL2~Lddtωγ̇t2,asL⟶+∞,γ̇L2=1λ1λ2γ̇3t2+λ122−e−γ3γ̇1t+eγ3γ̇2t2,∇γ̇Lγ̇,γ̇L2=O1asL⟶+∞.
Ifωγ̇t=0 and d/dtωγ̇t≠0, by (14), we get
(31)limL⟶∞κγLL=d/dtωγ̇t1/λ1λ2γ̇3t2+λ12/2−e−γ3γ̇1t+eγ3γ̇2t2.
☐☐


## 3. The Sub-Riemannian Limit of Geodesic Curvature of Curves on Surfaces inE1,1,gLλ1,λ2

In this section, we will compute the sub-Riemannian limit of the geodesic curvature of curves on surfaces inE1,1,gLλ1,λ2. We will say that a surface Σ⊂E1,1,gLλ1,λ2 is regular if Σ is a Euclidean C2-smooth compact and oriented surface. In particular, we will assume that there exists a Euclidean C2-smooth function u:E1,1⟶ℝ such that
(32)Σ=x,y,z∈E1,1:ux,y,z=0,and ux∂x+uy∂y+uz∂z≠0. Let ∇Hu=X1uX1+X2uX2. A point x,y,z∈Σ is called characteristic if ∇Hux,y,z=0. We define the characteristic set as follows:
(33)CΣ≔x,y,z∈Σ∣∇Hux,y,z=0.Note that the computations in the present paper will be local and away from characteristic points ofΣ. We define p≔X1u, q≔X2u, r≔X3~u, and
(34)l≔p2+q2,lL≔p2+q2+r2,p¯≔pl,q¯≔ql,pL¯≔plL,qL¯≔qlL,rL¯≔rlL.In particular,p¯2+q¯2=1. We remark that these functions are well defined at every noncharacteristic point. Let
(35)vL=pL¯X1+qL¯X2+rL¯X3~,e1=q¯X1−p¯X2,e2=rL¯p¯X1+rL¯q¯X2−llLX3~,then vL is the Riemannian unit normal vector to Σ, and e1, e2 are the orthonormal basis of Σ. On TΣ, we define a linear transformation JL:TΣ⟶TΣ such that
(36)JLe1≔e2,JLe2≔−e1.For everyU,V∈TΣ, we define ∇UΣ,LV=π∇ULV, where π:TE1,1⟶TΣ is the projection. Then, ∇Σ,L is the Levi-Civita connection on Σ with respect to the metric gLλ1,λ2. By (21), (34), and
(37)∇γ̇Σ,Lγ̇=∇γ̇Lγ̇,e1Le1+∇γ̇Lγ̇,e2Le2,we have
(38)∇γ̇Σ,Lγ̇=q¯1λ1λ2γ¨3t+λ132+λ1λ222L2−e−γ3γ̇1t+eγ3γ̇2tωγ̇t−p¯λ122γ¨2eγ3+γ̇2γ̇3eγ3−γ¨1e−γ3+γ̇1γ̇3e−γ3−Lγ̇3tωγ̇te1+r¯Lp¯1λ1λ2γ¨3t+λ132+λ1λ222L2−e−γ3γ̇1t+eγ3γ̇2tωγ̇t+r¯Lq¯λ122γ¨2eγ3+γ̇2γ̇3eγ3−γ¨1e−γ3+γ̇1γ̇3e−γ3−Lγ̇3tωγ̇t−llLL1/2ddtωγ̇t−λ1222λ2L−e−γ3γ̇1t+eγ3γ̇2tγ̇3te2.Moreover, ifωγ̇t=0, then
(39)∇γ̇Σ,Lγ̇=q¯1λ1λ2γ¨3t−p¯λ122γ¨2eγ3+γ̇2γ̇3eγ3−γ¨1e−γ3+γ̇1γ̇3e−γ3e1+r¯Lp¯1λ1λ2γ¨3t+r¯Lq¯λ122γ¨2eγ3+γ̇2γ̇3eγ3−γ¨1e−γ3+γ̇1γ̇3e−γ3−llLL1/2ddtωγ̇t−λ1222λ2L−e−γ3γ̇1t+eγ3γ̇2tγ̇3te2.Definition 7.
LetΣ⊂E1,1,gLλ1,λ2 be a regular surface, γ:a,b⟶Σ be a Euclidean C2-smooth regular curve. The geodesic curvature κγ,ΣL of γ at γt is defined as follows:
(40)κγ,ΣL≔∇γ̇Σ,Lγ̇Σ,L2γ̇Σ,L4−∇γ̇Σ,Lγ̇,γ̇Σ,L2γ̇Σ,L6.Definition 8.
LetΣ⊂E1,1,gLλ1,λ2 be a regular surface, γ:a,b⟶Σ be a Euclidean C2-smooth regular curve. We define the intrinsic geodesic curvature κγ,Σ∞ of γ at γt to be
(41)κγ,Σ∞≔limL⟶+∞κγ,ΣL,if the limit exists.Proposition 9.
LetΣ⊂E1,1,gLλ1,λ2 be a regular surface and γ:a,b⟶Σ be a Euclidean C2-smooth regular curve. Then, we have
(42)κγ,Σ∞=λ12λ24/2q¯2−e−γ3γ̇1t+eγ3γ̇2t2+p¯2γ̇3t2ωγ̇t,ifωγ̇t≠0,κγ,Σ∞=0,ifωγ̇t=0,ddtωγ̇t=0,limL⟶+∞κγ,ΣLL=d/dtωγ̇t1/λ1λ2γ̇3tq¯−λ12/2−e−γ3γ̇1t+eγ3γ̇2tp¯2,ifωγ̇t=0,ddtωγ̇t≠0.Proof.
By (17) and γ̇∈TΣ, we have
(43)γ̇t=1λ1λ2γ̇3tX1+λ122−e−γ3γ̇1t+eγ3γ̇2tX2+ωγ̇tX3.
On the other hand,(44)γ̇t=ae1+be2=aq¯X1−p¯X2+brL¯p¯X1+rL¯q¯X2−llLX3~=aq¯+brL¯p¯X1+−ap¯+brL¯q¯X2−bllLL−1/2X3.
Comparing the above equations, we get(45)aq¯+brL¯p¯=1λ1λ2γ̇3t,−ap¯+brL¯q¯=λ122−e−γ3γ̇1t+eγ3γ̇2t,−bllLL−1/2=ωγ̇t,from which
(46)a=1λ1λ2γ̇3tq¯−λ122−e−γ3γ̇1t+eγ3γ̇2tp¯,b=−lLlL1/2ωγ̇t.
This proves the following:(47)γ̇=1λ1λ2γ̇3tq¯−λ122−e−γ3γ̇1t+eγ3γ̇2tp¯e1−lLlL1/2ωγ̇te2.
By (37), we have
(48)∇γ̇Σ,Lγ̇Σ,L2=q¯1λ1λ2γ¨3t+λ132+λ1λ222L2−e−γ3γ̇1t+eγ3γ̇2tωγ̇t−p¯λ122γ¨2eγ3+γ̇2γ̇3eγ3−γ¨1e−γ3+γ̇1γ̇3e−γ3−Lγ̇3tωγ̇t2+r¯Lp¯1λ1λ2γ¨3t+λ132+λ1λ222L2−e−γ3γ̇1t+eγ3γ̇2tωγ̇t+r¯Lq¯λ122γ¨2eγ3+γ̇2γ̇3eγ3−γ¨1e−γ3+γ̇1γ̇3e−γ3−Lγ̇3tωγ̇t−llLL1/2ddtωγ̇t−λ1222λ2L−e−γ3γ̇1t+eγ3γ̇2tγ̇3t2~λ12λ242q¯2−e−γ3γ̇1t+eγ3γ̇2t2+p¯2γ̇3t2L2ωγ̇t2,asL⟶+∞.
Similarly, we get that whenωγ̇t≠0,
(49)γ̇Σ,L=1λ1λ2γ̇3tq¯−λ122−e−γ3γ̇1t+eγ3γ̇2tp¯2+LlLlωγ̇t21/2~L1/2ωγ̇t,asL⟶+∞.
By (37) and (46), we have
(50)∇γ̇Σ,Lγ̇,γ̇Σ,L~1λ1λ2γ̇3tq¯−λ122−e−γ3γ̇1t+eγ3γ̇2tp¯·q¯λ1λ2222−e−γ3γ̇1t+eγ3γ̇2t+p¯γ̇3tωγ̇tL+ddtωγ̇tωγ̇tL~N0L,asL⟶+∞,where N0 does not depend on L. By (39), we get
(51)kγ,Σ∞=limL⟶+∞kγ,ΣL=λ12λ24/2q¯2−e−γ3γ̇1t+eγ3γ̇2t2+p¯2γ̇3t2ωγ̇t,ifωγ̇t≠0.
Whenωγ̇t=0 and d/dtωγ̇t=0, we have
(52)∇γ̇Σ,Lγ̇Σ,L2=q¯1λ1λ2γ¨3t−p¯λ122γ¨2eγ3+γ̇2γ̇3eγ3−γ¨1e−γ3+γ̇1γ̇3e−γ32+r¯Lp¯1λ1λ2γ¨3t+r¯Lq¯λ122γ¨2eγ3+γ̇2γ̇3eγ3−γ¨1e−γ3+γ̇1γ̇3e−γ3+llLL1/2λ1222λ2L−e−γ3γ̇1t+eγ3γ̇2tγ̇3t2~q¯1λ1λ2γ¨3t−p¯λ122γ¨2eγ3+γ̇2γ̇3eγ3−γ¨1e−γ3+γ̇1γ̇3e−γ32,as L⟶+∞ and
(53)γ̇Σ,L2=1λ1λ2γ̇3tq¯−λ122−e−γ3γ̇1t+eγ3γ̇2tp¯2,(54)∇γ̇Σ,Lγ̇,γ̇Σ,L=1λ1λ2γ̇3tq¯−λ122−e−γ3γ̇1t+eγ3γ̇2tp¯·q¯1λ1λ2γ¨3t−p¯λ122γ¨2eγ3+γ̇2γ̇3eγ3−γ¨1e−γ3+γ̇1γ̇3e−γ3≔AB.
By (51), (52), (53), and (39), we get
(55)κγ,Σ∞=B2A4−A2B2A6=0.
Whenωγ̇t=0 and d/dtωγ̇t≠0, we have
(56)∇γ̇Σ,Lγ̇Σ,L2~Lddtωγ̇t2,asL⟶+∞,γ̇Σ,L=1λ1λ2γ̇3tq¯−λ122−e−γ3γ̇1t+eγ3γ̇2tp¯,∇γ̇Σ,Lγ̇,γ̇Σ,L=O1.
So, we get(57)limL⟶+∞κγ,ΣLL=d/dtωγ̇t1/λ1λ2γ̇3tq¯−λ12/2−e−γ3γ̇1t+eγ3γ̇2tp¯2.
☐☐Definition 10.
LetΣ⊂E1,1,gLλ1,λ2 be a regular surface. Let γ:a,b⟶Σ be a Euclidean C2-smooth regular curve. The signed geodesic curvature κγ,ΣL,s of γ at γt is defined as follows:
(58)κγ,ΣL,s≔∇γ̇Σ,Lγ̇,JLγ̇Σ,Lγ̇Σ,L3,where JL is defined by (35).Definition 11.
LetΣ⊂E1,1,gLλ1,λ2 be a regular surface. Let γ:a,b⟶Σ be a Euclidean C2-smooth regular curve. We define the intrinsic geodesic curvature κγ,Σ∞ of γ at the noncharacteristic point γt to be
(59)κγ,Σ∞,s≔limL⟶+∞κγ,ΣL,s,if the limit exists.Proposition 12.
LetΣ⊂E1,1,gLλ1,λ2 be a regular surface. Let γ:a,b⟶Σ be a Euclidean C2-smooth regular curve. Then, we have
(60)κγ,Σ∞,s=q¯λ1λ222/2−e−γ3γ̇1t+eγ3γ̇2t+p¯γ̇3tωγ̇t,ifωγ̇t≠0,κγ,Σ∞,s=0,ifωγ̇t=0,ddtωγ̇t=0,limL⟶+∞kγ,ΣL,sL=−1/λ1λ2γ̇3tq¯+λ12/2−e−γ3γ̇1t+eγ3γ̇2tp¯d/dtωγ̇t1/λ1λ2γ̇3tq¯−λ12/2−e−γ3γ̇1t+eγ3γ̇2tp¯3,ifωγ̇t=0,ddtωγ̇t≠0.Proof.
By (35)(46), we get
(61)JLγ̇=1λ1λ2γ̇3tq¯−λ122−e−γ3γ̇1t+eγ3γ̇2tp¯JLe1−lLlL1/2ωγ̇tJLe2=lLlL1/2ωγ̇te1+1λ1λ2γ̇3tq¯−λ122−e−γ3γ̇1t+eγ3γ̇2tp¯e2.
By (37) and the above equation, we have
(62)∇γ̇Σ,Lγ̇,JLγ̇=lLlL1/2ωγ̇t·q¯1λ1λ2γ¨3t+λ132+λ1λ222L2−e−γ3γ̇1t+eγ3γ̇2tωγ̇t−p¯λ122γ¨2eγ3+γ̇2γ̇3eγ3−γ¨1e−γ3+γ̇1γ̇3e−γ3−Lγ̇3tωγ̇t+1λ1λ2γ̇3tq¯−λ122−e−γ3γ̇1t+eγ3γ̇2tp¯·r¯Lp¯1λ1λ2γ¨3t+λ132+λ1λ222L2−e−γ3γ̇1t+eγ3γ̇2tωγ̇t+r¯Lq¯λ122γ¨2eγ3+γ̇2γ̇3eγ3−γ¨1e−γ3+γ̇1γ̇3e−γ3−Lγ̇3tωγ̇t−llLL1/2ddtωγ̇t−λ1222λ2L−e−γ3γ̇1t+eγ3γ̇2tγ̇3t~q¯λ1λ2222−e−γ3γ̇1t+eγ3γ̇2t+p¯γ̇3tωγ̇t2L3/2,asL⟶+∞.
So, we get(63)κγ,ΣL,s=∇γ̇Σ,Lγ̇,JLγ̇Σ,Lγ̇Σ,L3=L3/2q¯λ1λ222/2−e−γ3γ̇1t+eγ3γ̇2t+p¯γ̇3tωγ̇t2L3/2ωγ̇t3.
Furthermore,(64)κγ,Σ∞,s=limL⟶+∞kγ,ΣL,s=q¯λ1λ222/2−e−γ3γ̇1t+eγ3γ̇2t+p¯γ̇3tωγ̇t.
Whenωγ̇t=0 and d/dtωγ̇t=0, we get
(65)∇γ̇Σ,Lγ̇,JLγ̇L,Σ=1λ1λ2γ̇3tq¯−λ122−e−γ3γ̇1t+eγ3γ̇2tp¯·r¯Lp¯1λ1λ2γ¨3t+r¯Lq¯λ122γ¨2eγ3+γ̇2γ̇3eγ3−γ¨1e−γ3+γ̇1γ̇3e−γ3+llLL1/2λ1222λ2L−e−γ3γ̇1t+eγ3γ̇2tγ̇3t~N1L−1/2,asL⟶+∞,where N1 does not depend on L. So, κγ,Σ∞,s=0. When ωγ̇t=0 and d/dtωγ̇t≠0, we have
(66)∇γ̇Σ,Lγ̇,JLγ̇L,Σ~L1/2−1λ1λ2γ̇3tq¯+λ122−e−γ3γ̇1t+eγ3γ̇2tp¯ddtωγ̇t,asL⟶+∞.
We get(67)κγ,Σ∞,s=limL⟶+∞κγ,ΣL,sL=limL⟶+∞−1/λ1λ2γ̇3tq¯+λ12/2−e−γ3γ̇1t+eγ3γ̇2tp¯d/dtωγ̇tL1/21/λ1λ2γ̇3tq¯−λ12/2−e−γ3γ̇1t+eγ3γ̇2tp¯3L=−1/λ1λ2γ̇3tq¯+λ12/2−e−γ3γ̇1t+eγ3γ̇2tp¯d/dtωγ̇t1/λ1λ2γ̇3tq¯−λ12/2−e−γ3γ̇1t+eγ3γ̇2tp¯3.
☐☐


## 4. The Sub-Riemannian Limit of the Riemannian Gaussian Curvature of Surfaces inE1,1,gLλ1,λ2

In the following, we compute the sub-Riemannian limit of the Riemannian Gaussian curvature of surfaces inE1,1,gLλ1,λ2. We define the second fundamental form IIL of the embedding of Σ into E1,1,gLλ1,λ2 by
(68)IIL=∇e1LvL,e1L∇e1LvL,e2L∇e2LvL,e1L∇e2LvL,e2L.We have the following theorem similar to Theorem 4.3 in [10].Theorem 13.
The second fundamental formIIL of the embedding of Σ into E1,1,gLλ1,λ2 is given by
(69)IIL=h11h12h21h22,where
(70)h11=llLX1p¯+X2q¯−λ12r¯Lp¯q¯L−1/2,h12=h21=−lLle1,∇Hr¯LL−L2λ22+λ122Lq¯L2−p¯L2+r¯L2λ122Lq¯2−p¯2,h22=−l2lL2e2,∇HrlL+X3~r¯L+r¯Lp¯Lq¯LLλ12+r¯L3p¯q¯Lλ12.Proof.
Sincee1,vLL=0 and e2,vLL=0, we have
(71)∇e1LvL,e1L=−∇e1Le1,vLL,∇e2LvL,e2L=−∇e2Le2,vLL.
Using the definition of the connection, the identities in (8) and grouping terms, we have
(72)∇e1Le1=∇q¯X1−p¯X2Lq¯X1−p¯X2=q¯X1q¯X1−X1p¯X2−p¯∇X1LX2−p¯X2q¯X1−X2p¯X2+q¯∇X2LX1=q¯X1q¯−p¯X2q¯X1−q¯X1p¯−p¯X2p¯X2+p¯q¯λ12LX3,since p¯2+q¯2=1, we have p¯Xip¯+q¯Xiq¯=0, i=1,2,3. We have
(73)h11=−∇e1Le1,vLL=−p¯Lq¯X1q¯−p¯X2q¯−q¯Lq¯X1p¯−p¯X2p¯+r¯Lp¯q¯λ12L=llLX1p¯+X2q¯−λ12r¯Lp¯q¯L−1/2.
To computeh12 and h21, using the definition of the connection, we obtain
(74)∇e1Le2=∇q¯X1−p¯X2Lr¯Lp¯X1+r¯Lq¯X2−llLL−1/2X3=q¯X1r¯Lp¯X1+X1r¯Lq¯X2+r¯Lq¯∇X1LX2−L−1/2X1llLX3−llLL−1/2∇X1LX3−p¯X2r¯Lp¯X1+r¯Lp¯∇X2LX1+X2r¯Lq¯X2−L−1/2X2llLX3−llLL−1/2∇X2LX3=q¯X1r¯Lp¯−p¯X2r¯Lp¯+p¯llLL−1/2λ12+λ22L2X1+q¯X1r¯Lq¯−p¯X2r¯Lq¯−q¯llLL−1/2λ12−λ22L2X2+p¯X2L−1/2llL−q¯X1L−1/2llL+r¯L2λ22+λ12Lp¯2−q¯2X3.
Next, we compute the inner product of this withvL. Using the product rule and the identity q¯Lp¯=p¯Lq¯, we obtain
(75)∇e1Le2,vLL=p¯Lq¯X1r¯Lp¯−p¯X2r¯Lp¯+p¯llLL−1/2λ12+λ22L2+q¯Lq¯X1r¯Lq¯−p¯X2r¯Lq¯−q¯llLL−1/2λ12−λ22L2+r¯Lp¯X2llL−q¯X1llL+r¯L2λ22L+λ12Lp¯2−q¯2=p¯Lq¯p¯X1r¯L+r¯LX1p¯−p¯Lp¯p¯X2r¯L+r¯LX2p¯+q¯Lq¯q¯X1r¯L+r¯LX1q¯−q¯Lp¯q¯X2r¯L+r¯LX2q¯+L2λ22p¯L2+q¯L2+r¯L2+r¯Lp¯X2llL−r¯Lq¯X1llL+λ12Lp¯L2−q¯L2+r¯L22λ12Lp¯2−q¯2.
The identitiesp¯L2+q¯L2+r¯L2=1 and p¯2+q¯2=1 yield
(76)∇e1Le2,vLL=llLq¯X1r¯L−llLp¯X2r¯L+r¯Lp¯X2llL−r¯Lq¯X1llL+L2λ22+λ12Lp¯L2−q¯L2+r¯L22λ12Lp¯2−q¯2=llLe1,∇Hr¯LL−r¯Le1,∇HllLL+L2λ22+λ12Lp¯L2−q¯L2+r¯L22λ12Lp¯2−q¯2.
Finally, we use the identityl/lL−lL/l∇Hr¯L=r¯L∇Hl/lL, we have
(77)∇e1Le2,vLL=lLle1,∇Hr¯LL+L2λ22+λ12Lp¯L2−q¯L2+r¯L22λ12Lp¯2−q¯2.
Therefore, we have(78)h21=h12=−∇e1Le2,vLL=−lLle1,∇Hr¯LL−L2λ22+λ122Lq¯L2−p¯L2+r¯L2λ122Lq¯2−p¯2.
Since∇e2vL,e2L=−∇e2e2,vLL, using the definition of connection, the identities in (9) and grouping terms, we have
(79)∇e2Le2=∇r¯Lp¯X1+r¯Lq¯X2−l/lLX3~Lr¯Lp¯X1+r¯Lq¯X2−llLX3~=r¯Lp¯X1r¯Lp¯X1+X1r¯Lq¯X2+r¯Lq¯∇X1LX2−L−1/2X1llLX3−L−1/2llL∇X1LX3+r¯Lq¯X2r¯Lp¯X1+X2r¯Lq¯X2+r¯Lp¯∇X2LX1−L−1/2X2llLX3−L−1/2llL∇X2LX3−llLL−1/2X3r¯Lp¯X1+r¯Lp¯∇X3LX1+X3r¯Lq¯X2+r¯Lq¯∇X3LX2−L−1/2X3llLX3=r¯Lp¯X1r¯Lp¯+r¯Lq¯X2r¯Lp¯−L−1/2llLX3r¯Lp¯−r¯Lq¯L−1/2llLλ12+λ22LX1+r¯Lp¯X1r¯Lq¯+r¯Lq¯X2r¯Lq¯−L−1/2llLX3r¯Lq¯−r¯Lp¯L−1/2llL−λ22LX2+−r¯Lp¯L−1/2X1llL−r¯Lq¯X2L−1/2X2llL+L−1llLX3llL−r¯L2p¯q¯λ12LX3.
Taking the inner product withvL yields
(80)∇e2Le2,vLL=p¯Lr¯Lp¯X1r¯Lp¯+r¯Lq¯X2r¯Lp¯−L−1/2llLX3r¯Lp¯−r¯Lq¯L−1/2llLλ12+λ22L+q¯Lr¯Lp¯X1r¯Lq¯+r¯Lq¯X2r¯Lq¯−L−1/2llLX3r¯Lq¯−r¯Lp¯L−1/2llL−λ22L+r¯L−r¯Lp¯X1llL−r¯Lq¯X2X2llL+L−1/2llLX3llL−r¯L2p¯q¯λ12L.
To simplify this, first use the product rule for the terms involvingXip¯r¯L and Xiq¯r¯L together with the identities p¯Xip¯+q¯Xiq¯=0, r¯L∇Hl/lL=l/lL−lL/l∇Hr¯L, and p¯2+q¯2=1. Under these simplifications, terms involving Xip¯ and Xiq¯ cancel and one is left with terms involving components of ∇r¯L:
(81)∇e2Le2,vLL=p¯Lp¯L+p¯rlX1r¯L+q¯Lq¯L+q¯rlX2r¯L−X3~r¯L−r¯Lp¯Lq¯LLλ12−r¯L3p¯q¯Lλ12.
We conclude by rewriting the expressionXir¯L in terms of Xir/l. Therefore, we have
(82)h22=−∇e2Le2,vLL=−l2lL2e2,∇HrlL+X3~r¯L+r¯Lp¯Lq¯LLλ12+r¯L3p¯q¯Lλ12.☐☐The Riemannian mean curvatureHL of Σ is defined by
(83)HL≔trIIL=llLX1p¯+X2q¯−λ12r¯Lp¯q¯L−1/2−l2lL2e2,∇HrlL+X3~r¯L+r¯Lp¯Lq¯LLλ12+r¯L3p¯q¯Lλ12.Define the curvature of a connection∇ by
(84)RX,YZ=∇X∇YZ−∇Y∇XZ−∇X,YZ.Let(85)KΣ,Le1,e2=−RΣ,Le1,e2e1,e2Σ,L,KLe1,e2=−RLe1,e2e1,e2L.By the Gauss equation, we have(86)KΣ,Le1,e2=KLe1,e2+detIIL.Proposition 14.
Away from characteristic point, the horizontal mean curvatureH∞ of Σ∈E1,1,gLλ1,λ2 is given by
(87)H∞=limL⟶∞HL=X1p¯+X2q¯.Proof.
By(88)l2lL2e2,∇HrlL=p¯rlX1r¯L+q¯rlX2r¯L=OL−1,llLX1p¯+X2q¯⟶X1p¯+X2q¯,X3~r¯L⟶0,λ12r¯Lp¯q¯L−1/2⟶0,r¯Lp¯Lq¯LLλ12+r¯L3p¯q¯Lλ12⟶0,we get (86) ☐☐By Proposition1 and (83), we have the following lemma.Lemma 15.
LetE1,1,gLλ1,λ2 be the group of rigid motions of the Minkowski plane with a general left-invariant metric. Then, we have
(89)RLX1,X2X1=−λ14+2λ12λ22L+3λ24L24LX2,RLX1,X2X2=λ14−2λ12λ22L−3λ24L24LX1,RLX1,X2X3=0,RLX1,X3X1=3λ14+2λ12λ22L−λ24L24LX3,RLX1,X3X2=0,RLX1,X3X3=−3λ14−2λ12λ22L+λ24L24X1,RLX2,X3X1=0,RLX2,X3X2=−λ14−2λ12λ22L−λ24L24LX3,RLX2,X3X3=λ14+2λ12λ22L+λ24L24X2.Proposition 16.
Away from characteristic points, we have(90)KΣ,Le1,e2=−e1,∇HX3u∇HuL−λ24X3u2l2,asL⟶∞.Proof.
We compute(91)RLe1,e2e1=RLq¯X1−p¯X2,r¯Lp¯X1+r¯Lq¯X2−llLLX3q¯X1−p¯X2=r¯Lp¯q¯2RLX1,X1X1+r¯Lq¯3RLX1,X2X1−lq¯2lLLRLX1,X3X1−r¯Lp¯2q¯RLX2,X1X1−r¯Lp¯q¯2RLX2,X2X1+lp¯q¯lLLRLX2,X3X1−r¯Lp¯2q¯RLX1,X1X2−r¯Lp¯q¯2RLX1,X2X2+lp¯q¯lLLRLX1,X3X2+r¯Lp¯3RLX2,X1X2+r¯Lp¯2q¯RLX2,X2X2−lp¯2lLLRLX2,X3X2=r¯Lq¯3RLX1,X2X1−q¯2llLLRLX1,X3X1−r¯Lp¯2q¯RLX2,X1X1−r¯Lq¯2p¯RLX1,X2X2+r¯Lp¯3RLX2,X1X2−p¯2llLLRLX2,X3X2=−r¯Lp¯λ14−2λ12λ22L−3λ24L24LX1+r¯Lq¯−λ14+2λ12λ22L+3λ24L24LX2−llLLq¯23λ14+2λ12λ22L−λ24L24L+p¯2−λ14−2λ12λ22L−λ24L24LX3,(92)KLe1,e2=−RLe1,e2e1,e2L=−r¯L2λ12λ222−λ144L+3λ24L4−llLq¯2λ12λ222−λ24L4+3λ144L+llLp¯2λ12λ222+λ144L+λ24L4~llL2λ24L4−3λ244X3u2l2−λ12λ22q¯2+λ12λ222,asL⟶∞.
By Theorem13 and ∇Hr¯L=L−1/2∇HX3u/∇Hu+OL−1 as L⟶+∞, we get
(93)detIIL=h11h22−h122=−λ24L4−e1,∇HX3u∇HuL+λ12λ222q¯2−p¯2+OL−1/2asL⟶∞.
By (85), (91), and (92), we get the desired equation. ☐☐


## 5. A Gauss-Bonnet Theorem inE1,1,gLλ1,λ2

In this section, we will prove the Gauss-Bonnet theorem in the group of rigid motions of the Minkowski planeE1,1,gLλ1,λ2. Let us first consider the case of a regular curve γ:a,b⟶E1,1,gLλ1,λ2, and we define the Riemannian length measure as follows:
(94)dsL=γ̇Ldt.Lemma 17.
Letγ:a,b⟶E1,1,gLλ1,λ2 be a Euclidean C2-smooth regular curve. Let
(95)ds≔ωγ̇tdt,ds¯≔121ωγ̇t1λ1λ2γ̇3t2+λ122−eγ3γ̇1t+eγ3γ̇2t2dt.
Then, we have(96)limL⟶∞1L∫γdsL=∫abds.
Whenωγ̇t≠0, we have
(97)1LdsL=ds+ds¯L−1+OL−2,asL⟶+∞.
Whenωγ̇t=0, we have
(98)1LdsL=1L1λ1λ2γ̇3t2+λ122−eγ3γ̇1t+eγ3γ̇2t2dt.Proof.
We know that(99)γ̇tL=1λ1λ2γ̇3t2+λ122−eγ3γ̇1t+eγ3γ̇2t2+Lωγ̇t2.
Similar to the proof of Lemma 6.1 in [4], we can prove
(100)limL⟶∞1L∫γγ̇tLdt=∫ablimL⟶∞1Lγ̇tLdt=∫ablimL⟶∞1L1λ1λ2γ̇3t2+λ122−eγ3γ̇1t+eγ3γ̇2t2+Lωγ̇t2dt=∫abωγ̇tdt=∫abds,is desired. When ωγ̇t≠0, we have
(101)1LdsL=L−11λ1λ2γ̇3t2+λ122−eγ3γ̇1t+eγ3γ̇2t2+ωγ̇t2dt.
Using the Taylor expansion, we can prove(102)1LdsL=ds+ds¯L−1+OL−2,asL⟶+∞.
From the definition ofdsL and ωγ̇t=0, we get
(103)1LdsL=1L1λ1λ2γ̇3t2+λ122−eγ3γ̇1t+eγ3γ̇2t2dt.
☐☐Proposition 18.
LetΣ⊂E1,1,gLλ1,λ2 be a Euclidean C2-smooth surface and Σ=u=0 and dσΣ,L denote the surface measure on Σ with respect to the Rimannian metric gLλ1,λ2. Let
(104)dσΣ≔p¯ω2−q¯ω1∧ω,dσ¯Σ≔X3ulω1∧ω2−X3u22l2p¯ω2−q¯ω1∧ω.
Then, we have(105)1LdσΣ,L=dσΣ+dσ¯ΣL−1+OL−2,asL⟶+∞.
IfΣ=fD with f=fu1,u2=f1,f2,f3:D⊂ℝ2⟶E1,1,gLλ1,λ2, then
(106)limL⟶∞1L∫ΣdσΣ,L=∫Dσ12+σ221/2du1du2,where
(107)σ1=f1u1f2u2λ1λ2−f2u1f1u2λ1λ2,σ2=f3u2f1u1−f3u1f1u222e−z1λ1+f3u2f2u1−f3u1f2u222ez1λ12.Proof.
It is well known thatgLX1,·=ω1, gLX2,·=ω2, and gLX3,·=Lω. We define e1∗≔gLe1,· and e2∗≔gLe2,·; then, we have
(108)e1∗=q¯ω1−p¯ω2,e2∗=r¯Lp¯ω1+r¯Lq¯ω2−llLL1/2ω.
Therefore, we have(109)1LdσΣ,L=1Le1∗∧e2∗=llLp¯ω2−q¯ω1∧ω+1Lr¯Lω1∧ω2.
Recalling, we know that(110)r¯L=X3uL−1/2p2+q2+L−1X3u2,and with the Taylor expansion, we have
(111)1lL=1l−12l3X3uL−1+OL−2,asL⟶+∞,and we get (104). By (4), we have
(112)fu1=f1u1∂x+f2u1∂y+f3u1∂z=f3u11λ1λ2X1+f1u1−22e−zλ1+f2u122ezλ1X2+Lf1u1−22e−zλ2+f2u1−22ezλ2X3~=fu11X1+fu12X2+fu13X3~,fu2=f1u2∂x+f2u2∂y+f3u2∂z=f3u21λ1λ2X1+f1u2−22e−zλ1+f2u222ezλ1X2+Lf1u2−22e−zλ2+f2u2−22ezλ2X3~=fu21X1+fu22X2+fu23X3~.
Let(113)v¯L=X1X2X3~fu11fu12fu13fu21fu22fu23=Lf1u1f2u2λ1λ2−f2u1f1u2λ1λ2X1+Lf3u2f1u1−f3u1f1u222e−z1λ1+f3u2f2u1−f3u1f2u222ez1λ1X2+f3u2f1u1−f3u1f1u222e−z1λ2+f3u1f2u2−f3u2f2u122ez1λ2X3~.
We know thatdσΣ,L=detgijdu1du2,gij=gLfui,fuj, and
(114)detgij=v¯LL2=Lf1u1f2u2λ1λ2−f2u1f1u2λ1λ22+Lf3u2f1u1−f3u1f1u222e−z1λ1+f3u2f2u1−f3u1f2u222ez1λ12+f3u2f1u1−f3u1f1u222e−z1λ2+f3u1f2u2−f3u2f2u122ez1λ22.
So by the dominated convergence theorem, we get (105).☐☐Similar to the proof of Theorem 4.3 in [6], we get a generalized Gauss-Bonnet theorem in E1,1,gLλ1,λ2 as shown in the following theorem.Theorem 19.
LetΣ⊂E1,1,gLλ1,λ2 be a regular surface with finitely many boundary components ∂Σi, i∈1,⋯,n, given by Euclidean C2-smooth regular and closed curves γi:0,2π⟶∂Σi. Suppose that the characteristic set CΣ satisfies H1CΣ=0, where H1CΣ denotes the Euclidean 1-dimensional Hausdorff measure of CΣ and that ∇HuH−1 is locally summable with respect to the Euclidean 2-dimensional Hausdorff measure near the characteristic set CΣ. Then, we have
(115)∫ΣAdσΣ+∑i=1n∫γiκγi,Σ∞,sds=0,where A=−e1,∇HX3u/∇HuL−λ24X3u2/l2.Proof.
Using similar discussions in [4, 5], we assume that all points satisfy ωγ̇it≠0 on the curve γi. Recalling the result in Proposition 12 indicates
(116)κγi,ΣL,s=κγi,Σ∞,s+OL−1/2.
According to the Gauss-Bonnet theorem, we get(117)∫ΣKΣ,L1LdσΣ,L+∑i=1n∫γiκγi,ΣL,s1LdsL=2πχΣL.
So by (115), (116), (104), (89), (96), and (97), we get
(118)∫ΣAdσΣ+∑i=1n∫γiκγi,Σ∞,sdsL+OL−1/2=2πχΣL,where A=−e1,∇HX3u/∇HuL−λ24X3u2/l2. Let L go to the infinity, and use the dominated convergence theorem, we get the desired result.☐☐



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*Source: 1431082-2021-08-10.xml*