# Sub-Lorentzian Geometry of Curves and Surfaces in a Lorentzian Lie Group

Authors: Haiming Liu; Jianyun Guan
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

We consider the sub-Lorentzian geometry of curves and surfaces in the Lie groupE1,1. Firstly, as an application of Riemannian approximants scheme, we give the definition of Lorentzian approximants scheme for E1,1 which is a sequence of Lorentzian manifolds denoted by Eλ1,λ2L. By using the Koszul formula, we calculate the expressions of Levi-Civita connection and curvature tensor in the Lorentzian approximants of Eλ1,λ2L in terms of the basis E1,E2,E3. These expressions will be used to define the notions of the intrinsic curvature for curves, the intrinsic geodesic curvature of curves on surfaces, and the intrinsic Gaussian curvature of surfaces away from characteristic points. Furthermore, we derive the expressions of those curvatures and prove two generalized Gauss-Bonnet theorems in Eλ1,λ2L.


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## Body

## 1. Introduction

In recent years, there has been done a lot of research concerning Gauss-Bonnet theorems in three-dimensional Lie groups with sub-Riemannian geometric structures. One of the reasons for this success is the discovery of the method of Riemannian approximations. Initial work by Balogh et al. proved a Heisenberg version of the Gauss-Bonnet theorem with the help of the method of Riemannian approximations [1, 2]. Hereafter the above work, many research on Gauss-Bonnet theorem have been gained; we refer the monograph [3–6]. In particular, Wang and Wei proved Gauss-Bonnet theorems on the affine group [3], the group of rigid motions of the Minkowski plane [3, 4], the BCV spaces [5], and the Lorentzian Heisenberg group [4, 6]. Inspired by their work, we proved Gauss-Bonnet theorems in the rototranslation group [7, 8], Lorentzian Sasakian space forms, and the group of rigid motions of Minkowski plane with the general left-invariant metric [9, 10]. Riemannian approximations can be extended to the case for any Lie group equipped with left-invariant Lorentzian metric g named Lorentzian approximations. In particular, one can consider a sequence of Lorentzian manifolds denoted by G,gL, where a family of metrics gL=−ω1⊗ω1+ω2⊗ω2+Lω⊗ω, for L>0, is essentially obtained as an anisotropic blow-up of the Lorentzian metric g. Then, one can define sub-Lorentzian objects as limits of horizontal objects in G,gL, since the intrinsic horizontal geometry does not change with L. Some typical works of Lorentzian approximations in the Lorentzian Heisenberg group are obtained in [4, 6]. For Lorentzian Sasakian space forms, see [9].In this paper, we consider sub-Lorentzian geometry of curves and surfaces onE1,1. The group of rigid motions of Minkowski plane E1,1 has two left-invariant Lorentzian metrics g1 and g2 [11, 12]. Onda proved that the metric g1 is a Lorentz Ricci soliton [12]. In [13], Patrangenaru proved that any left-invariant metric on E1,1 is isometric to one of the metric gλ1,λ2,λ3 with λ1≥λ2>0 and λ3=1/λ1λ2. In [14], the metric gλ1,λ2,λ3 was denoted by gλ1,λ2=gλ1,λ2,1/λ1λ2.E1,1 with the metric gλ1,λ2 provides a natural 2-parametric deformation family of Sol3=E1,1,g1,1 which is the model space of solve geometry in the eight model geometries of Thurston, which makes E1,1,gλ1,λ2 very interesting and important [14, 15]. However, very little is known about sub-Lorentzian geometry of E1,1. In this paper, we focus on the general left-invariant Lorentzian metric g=−ω1⊗ω1+ω2⊗ω2+ω⊗ω on E1,1, where the coframe ω1=1/λ1λ2du3,ω2=λ1/2−e−u3du1+eu3du2,ω=−λ2/2e−u3du1+eu3du2. By using the method of Lorentzian approximations, we consider a sequence of Lorentzian manifolds Eλ1,λ2L=E1,1,gL, where gL=−ω1⊗ω1+ω2⊗ω2+Lω⊗ω, for L>0, is an anisotropic blow-up of the Lorentzian metric g. Using the Koszul formula, we calculate the expressions of Levi-Civita connection and curvature tensor in the Lorentzian approximants of Eλ1,λ2L in terms of the basis E1,E2,E3. We define the notions of the intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on spacelike surfaces and Lorentzian surfaces, and the intrinsic Gaussian curvature of spacelike surfaces and Lorentzian surfaces away from characteristic points. Furthermore, we derive the expressions of those curvatures and prove two generalized Gauss-Bonnet theorems in Eλ1,λ2L.This paper is organized as follows. In Section2, we introduce the Lorentzian approximations of E1,1,g and calculate the expressions of corresponding Levi-Civita connection in terms of the basis E1,E2,E3. Furthermore, we define the notions of geodesic curvature and intrinsic geodesic curvature of curves in Eλ1,λ2L. We get the expressions of those curvatures and give an example. In Sections 3 and 4, we compute intrinsic geodesic curvatures of regular curves on Lorentzian surfaces and the intrinsic Gaussian curvature of Lorentzian surfaces in Eλ1,λ2L. We also give two examples. In Section 5, we get the first Gauss-Bonnet theorem in Eλ1,λ2L. In Section 6, we compute intrinsic geodesic curvature of curves on spacelike surfaces and the intrinsic Gaussian curvature of spacelike surfaces in Eλ1,λ2L and we get the second Guass-Bonnet theorem.


## 2. Curvatures for Curves in Lorentzian ApproximationsEλ1,λ2L

In this section, some basic notions in the Lorentzian group of rigid motions of the Minkowski plane group will be introduced. LetE1,1 be the motion group of the Minkowski 2-space. This consists of all matrices of the form
(1)E1,1=eu30u10e−u3u2001u1,u2,u3∈ℝ.Topologically,E1,1 is a Lie group which is diffeomorphic to ℝ3. Its Lie algebra has a basis consisting of
(2)E1=λ1λ2∂∂u3,(3)E2=1λ12−eu3∂∂u1+e−u3∂∂u2,(4)E3=−1λ22eu3∂∂u1+e−u3∂∂u2,for which
(5)E1,E2=λ22E3,(6)E2,E3=0,(7)E1,E3=λ12E2.From the above equations in Equation (2), we get that
(8)∂∂u1=−22e−u3λ1E2+λ2E3,(9)∂∂u2=22eu3λ1E2−λ2E3,(10)∂∂u3=1λ1λ2E1.We denotespanE1,E2,E3=TE1,1. Let H=spanE1,E2 be the horizontal distribution on E1,1 and
(11)ω1=1λ1λ2du3,(12)ω2=λ12−e−u3du1+eu3du2,(13)ω=−λ22e−u3du1+eu3du2.Then,H=kerω. We consider the left-invariant Lorentzian metrics given by g=−ω1⊗ω1+ω2⊗ω2+ω⊗ω and its anisotropic blow-up gL=−ω1⊗ω1+ω2⊗ω2+Lω⊗ω, for L>0. We call E1,1,gL the Lorentzian approximants of E1,1 and denote by Eλ1,λ2L=E1,1,gL throughout the paper. It is easy to check that g=g1 is the Lorentzian metric on E1,1 and E1,E2,E3~≔L−1/2E3 are pseudo-orthonormal basis on TE1,1 with respect to gL. Then, the Levi-Civita connection ∇E,L of Eλ1,λ2L is given by following proposition.Proposition 1.
The Levi-Civita connection∇E,L relative to the coordinate frame E1,E2,E3~ of Eλ1,λ2L is described as follows:
(14)∇EjE,LEj=0,1≤j≤3,∇E1E,LE2=λ22L−λ122LE3,∇E2E,LE1=−λ22L−λ122LE3,∇E1E,LE3=λ12−λ22L2E2,∇E3E,LE1=−λ12−λ22L2E2,∇E2E,LE3=∇E3E,LE2=−λ12+λ22L2E1.Proof.
To derive the above expressions, it is useful to recall the following Koszul identity from the famous proof of the unique of Riemannian connection:(15)2∇EiE,LEj,EkL=Ei,Ej,EkL−Ej,Ek,EiL+EkEi,EjL,where i,j,k=1,2,3. By Equations (3) and (14), the first formula in Equation (11) is as follows:
(16)2∇EjE,LEj,EkL=Ej,Ej,EkL−Ej,Ek,EjL+Ek,Ej,EjL=−Ej,Ek,EjL+Ek,Ej,EjL=2Ek,Ej,EjL.
Whenj=1, we compute ∇E1E,LE1,EkL=Ek,E1,E1L. It follows that ∇E1E,LE1,E1L=0,∇E1E,LE1,E2L=E2,E1,E1L=−λ22E3,E1L=0 and ∇E1E,LE1,E3L=E3,E1,E1L=0. Hence, ∇E1E,LE1=0. Similarly, ∇E2E,LE2=0 and ∇E3E,LE3=0. Moreover, other expressions follow some similar computations.
A parametrized curve inEλ1,λ2L is a map α:I⟶Eλ1,λ2L, where I is an open interval in ℝ, where αt=α1t,α2t,α3t and each function αit has derivatives of two orders, for all t∈I. Such α is called C2-smooth. The regular curve α is called regular provided that there does not exist t∈I with α̇t=dαt/dt=0. We call α a spacelike curve, timelike curve, or null curve if α̇t=dαt/dt is a spacelike vector, timelike vector, or null vector at any t∈I, respectively.Definition 2.
For an arbitraryC1-smooth regular curve α:I⟶Eλ1,λ2L, αt=α1t,α2t,α3t, we say that αt is a horizontal point of α provided the following function ωα̇t satisfies the following:
(17)ωα̇t=−λ22e−α3du1+eα3du2α̇1t∂∂u1+α̇2t∂∂u2+α̇3t∂∂u3=−λ222e−α3α̇1t+eα3α̇2t=0.Definition 3.
For an arbitraryC2-smooth regular curve α:I⟶Eλ1,λ2L, αt=α1t,α2t,α3t, we define the geodesic curvature καE,L of α at t in the following way.
(18)καE,L≔∇α̇E,Lα̇L2α̇L4−∇α̇E,Lα̇,α̇L2α̇,α̇L3,if ∇α̇E,Lα̇ is a spacelike vector.
(19)καE,L≔∇α̇E,Lα̇L2α̇L4+∇α̇E,Lα̇,α̇L2α̇,α̇L3,if ∇α̇E,Lα̇ is a timelike vector.Proposition 4.
There are formulae of curvature forC2-smooth regular curve α:I⟶Eλ1,λ2L.(1)
If∇α̇E,Lα̇ is a spacelike vector, then
(20)καE,L=−1λ1λ2α¨3t−λ132+λ1λ222L2−e−α3α̇1t+eα3α̇2tωα̇t2+λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3−Lωα̇tα̇3t2+Lddtωα̇t−λ1222λ2L−e−α3α̇1t+eα3α̇2tα̇3t2·−1λ1λ2α̇3t2+λ122−e−α3α̇1t+eα3α̇2t2+Lωα̇t2−2−κ21/2,where
(21)κ2=−1λ1λ2α̇3t1λ1λ2α¨3t+λ132+λ1λ222L2−e−α3α̇1t+eα3α̇2tωα̇t+λ122−e−α3α̇1+eα3α̇2λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3−Lωα̇α̇3+Lωα̇tddtωα̇t−λ1222λ2L−e−α3α̇1t+eα3α̇2tα̇3t2·−1λ1λ2α̇3t2+λ122−e−α3α̇1t+eα3α̇2t2+Lωα̇t2−3.In particular, ifαt is a horizontal point of α,
(22)καE,L=−1λ1λ2α¨3t2+λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α32+Lddtωα̇t−λ1222λ2L−e−α3α̇1t+eα3α̇2tα̇3t2·−1λ1λ2α̇3t2+λ122−e−α3α̇1t+eα3α̇2t2−2−1λ1λ2α̇3t1λ1λ2α¨3t+λ122−e−α3α̇1t+eα3α̇2tλ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α32·−1λ1λ2α̇3t2+λ122−e−α3α̇1t+eα3α̇2t2−31/2.(2)
If∇α̇E,Lα̇ is a timelike vector, then
(23)καE,L=1λ1λ2α¨3t−λ132+λ1λ222L2−e−α3α̇1t+eα3α̇2tωα̇t2−λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3−Lωα̇tα̇3t2−Lddtωα̇t−λ1222λ2L−e−α3α̇1t+eα3α̇2tα̇3t2·−1λ1λ2α̇3t2+λ122−e−α3α̇1t+eα3α̇2t2+Lωα̇t2−2+1λ1λ2α̇3t1λ1λ2α¨3t+λ132+λ1λ222L2−e−α3α̇1t+eα3α̇2tωα̇t+λ122−e−α3α̇1+eα3α̇2λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3−Lωα̇α̇3+Lωα̇tddtωα̇t−λ1222λ2L−e−α3α̇1t+eα3α̇2tα̇3t2·−1λ1λ2α̇3t2+λ122−e−α3α̇1t+eα3α̇2t2+Lωα̇t2−31/2.In particular, ifαt is a horizontal point of α,
(24)καE,L=1λ1λ2α¨3t2−λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α32−Lddtωα̇t−λ1222λ2L−e−α3α̇1t+eα3α̇2tα̇3t2·−1λ1λ2α̇3t2+λ122−e−α3α̇1t+eα3α̇2t2−2+1λ1λ2α̇3t1λ1λ2α¨3t+λ122−e−α3α̇1t+eα3α̇2tλ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α32·−1λ1λ2α̇3t2+λ122−e−α3α̇1t+eα3α̇2t2−31/2.Proof.
By Equation (8), we have the following:
(25)α̇t=α̇1t∂∂x+α̇2t∂∂y+α̇3t∂∂z=−22e−α3λ1E2+λ2E3α̇1t+22eα3λ1E2−λ2E3α̇2t+1λ1λ2E1α̇3t=1λ1λ2α̇3tE1+λ122−e−α3α̇1t+eα3α̇2tE2+ωα̇tE3.
By Proposition1 and Equation (25), we get the following:
(26)∇α̇E,LE1=1λ1λ2α̇3t∇E1E,LE1+λ122−e−α3α̇1t+eα3α̇2t∇E2E,LE1+ωα̇t∇E3E,LE1=−λ12−λ22L2ωα̇tE2+−λ1λ222L−λ1324L−e−α3α̇1t+eα3α̇2tE3,(27)∇α̇E,LE2=1λ1λ2α̇3t∇E1E,LE2+λ122−e−α3α̇1t+eα3α̇2t∇E2E,LE2+ωα̇t∇E3E,LE2=−λ12+λ22L2ωα̇tE1+λ22L−λ122λ1λ2Lα̇3tE3,(28)∇α̇E,LE3=1λ1λ2α̇3t∇E1E,LE3+λ122−e−α3α̇1t+eα3α̇2t∇E2E,LE3+ωα̇t∇E3E,LE3=−λ132+λ1λ222L4−e−α3α̇1t+eα3α̇2tE1+λ12−λ22L2λ1λ2α̇3tE2.
Using Equations (25) and (26), we have the following:
(29)∇α̇E,Lα̇=∇α̇E,L1λ1λ2α̇3tE1+λ122−e−α3α̇1t+eα3α̇2tE2+ωα̇tE3=−1λ1λ2α¨3t−λ132+λ1λ222L2−e−α3α̇1t+eα3α̇2tωα̇tE1+λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3−Lωα̇tα̇3tE2+ddtωα̇t−λ1222λ2L−e−α3α̇1t+eα3α̇2tα̇3tE3.
By Equations (14), (25), and (29), we obtain the following:
(30)∇α̇E,Lα̇L2=−1λ1λ2α¨3t−λ132+λ1λ222L2−e−α3α̇1t+eα3α̇2tωα̇t2+λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3−Lωα̇tα̇3t2+Lddtωα̇t−λ1222λ2L−e−α3α̇1t+eα3α̇2tα̇3t2,α̇L4=−1λ1λ2α̇3t2+λ122−e−α3α̇1t+eα3α̇2t2+Lωα̇t22,∇α̇E,Lα̇,α̇L2=1λ1λ2α̇3t1λ1λ2α¨3t−λ132+λ1λ222L2−e−α3α̇1t+eα3α̇2tωα̇t+λ122−e−α3α̇1t+eα3α̇2t·λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3−Lωα̇tα̇3t+Lωα̇tddtωα̇t−λ1222λ2L−e−α3α̇1t+eα3α̇2tα̇3t2,α̇,α̇L3=−1λ1λ2α̇3t2+λ122−e−α3α̇1t+eα3α̇2t2+Lωα̇t23.By the definition ofκαE,L, we get the desired formulae of curvature.Definition 5.
Letα:I⟶Eλ1,λ2L be a C2-smooth regular curve, we define the intrinsic curvature καE,∞ of α at αt to be
(31)καE,∞≔limL⟶∞καE,L,if the limit exists.
To derive the expression of intrinsic curvature, we need the following notion: for continuous functionsh1,h2:0,+∞⟶ℝ,
(32)h1L~h2L,asL⟶+∞⇔limL⟶∞h1Lh2L=1.Proposition 6.
Letα:I⟶Eλ1,λ2L be a C2-smooth regular curve in Eλ1,λ2L.
(1)
If∇α̇E,Lα̇ is a spacelike vector and −λ12λ24/2−e−α3α̇1t+eα3α̇2t2+α̇3t2>0, then we have the following formula of καE,∞. If ωα̇t≠0, then(33)καE,∞=−λ12λ24/2−e−α3α̇1t+eα3α̇2t2+α̇3t2ωα̇t.
Ifωα̇t=0 and d/dtωα̇t=0, then
(34)καE,∞=−1λ1λ2α¨3t2+λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α32·−1λ1λ2α̇3t2+λ122−e−α3α̇1t+eα3α̇2t2−2−−1λ1λ2α̇3t1λ1λ2α¨3t+λ122−e−α3α̇1t+eα3α̇2tα¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α32·−1λ1λ2α̇3t2+λ122−e−α3α̇1t+eα3α̇2t2−31/2.
Ifωα̇t=0 and d/dtωα̇t≠0, then
(35)limL⟶∞καE,LL=d/dtωα̇t−1/λ1λ2α̇3t2+λ12/2−e−α3α̇1t+eα3α̇2t2.(2)
If∇α̇E,Lα̇ is a timelike vector and λ12λ24/2−e−α3α̇1t+eα3α̇2t2−α̇3t2>0, then we have the following formula of καE,∞. If ωα̇t≠0, then(36)καE,∞=λ12λ24/2−e−α3α̇1t+eα3α̇2t2−α̇3t2ωα̇t.
Ifωα̇t=0 and d/dtωα̇t=0, then
(37)καE,∞=−−1λ1λ2α¨3t2+λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α32·−1λ1λ2α̇3t2+λ122−e−α3α̇1t+eα3α̇2t2−2+−1λ1λ2α̇3t1λ1λ2α¨3t+λ122−e−α3α̇1t+eα3α̇2tα¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α32·−1λ1λ2α̇3t2+λ122−e−α3α̇1t+eα3α̇2t2−31/2.Proof.
(1)
If∇α̇Lα̇ is a spacelike vector, when ωα̇t≠0, we have the following:(38)∇α̇E,Lα̇,∇α̇E,Lα̇~−λ12λ242−e−α3α̇1t+eα3α̇2t2+α̇3t2ωα̇t2L2asL⟶+∞,α̇,α̇L~Lωα̇t2,∇α̇E,Lα̇,α̇L2~OL2asL⟶+∞.
Therefore,(39)∇α̇E,Lα̇L2α̇L4⟶−λ12λ24/2−e−α3α̇1t+eα3α̇2t2+α̇3t2ωα̇t2asL⟶+∞,∇α̇E,Lα̇,α̇L2α̇,α̇L3⟶0asL⟶+∞.
Ifωα̇t≠0, by Equation (7), we have the following:
(40)καE,∞=−λ12λ24/2−e−α3α̇1t+eα3α̇2t2+α̇3t2ωα̇t.
By Equation (20) and d/dtωα̇t=0, we have the following:
(41)καE,∞=−1λ1λ2α¨3t2+λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α32·−1λ1λ2α̇3t2+λ122−e−α3α̇1t+eα3α̇2t2−2−1λ1λ2α̇3t1λ1λ2α¨3t+λ122−e−α3α̇1t+eα3α̇2tα¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α32·−1λ1λ2α̇3t2+λ122−e−α3α̇1t+eα3α̇2t2−31/2.
Whenωα̇t=0 and d/dtωα̇t≠0, we have the following:
(42)∇α̇E,Lα,∇α̇E,LαL~Lddtωα̇t2asL⟶+∞,α̇,α̇L=−1λ1λ2α̇3t2+λ122−e−α3α̇1t+eα3α̇2t2,∇α̇E,Lα̇,α̇L2=O1asL⟶+∞. If ωα̇t=0 and d/dtωα̇t≠0, by (Equation (15)), we get the following:
(43)limL⟶∞καE,LL=d/dtωα̇t−1/λ1λ2α̇3t2+λ12/2−e−α3α̇1t+eα3α̇2t2.(2)
By some similar computations, we get (2).Example 7.
Letα:I⟶Eλ1,λ2L be a C2-smooth regular curve, where I is an open interval in ℝ and αt=α1t,α2t,α3t. We compute the following:
(44)ωα̇t=−λ222e−α3α̇1t+eα3α̇2t.
By Equation (29), we have the following:
(45)∇α̇E,Lα̇=−1λ1λ2α¨3t−λ132+λ1λ222L2−e−α3α̇1t+eα3α̇2tωα̇tE1+λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3−Lωα̇tα̇3tE2+ddtωα̇t−λ1222λ2L−e−α3α̇1t+eα3α̇2tα̇3tE3,gLE1,E1=−1,gLE2,E2=1,gLE3,E3=L.
Suppose thatαt=1,t,0,λ1=λ2=1 be a C2-smooth regular curve, and α1t=1,α2t=t,α3t=0. Then, we have the following:
(46)ωα̇t=−22e0×0+e0×1=−22≠0,∇α̇E,Lα̇=−0−2+2L2−e0×0+e0×1×−22E1+220×e0+1×0×e0−0×e0+0×0×e0−L×−22×0E2+ddt−22−22L−e0×0+e0×1×0E3=−1+L2E1.
Therefore, we have the following:(47)∇α̇E,Lα̇,∇α̇E,Lα̇=−1+L24<0.
It implies that∇α̇E,Lα̇ is a timelike vector and λ12λ24/2−e−α3α̇1t+eα3α̇2t2−α̇3t2=1/2>0. By Proposition 6, we have the following:
(48)καE,∞=1/22/2=1.
We can use the above example to illustrate the meaning of Definition5. We assume that αt=1,t,0,λ1=1,λ2=2. Then, we have the following:
(49)ωα̇t=2e0×0+e0×1=−2≠0,∇α̇E,Lα̇=−−12×0−2+42L2−e0×0+e0×1×−2E1+220×e0+1×0×e0−0×e0+0×0×e0−L×−2×0E2+ddt−2−24L−e0×0+e0×1×0E3=−1+4LE1.
Therefore, we have the following:(50)∇α̇E,Lα̇,∇α̇E,Lα̇=−1+4L2<0.
It implies that∇α̇E,Lα̇ is a timelike vector and λ12λ24/2−e−α3α̇1t+eα3α̇2t2−α̇3t2=8>0. By Proposition 6, we have the following:
(51)καE,∞=82=2.


## 3. Geodesic Curvatures for Curves on Lorentzian Surfaces in Lorentzian ApproximationsEλ1,λ2L

In this section, we will compute geodesic curvatures for curves on Lorentzian surfaces in Lorentzian approximationsEλ1,λ2L. We consider a regular surface S⊂Eλ1,λ2L such that S is a C2-smooth compact and oriented surface. Supposed that
(52)S=u1,u2,u3∈Eλ1,λ2L:Fu1,u2,u3=0,where F:Eλ1,λ2L⟶ℝ is a C2-smooth function and Fu1∂u1+Fu2∂u2+Fu3∂u3≠0. A point u∈S is called characteristic if ∇HFu=0, where ∇HF=−E1FE1+E2FE2. We call the set CS≔u1,u2,u3∈S∇HFu1,u2,u3=0 characteristic set of S. We will give some symbols away from characteristic points of S. We define p1≔E1F, q1≔E2F, and r≔E3~F. If −p12+q12>0, we say that S⊂Eλ1,λ2L is a horizontal spacelike surface. Under this assumption, −p12+q12+r2>0 for L⟶+∞. Therefore, we can define the following functions:
(53)l≔−p12+q12,(54)lL≔−p12+q12+r12,(55)p¯1≔p1l,(56)q¯1≔q1l,(57)p¯1,L≔p1lL,(58)q¯1,L≔q1lL,(59)r¯1,L≔r1lL,which will be used to define a frame of S. We construct the following:
(60)NL=−p¯1,LE1+q¯1,LE2+r¯1,LE~3,T1=q¯1T1−p¯1E2,T2=r¯1,Lp¯1T1−r¯1,Lq¯1E2+llLE~3,where NL is the unit spacelike normal vector to S, T1 is the unit timelike vector, and T2 is the unit spacelike vector. One can check that T1,T2 are a pseudo-orthonomal basis for tangent space of S. We call S a Lorentzian surface in Eλ1,λ2L. For the case that α:I⟶Eλ1,λ2L is a C2-smooth timelike curve and α̇=mT1+nT2, we define the following:
(61)JLα̇≔mT2+nT1.For the case thatα:I⟶Eλ1,λ2L is a C2-smooth spacelike curve, we define the following:
(62)JLα̇≔−mT2−nT1.That makesα̇,JLα̇=0 and α̇,JLα̇ have the same orientation with T1,T2. For every X,Y∈TS, we define the Levi-Civita connection with respect to the metric gL on S by ∇XS,E,LY=π∇XE,LY, where π:TEλ1,λ2L⟶TS is the projection. Furthermore,
(63)∇α̇S,E,Lα̇=−∇α̇E,Lα̇,T1LT1+∇α̇E,Lα̇,T2LT2.In particular, we get the following:(64)∇α̇S,E,Lα̇=−−q¯11λ1λ2α¨3t−λ132+λ1λ222L2−e−α3α̇1t+eα3α̇2tωα̇t−p¯1λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3−Lα̇3tωα̇tT1+−r¯1,Lp¯11λ1λ2α¨3t−λ132+λ1λ222L2−e−α3α̇1t+eα3α̇2tωα̇t−r¯1,Lq¯1λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3−Lα̇3tωα̇t+llLL1/2ddtωα̇t−λ1222λ2L−e−α3α̇1t+eα3α̇2tα̇3tT2.Moreover, ifωα̇t=0, then
(65)∇α̇S,E,Lα̇=−−q¯11λ1λ2α¨3t−p¯1λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3T1+−r¯1,Lp¯11λ1λ2α¨3t−r¯1,Lq¯1λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3+llLL1/2ddtωα̇t−λ1222λ2L−e−α3α̇1t+eα3α̇2tα̇3tT2.Definition 8.
Letα:I⟶S be a C2-smooth regular curve, where S⊂Eλ1,λ2L is a regular Lorentzian surface. We define the geodesic curvature as follows.
(1)
If∇α̇S,E,Lα̇ is spacelike vector, the geodesic curvature κα,SE,L of α at αt is defined by the following:
(66)κα,SE,L≔∇α̇S,E,Lα̇S,L2α̇S,L4−∇α̇S,E,Lα̇,α̇S,L2α̇,α̇S,L3.(2)
If∇α̇S,E,Lα̇ is timelike vector, the geodesic curvature κα,SE,L of α at αt is defined by the following:
(67)κα,SE,L≔∇α̇S,E,Lα̇S,L2α̇S,L4+∇α̇S,E,Lα̇,α̇S,L2α̇,α̇S,L3.Definition 9.
Letα:I⟶S be a C2-smooth regular curve, where S⊂Eλ1,λ2L is a regular Lorentzian surface. We define the intrinsic geodesic curvature κα,SE,∞ of α at αt to be
(68)κα,SE,∞≔limL⟶+∞κα,SE,L,if the limit exists.Proposition 10.
Letα:I⟶S be a C2-smooth regular curve, where S⊂Eλ1,λ2L is a regular Lorentzian surface.
(1)
If∇α̇S,E,Lα̇ is a timelike vector, then
(69)κα,SE,∞=λ12λ24/2−e−α3α̇1t+eα3α̇2t2+p¯12α̇3t2ωα̇t,ifωα̇t≠0,(70)κα,SE,∞=0,ifωα̇t=0,(71)ddtωα̇t=0.(2)
If∇α̇S,E,Lα̇ is a spacelike vector, then
(72)limL⟶+∞κα,SE,LL=d/dtωα̇t1/λ1λ2α̇3tq¯1+λ12/2−e−α3α̇1t+eα3α̇2tp¯12,ifωα̇t=0,ddtωα̇t≠0.Proof.
On the one hand, by Equation (25), we get the following:
(73)α̇t=1λ1λ2α̇3tE1+λ122−e−α3α̇1t+eα3α̇2tE2+ωα̇tE3.
On the other hand, sinceα̇∈TS, we denote the following:
(74)α̇t=mT1+nT2=mq¯1E1−p¯1E2+nr¯1,Lp¯1E1−r¯1,Lq¯1E2+llLE3~=mq¯1+nr¯1,Lp¯1E1−mp¯1+nr¯1,Lq¯1E2+nllLL−1/2E3.
The above equations yield the following:(75)mq¯1+nr¯1,Lp¯1=1λ1λ2α̇3t,−mp¯1−nr¯1,Lq¯1=λ122−e−α3α̇1t+eα3α̇2t,nllLL−1/2=ωα̇t.
By solving Equation (75), we get the following:
(76)m=1λ1λ2α̇3tq¯1+λ122−e−α3α̇1t+eα3α̇2tp¯1,n=lLlL1/2ωα̇t.
Therefore,α̇ takes the following form:
(77)α̇=1λ1λ2α̇3tq¯1+λ122−e−α3α̇1t+eα3α̇2tp¯1T1+lLlL1/2ωα̇tT2.
We denote the following:(78)C=−−q¯11λ1λ2α¨3t−λ132+λ1λ222L2−e−α3α̇1t+eα3α̇2tωα̇t−p¯1λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3−Lα̇3tωα̇t2+−r¯1,Lp¯11λ1λ2α¨3t−λ132+λ1λ222L2−e−α3α̇1t+eα3α̇2tωα̇t−r¯1,Lq¯1λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3−Lα̇3tωα̇t+llLL1/2ddtωα̇t−λ1222λ2L−e−α3α̇1t+eα3α̇2tα̇3t2.
Then, by Equation (64), we get ∇α̇S,E,Lα̇,∇α̇S,E,Lα̇S,L=C, where
(79)C~−λ12λ242q¯12−e−α3α̇1t+eα3α̇2t2+p¯1α̇3t2ωα̇t2L2asL⟶+∞.
In caseωα̇t≠0, we obtain the following:
(80)α̇,α̇S,L=−1λ1λ2α̇3tq¯1+λ122−e−α3α̇1t+eα3α̇2tp¯12+LllLωα̇t22~L2ωα̇t2asL⟶+∞.
By Equations (64) and (77), we obtain the following:
(81)∇α̇S,E,Lα̇,α̇S,L=D~1λ1λ2α̇3tq¯1+λ122−e−α3α̇1t+eα3α̇2tp¯1·−q¯1λ1λ2222−e−α3α̇1t+eα3α̇2t−p¯1α̇3tωα̇tL+ddtωα̇tωα̇tL~N0LasL⟶+∞,where
(82)D=1λ1λ2α̇3tq¯1+λ122−e−α3α̇1t+eα3α̇2tp¯1q¯11λ1λ2α¨3−λ132+λ1λ222L2−e−α3α̇1+eα3α̇2ωα̇+p¯1λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3−Lα̇3tωα̇t+llLL1/2ωα̇·−r¯1,Lp¯11λ1λ2α¨3−λ132+λ1λ222L2−e−α3α̇1+eα3α̇2ωα̇−r¯1,Lq¯1λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3−Lα̇3tωα̇t+llLL1/2ddtωα̇t−λ1222λ2L−e−α3α̇1t+eα3α̇2tα̇3t,and N0 does not depend on L. By Equation (64), we have the following:
(83)κα,SE,∞=limL⟶+∞κα,SE,L=λ12λ24/2q¯12−e−α3α̇1t+eα3α̇2t2+p¯12α̇3t2ωα̇tifωα̇t≠0.
Ifωα̇t=0 and d/dtωα̇t=0, we have the following:
(84)∇α̇S,E,Lα̇,∇α̇S,E,Lα̇S,L~−q¯11λ1λ2α¨3t+p¯1E2asL⟶+∞,where E=λ12/2α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3, and
(85)α̇,α̇S,L=−1λ1λ2α̇3tq¯1+λ122−e−α3α̇1t+eα3α̇2tp¯12,(86)∇α̇S,E,Lα̇,α̇S,L=−1λ1λ2α̇3tq¯1+λ122−e−α3α̇1t+eα3α̇2tp¯1·q¯11λ1λ2α¨3t+p¯1λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3≔−A1A2.
By Equations (79)–(86) and Equation (65), we get the following:
(87)κα,SE,∞=A22A14−A12A22A16=0.
Whenωα̇t=0, and d/dtωα̇t≠0, we have the following:
(88)∇α̇S,E,Lα̇,∇α̇S,E,Lα̇S,L~Lddtωα̇t2asL⟶+∞,α̇,α̇S,L=−1λ1λ2α̇3tq¯1+λ122−e−α3α̇1t+eα3α̇2tp¯12,∇α̇S,E,Lα̇,α̇S,L=O1.
It follows that(89)limL⟶+∞κα,SE,LL=d/dtωα̇t1/λ1λ2α̇3tq¯1+λ12/2−e−α3α̇1t+eα3α̇2tp¯12,if ωα̇t=0, and d/dtωα̇t≠0. This completes the proof.Example 11.
We assume that there exists aC2-smooth function F=y:Eλ1,λ2L⟶ℝ such that
(90)S=u1,u2,u3∈Eλ1,λ2L:y=0.
Then,Fu1∂u1+Fu2∂u2+Fu3∂u3=∂y≠0. Let
(91)E1=λ1λ2∂∂u3,E2=1λ12−eu3∂∂u1+e−u3∂∂u2,E3=−1λ22eu3∂∂u1+e−u3∂∂u2.
So, we have the following:(92)p1≔E1F=λ1λ2∂∂u3y=0,q1≔E2F=1λ12−eu3∂∂u1+e−u3∂∂u2y=1λ12e−u3,r1≔E~3F=−L−1/21λ22eu3∂∂u1+e−u3∂∂u2y=−L−1/21λ22e−u3.
Therefore,−p12+q12=1/λ12e−u32>0, so S⊂Eλ1,λ2L is a horizontal spacelike surface. By Equation (53), we have the following:
(93)l≔−p12+q12=1λ12e−u3,lL≔−p12+q12+r12=e−u321λ12+1λ22L,p¯1≔p1l=0,q¯1≔q1l=1,p¯1,L≔p1lL=0,q¯1,L≔qlL=1/λ11/λ12+1/λ22L,r¯1,L≔r1lL=−L−1/21/λ21/λ12+1/λ22L.
By Equation (60), we have the following:
(94)NL=1/λ11/λ12+1/λ22LE2−−L−1/21/λ21/λ12+1/λ22LE~3,T1=E1,T2=L−1/21/λ21/λ12+1/λ22LE2+1/λ11/λ12+1/λ22LE~3.
Then,T1,T2=E1L−1/21/λ2/1/λ12+1/λ22LE2+1/λ1/1/λ12+1/λ22LE~3. Thus, it is concluded that S is a Lorentzian surface in Eλ1,λ2L.
Let(95)α:0,2π⟶S,θ⟶cosθ,0,sinθ,λ1=λ2=1be the circle centered at the origin on y=0. By
(96)ωα̇θ=−λ222e−α3α̇1θ+eα3α̇2θ
andα1θ=cosθ,α2θ=0,α3θ=sinθ, we have the following:
(97)ωα̇θ=22e−sinθsinθ.
By Equation (64), we have the following:
(98)∇α̇S,E,Lα̇=−sinθ−1+L2e−2sinθsin2θT1+2L+1L−112cosθe−sinθT2.
Then,(99)∇α̇S,E,Lα̇,∇α̇S,E,Lα̇=−sinθ−1+L2e−2sinθsin2θ2+2L+1L−112cosθe−sinθ2.
Ifcosθ=0, then sinθ=±1. In this case, ωα̇θ≠0. Then, we have ∇α̇S,E,Lα̇ is a timelike vector. By Proposition 10 and Equation (69), we have at the point θ which satisfies cosθ=0:
(100)κα,SE,∞=λ12λ24/2q¯12−e−α3α̇1t+eα3α̇2t2+p¯12α̇3t2ωα̇t=22.
When we assume thatα:0,2π⟶S,θ⟶cosθ,0,sinθ,λ1=1,λ2=2 be a circle centered at the origin on y=0. We get the following:
(101)ωα̇θ=−2e−sinθsinθ.
By Equation (64), we have the following:
(102)∇α̇S,E,Lα̇=−12sinθ+1+4Le−2sinθsin2θT1+224L+1cosθe−sinθ1+4L−cosθsinθe−sinθ2+6LT2.
So,(103)∇α̇S,E,Lα̇,∇α̇S,E,Lα̇=−−12sinθ+1+4Le−2sinθsin2θ2+224L+1cosθe−sinθ1+4L−cosθsinθe−sinθ2+6L2.
Ifcosθ=0, then sinθ=±1. In this case, ωα̇θ≠0. Then, we have ∇α̇S,E,Lα̇ is a timelike vector. By Proposition 10 and Equation (69), we have at the point θ which satisfies cosθ=0.(104)κα,SE,∞=λ12λ24/2q¯12−e−α3α̇1t+eα3α̇2t2+p¯12α̇3t2ωα̇t=2.Definition 12.
Letα:I⟶S be a C2-smooth regular curve, where S⊂Eλ1,λ2L is a regular Lorentzian surface. The signed geodesic curvature κα,SE,L,s of α at αt is defined as follows:
(105)κα,SE,L,s≔∇α̇S,E,Lα̇,JLα̇S,Lα̇S,L3,where JL is defined by Equations (61) and (62).Definition 13.
Letα:I⟶S be a C2-smooth regular curve, where S⊂Eλ1,λ2L is a regular Lorentzian surface. We define the intrinsic geodesic curvature κα,SE,∞,s of α at the noncharacteristic point αt to be
(106)κα,SE,∞,s≔limL⟶+∞κα,SE,L,s,if the limit exists.Proposition 14.
Letα:I⟶S be a C2-smooth regular curve, where S⊂Eλ1,λ2L is a regular Lorentzian surface.
(1)
Ifα:I⟶S be a spacelike C2-smooth curve, then ωα̇t≠0 and(107)κα,SE,∞,s=−q¯1λ1λ222/2−e−α3α̇1t+eα3α̇2t+p¯1α̇3tωα̇t.(2)
Ifα:I⟶S be a timelike C2-smooth curve, then ωα̇t=0 and(108)κα,SE,∞,s=0,ifddtωα̇t=0,limL⟶+∞κα,SE,L,sL=d/dtωα̇t1/λ1λ2α̇3tq¯1+λ12/2−e−α3α̇1t+eα3α̇2tp¯12,if d/dtωα̇t≠0.Proof.
By Equations (61), (62), and (77), we get the following:
(109)JLα̇=1λ1λ2α̇3tq¯1+λ122−e−α3α̇1t+eα3α̇2tp¯1JLT1−lLlL1/2ωα̇tJLT2=lLlL1/2ωα̇tT1+1λ1λ2α̇3tq¯1+λ122−e−α3α̇1t+eα3α̇2tp¯1T2.
By Equation (64) and the above equation, we have the following:
(110)∇α̇S,E,Lα̇,JLα̇=lLlL1/2ωα̇t·−q¯11λ1λ2α¨3t−λ132+λ1λ222L2−e−α3α̇1t+eα3α̇2tωα̇t−p¯1λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3−Lα̇3tωα̇t+1λ1λ2α̇3tq¯1+λ122−e−α3α̇1t+eα3α̇2tp¯1·−r¯1,Lp¯11λ1λ2α¨3t−λ132+λ1λ222L2−e−α3α̇1t+eα3α̇2tωα̇t−r¯1,Lq¯1λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3−Lα̇3tωα̇t+llLL1/2ddtωα̇t−λ1222λ2L−e−α3α̇1t+eα3α̇2tα̇3t~q¯1λ1λ2222−e−α3α̇1t+eα3α̇2t+p¯1α̇3tωα̇t2L3/2asL⟶+∞.
So, we get the following:(111)κα,SE,L,s=∇α̇S,E,Lα̇,JLα̇S,Lα̇S,L3=L3/2q¯1λ1λ222/2−e−α3α̇1t+eα3α̇2t+p¯1α̇3tωα̇t2L3/2ωα̇t3.
Furthermore,(112)κα,SE,∞,s=limL⟶+∞κα,SE,L,s=−q¯1λ1λ222/2−e−α3α̇1t+eα3α̇2t+p¯1α̇3tωα̇t.
Whenωα̇t=0, and d/dtωα̇t=0, we get the following:
(113)∇α̇S,E,Lα̇,JLα̇L,S=1λ1λ2α̇3tq¯1+λ122−e−α3α̇1t+eα3α̇2tp¯1·−r¯1,Lp¯11λ1λ2α¨3t−r¯1,Lq¯1λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3+llLL1/2−λ1222λ2L−e−α3α̇1t+eα3α̇2tα̇3t~N1L−1/2asL⟶+∞,where N1 does not depend on L. So, κα,SE,∞,s=0. When ωα̇t=0, and d/dtωα̇t≠0, we have the following:
(114)∇α̇S,E,Lα̇,JLα̇L,S=1λ1λ2α̇3tq¯1+λ122−e−α3α̇1t+eα3α̇2tp¯1·−r¯1,Lp¯11λ1λ2α¨3t−r¯1,Lq¯1λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3+llLL1/2ddtωα̇t−λ1222λ2L−e−α3α̇1t+eα3α̇2tα̇3t~L1/21λ1λ2α̇3tq¯1+λ122−e−α3α̇1t+eα3α̇2tp¯1ddtωα̇tasL⟶+∞.
We get the following:(115)κα,SE,∞,s=limL⟶+∞κα,SE,LL=limL⟶+∞1/λ1λ2α̇3tq¯1+λ12/2−e−α3α̇1t+eα3α̇2tp¯1d/dtωα̇tL1/21/λ1λ2α̇3tq¯1+λ12/2−e−α3α̇1t+eα3α̇2tp¯13L=d/dtωα̇t1/λ1λ2α̇3tq¯1+λ12/2−e−α3α̇1t+eα3α̇2tp¯12.Example 15.
We takeF=y:Eλ1,λ2L⟶ℝ and αθ=cosθ,0,sinθ,λ1=λ2=1 as in Example 11. Then,
(116)ωα̇θ=22e−sinθsinθ.
By Equation (77), we have the following:
(117)α̇=1λ1λ2α̇3tq¯1+λ122−e−α3α̇1t+eα3α̇2tp¯1T1+lLlL1/2ωα̇tT2=cosθT1+L+122e−sinθsinθT2.
Then,(118)α̇,α̇=−cos2θ+L+12e−2sinθsin2θ.
So, we have whensinθ≠0, then α̇2>0 for the large L and we have α̇ is a spacelike vector. If sinθ≠0, by Proposition 14 (1), we have the following:
(119)κα,SE,∞,s=−q¯1λ1λ222/2−e−α3α̇1t+eα3α̇2t+p¯1α̇3tωα̇t=−1.
When we takeλ1=1,λ2=2, we have the following:
(120)ωα̇θ=−2e−sinθsinθ.
By Equation (77), we have the following:
(121)α̇=12cosθT1−24L+12e−sinθsinθT2.
Then,(122)α̇,α̇=−14cos2θ+4L+12e−2sinθsin2θ.
So, we have whensinθ≠0, then α̇2>0 for the large L and we have α̇ is a spacelike vector. If sinθ≠0, by Proposition 14 (1), we have the following:
(123)κα,SE,∞,s=−q¯1λ1λ222/2−e−α3α̇1t+eα3α̇2t+p¯1α̇3tωα̇t=−2.


## 4. Curvatures for Lorentzian Surfaces in Lorentzian ApproximationsEλ1,λ2L

In this section, we will compute the intrinsic Gaussian curvature of Lorentzian surfaces inEλ1,λ2L. We define the second fundamental form IIE,L of the embedding of S into Eλ1,λ2L by the following:
(124)IIE,L=∇T1E,LvL,T1L∇T1E,LvL,T2L∇T2E,LvL,T1L∇T2E,LvL,T2L.We have the following theorem.Theorem 16.
The second fundamental formIIE,L of the embedding of S into Eλ1,λ2L is given by the following:
(125)IIE,L=h11h12h21h22,where
(126)h11=llLE1p¯1−E2q¯1−λ12r¯Lp¯1q¯1L−1/2,h21=h12=lLlT1,∇Hr¯1,LL+L2λ22−λ122Lq¯1,L2+p¯1,L2−r¯L2λ122Lq¯12+p¯12,h22=l2lL2T2,∇HrlL+E3~r¯1,L−r¯1,Lp¯1,Lq¯1,LLλ12−r¯L3p¯1q¯1Lλ12.Proof.
SinceT1,NLL=0,T2,NLL=0, we have the following:
(127)∇T1E,LNL,T1L=−∇T1E,LT1,NLL,∇T2E,LNL,T2L=−∇T2E,LT2,NLL.
Using the definition of the connection, identities in Equation (11), and grouping terms, we have the following:
(128)∇T1E,LT1=∇q¯1E1−p¯1E2E,Lq¯1E1−p¯1E2=q¯1E1q¯1E1−E1p¯1E2−p¯1∇E1E,LE2−p¯1E2q¯1E1−E2p¯1E2+q¯1∇E2E,LE1=q¯1E1q¯1−p¯1E2q¯1E1−q¯1E1p¯1−p¯1E2p¯1E2+p¯1q¯1λ12LE3.
Sincep¯12−q¯12=−1, we have p¯1Eip¯1−q¯1Eiq¯1=0,i=1,2,3. Furthermore, we obtain the following:
(129)h11=−∇T1E,LT1,NLL=−p¯1,Lq¯1E1q¯1−p¯1E2q¯1−q¯1,Lq¯1E1p¯1−p¯1E2p¯1+r¯1,Lp¯1q¯1λ12L=llLE1p¯1−E2q¯1−λ12r¯1,Lp¯1q¯1L−1/2.
To computeh12 and h21, using the properties of connection, we compute the following:
(130)∇T1E,LT2=∇q¯1E1−p¯1E2E,Lr¯1,Lp¯1E1−r¯1,Lq¯1E2+llLL−1/2E3=q¯1E1r¯1,Lp¯1E1−E1r¯1,Lq¯1E2−r¯1,Lq¯1∇E1E,LE2+L−1/2E1llLE3+llLL−1/2∇E1E,LE3−p¯1E2r¯1,Lp¯1E1+r¯1,Lp¯1∇E2E,LE1−E2r¯1,Lq¯1E2+L−1/2E2llLE3+llLL−1/2∇E2E,LE3=q¯1E1r¯1,Lp¯1−p¯1E2r¯1,Lp¯1+p¯1llLL−1/2λ12+λ22L2E1+−q¯1E1r¯1,Lq¯1+p¯1E2r¯1,Lq¯1+q¯1llLL−1/2λ12−λ22L2E2+−p¯1E2L−1/2llL+q¯1E1L−1/2llL+r¯1,L2−λ22+λ12Lp¯12+q¯12E3.
Next, we compute the inner product of this withNL. Using the product rule and the identity q¯1,Lp¯1=p¯1,Lq¯1, we obtain the following:
(131)∇T1E,LT2,NLL=p¯1,Lq¯1p¯1E1r¯1,L+r¯1,LE1p¯1−p¯1,Lp¯1p¯1E2r¯1,L+r¯1,LE2p¯1−q¯1,Lq¯1q¯1E1r¯1,L+r¯1,LE1q¯1+q¯1,Lp¯1q¯1E2r¯1,L+r¯1,LE2q¯1−L2λ22−p¯1,L2+q¯1,L2+r¯1,L2−r¯1,Lp¯1E2llL+r¯1,Lq¯1E1llL+λ122Lp¯1,L2+q¯1,L2+r¯1,L22λ12Lp¯12+q¯12.
The identities−p¯1,L2+q¯1,L2+r¯L2=1 and −p¯12+q¯12=1 yield the following:
(132)∇T1E,LT2,NLL=−llLE1,∇Hr¯1,LL+r¯1,LE1,∇HllLL−L2λ22+λ122Lp¯1,L2+q¯1,L2+r¯1,L22λ12Lp¯12+q¯12.
Finally, by using the identityl/lL−lL/l∇Hr¯1,L=r¯1,L∇Hl/lL, we obtain the following:
(133)∇T1E,LT2,NLL=−lLlT1,∇Hr¯1,LL−L2λ22+λ12Lp¯1,L2+q¯1,L2+r¯1,L22λ12Lp¯12+q¯12.
Therefore,(134)h21=h12=−∇T1E,LT2,NLL=lLlT1,∇Hr¯1,LL+L2λ22−λ122Lq¯1,L2+p¯1,L2−r¯1,L2λ122Lq¯12+p¯12.
Since∇T2E,LNL,T2L=−∇T2E,LT2,NLL, by similar computation as above, we get the following:
(135)∇T2E,LT2=∇r¯1,Lp¯1E1−r¯1,Lq¯1E2+l/lLE3~E,Lr¯1,Lp¯1E1−r¯1,Lq¯1E2+llLE3~=r¯1,Lp¯1E1r¯1,Lp¯1E1−E1r¯1,Lq¯1E2−r¯1,Lq¯1∇E1E,LE2+L−1/2E1llLE3+L−1/2llL∇E1E,LE3−r¯1,Lq¯1E2r¯1,Lp¯1E1−E2r¯1,Lq¯1E2+r¯1,Lp¯1∇E2E,LE1+L−1/2E2llLE3+L−1/2llL∇E2E,LE3+llLL−1/2E3r¯1,Lp¯1E1+r¯1,Lp¯1∇E3E,LE1−E3r¯1,Lq¯1E2−r¯1,Lq¯1∇E3E,LE2+L−1/2E3llLE3=r¯1,Lp¯1E1r¯1,Lp¯1−r¯1,Lq¯1E2r¯1,Lp¯1+L−1/2llLE3r¯1,Lp¯1+r¯1,Lq¯1L−1/2llLλ12+λ22LE1+−r¯1,Lp¯1E1r¯1,Lq¯1+r¯1,Lq¯1E2r¯1,Lq¯1−L−1/2llLE3r¯1,Lq¯1+r¯1,Lp¯1L−1/2llL−λ22LE2+r¯1,Lp¯1L−1/2E1llL−r¯1,Lq¯1E2L−1/2E2llL+L−1llLE3llL+r¯1,L2p¯1q¯1λ12LE3.
Taking the inner product withNL yields the following:
(136)∇T2E,LT2,NLL=p¯1,Lr¯1,Lp¯1E1r¯1,Lp¯1−r¯1,Lq¯1E2r¯1,Lp¯1+L−1/2llLE3r¯1,Lp¯1+r¯1,Lq¯1L−1/2llLλ12+λ22L+q¯1,L−r¯1,Lp¯1E1r¯1,Lq¯1+r¯1,Lq¯1E2r¯1,Lq¯1−L−1/2llLE3r¯1,Lq¯1+r¯1,Lp¯1L−1/2llL−λ22L+r¯1,Lr¯1,Lp¯1E1llL−r¯1,Lq¯1E2E2llL+L−1/2llLE3llL+r¯1,L2p¯1q¯1λ12L.
Under some simplifications, one can get the following:(137)h22=−∇T2E,LT2,NLL=l2lL2T2,∇HrlL+E3~r¯1,L−r¯1,Lp¯1,Lq¯1,LLλ12−r¯1,L3p¯1q¯1Lλ12.The Riemannian mean curvatureHE,L of S is defined by the following:
(138)HE,L≔trIIE,L=llLE1p¯1−E2q¯1−λ12r¯1,Lp¯1q¯1L−1/2+l2lL2T2,∇HrlL+E3~r¯1,L−r¯1,Lp¯1,Lq¯1,LLλ12−r¯1,L3p¯1q¯1Lλ12.Proposition 17.
Away from characteristic point, the horizontal mean curvatureHE,∞ of S∈Eλ1,λ2L is given by the following:
(139)HE,∞=limL⟶∞HE,L=E1p¯1−E2q¯1.Proof.
By(140)l2lL2T2,∇HrlL=p¯1rlE1r¯1,L+q¯1rlE2r¯1,L=OL−1,llLE1p¯1−E2q¯1⟶E1p¯1−E2q¯1,E3~r¯1,L⟶0,λ12r¯1,Lp¯1q¯1L−1/2⟶0,−r¯1,Lp¯1,Lq¯1,LLλ12−r¯1,L3p¯1q¯1Lλ12⟶0,we get Equation (139).Recalling the definition of curvature tensor for a connection∇ is defined by the following:
(141)RX,YZ=∇X∇YZ−∇Y∇XZ−∇X,YZ.By Proposition1 and Equation (141), we have the following lemma.Lemma 18.
The curvature tensor ofEλ1,λ2L is given by the following:
(142)RE,LE1,E2E1=−λ14+2λ12λ22L+3λ24L24LE2,RE,LE1,E2E2=−λ14+2λ12λ22L+3λ24L24LE1,RE,LE1,E2E3=0,RE,LE1,E3E1=3λ14+2λ12λ22L−λ24L24LE3,RE,LE1,E3E2=0,RE,LE1,E3E3=3λ14+2λ12λ22L−λ24L24E1,RE,LE2,E3E1=0,RE,LE2,E3E2=λ14+2λ12λ22L+λ24L24LE3,RE,LE2,E3E3=−λ14+2λ12λ22L+λ24L24E2.
Let(143)KS,E,LT1,T2=−RS,E,LT1,T2T1,T2S,L,KE,LT1,T2=−RE,LT1,T2T1,T2L.By the Gauss equation, we have the following:(144)KS,E,LT1,T2=KE,LT1,T2+detIIE,L.Proposition 19.
Away from characteristic points, we have the following:(145)KS,E,∞T1,T2=−T1,∇HE3F∇HFL−λ24E3F2l2,asL⟶∞.Proof.
We compute the following:(146)RE,LT1,T2T1=RE,Lq¯1E1−p¯1E2,r¯1,Lp¯1E1−r¯1,Lq¯1E2+llLLE3q¯1E1−p¯1E2=r¯1,Lp¯1q¯12RE,LE1,E1E1−r¯1,Lq¯13RE,LE1,E2E1+lq¯12lLLRE,LE1,E3E1−r¯1,Lp¯12q¯1RE,LE2,E1E1+r¯1,Lp¯1q¯12RE,LE2,E2E1−lp¯1q¯1lLLRE,LE2,E3E1−r¯1,Lp¯12q¯1RE,LE1,E1E2+r¯1,Lp¯1q¯12RE,LE1,E2E2−lp¯1q¯1lLLRE,LE1,E3E2+r¯1,Lp¯13RE,LE2,E1E2−r¯1,Lp¯12q¯1RE,LE2,E2E2+lp¯12lLLRE,LE2,E3E2=−r¯1,Lq¯13RE,LE1,E2E1+q¯12llLLRE,LE1,E3E1−r¯1,Lp¯12q¯1RE,LE2,E1E1+r¯1,Lq¯12p¯1RE,LE1,E2E2+r¯1,Lp¯13RE,LE2,E1E2+p¯12llLLRE,LE2,E3E2=r¯1,Lp¯1−λ14+2λ12λ22L+3λ24L24LE1−r¯1,Lq¯1−λ14+2λ12λ22L+3λ24L24LE2+llLLq¯123λ14+2λ12λ22L−λ24L24L+p¯12λ14+2λ12λ22L+λ24L24LE3,(147)KE,LT1,T2=−RE,LT1,T2T1,T2L=−r¯1,L2λ12λ222−λ144L+3λ24L4−llLq¯12λ12λ222−λ24L4+3λ144L−llLp¯12λ12λ222+λ144L+λ24L4~llL2λ24L4−3λ244E3u2l2−λ12λ22q¯12+λ12λ222asL⟶∞.
By Theorem16 and ∇Hr¯1,L=L−1/2∇HE3F/∇HF+OL−1 as L⟶+∞, we get the following:
(148)detIIE,L=h11h22−h122=−λ24L4−T1,∇HE3F∇HFL+λ12λ222q¯12+p¯12+OL−1/2asL⟶∞.
By Equations (144), (147), and (148), we get the desired equation.


## 5. The First Gauss-Bonnet Theorem inEλ1,λ2L

In this section, we will prove Gauss-Bonnet theorem inEλ1,λ2L. We consider a spacelike curve α:I⟶Eλ1,λ2L, and define the Riemannian length measure by dsE,L=α̇Ldt.Lemma 20.
Letα:I⟶Eλ1,λ2L be a spacelike C2-smooth. Let dsE≔ωα̇tdt,(149)ds¯E≔121ωα̇t1λ1λ2α̇3t2+λ122−eα3α̇1t+eα3α̇2t2dt.
Then,(150)limL⟶∞1L∫αdsE,L=∫abdsE.
Whenωα̇t≠0, we have the following:
(151)1LdsE,L=dsE+ds¯EL−1+OL−2asL⟶+∞.
Whenωα̇t=0, we have the following:
(152)1LdsE,L=1L1λ1λ2α̇3t2+λ122−eα3α̇1t+eα3α̇2t2dt.Proof.
Since(153)α̇tL=1λ1λ2α̇3t2+λ122−eα3α̇1t+eα3α̇2t2+Lωα̇t2,we get the following:
(154)limL⟶∞1L∫αα̇tLdt=∫ablimL⟶∞1Lα̇tLdt=∫ablimL⟶∞1L1λ1λ2α̇3t2+λ122−eα3α̇1t+eα3α̇2t2+Lωα̇t2dt=∫abωα̇tdt=∫abdsE.
Whenωα̇t≠0, we have the following:
(155)1LdsE,L=L−11λ1λ2α̇3t2+λ122−eα3α̇1t+eα3α̇2t2+ωα̇t2dt.
Using the Taylor expansion, we can prove the following:(156)1LdsE,L=dsE+ds¯EL−1+OL−2asL⟶+∞.
From the definition ofdsE,L and ωα̇t=0, we get the following:
(157)1LdsE,L=1L1λ1λ2α̇3t2+λ122−eα3α̇1t+eα3α̇2t2dt.Proposition 21.
LetS⊂Eλ1,λ2L be a Euclidean C2-smooth surface and S=F=0 and dσS,E,L denote the surface measure on S with respect to the Lorentzian metric gL. Let
(158)dσS,E≔−p¯1ω2−q¯1ω1∧ω,dσ¯S,E≔−E3Flω1∧ω2+E3F22l2p¯1ω2−q¯1ω1∧ω.
Then,(159)1LdσS,E,L=dσS,E+dσ¯S,EL−1+OL−2asL⟶+∞.
IfS=fD with f=fx1,x2=f1,f2,f3:D⊂ℝ2⟶Eλ1,λ2L, then
(160)limL⟶∞1L∫SdσS,E,L=∫DAx1,x2,λ1,λ1dx1dx2,where Ax1,x2,λ1,λ1=f1x1f2x2λ1λ2−f2x1f1x2λ1λ22−f3x2f1x1−f3x1f1x22/2e−u31/λ1+f3x2f2x1−f3x1f2x22/2eu31/λ121/2.Proof.
LetgLE1,·=−ω1,gLE2,·=ω2,gLE3,·=Lω. We define T1∗≔gLT1,·,T2∗≔gLT2,·; then,
(161)T1∗=q¯1ω1−p¯1ω2,T2∗=r¯Lp¯1ω1−r¯Lq¯1ω2+llLL1/2ω.
Therefore,(162)1LdσS,E,L=1LT1∗∧T2∗=−llLp¯1ω2−q¯1ω1∧ω−1Lr¯Lω1∧ω2.
Recalling(163)r¯1,L=E3FL1/2−p2+q2+L−1E1F2and the Taylor expansion
(164)1lL=1l−12l3E3F2L−1+OL−2asL⟶+∞,we get Equation (159). By Equation (8), we have the following:
(165)fx1=f1x1∂u1+f2x1∂u2+f3x1∂u3=a11E1+a12E2+a13E3~,fx2=f1x2∂u1+f2x2∂u2+f3x2∂u3=a21E1+a22E2+a23E3~,where a11=f3x11/λ1λ2,a12=f1x1−2/2e−u3λ1+f2x12/2eu3λ1,a21=f3x21/λ1λ2,a13=Lf1x1−2/2e−u3λ2+f2x1−2/2eu3λ2.a22=f1x2−2/2e−u3λ1+f2x22/2eu3λ1 and a23=Lf1x2−2/2e−u3λ2+f2x2−2/2eu3λ2. Let
(166)N¯L=−E1E2E3~a11a12a13a21a22a23=−Lf1x1f2x2λ1λ2−f2x1f1x2λ1λ2E1+Lf3x2f1x1−f3x1f1x222e−u31λ1+f3x2f2x1−f3x1f2x222eu31λ1E2+f3x2f1x1−f3x1f1x222e−u31λ2+f3x1f2x2−f3x2f2x122eu31λ2E3~.
We know thatdσS,E,L=detgijdx1dx2,gij=gLfxi,fxj, and
(167)detgij=N¯LL2=−NL¯,NL¯=Lf1x1f2x2λ1λ2−f2x1f1x2λ1λ22−Lf3x2f1x1−f3x1f1x222e−u31λ1+f3x2f2x1−f3x1f2x222eu31λ12−f3x2f1x1−f3x1f1x222e−u31λ2+f3x1f2x2−f3x2f2x122eu31λ22,so by the dominated convergence theorem, we get the following:
(168)limL⟶∞1L∫SdσS,E,L=∫DAx1,x2,λ1,λ1dx1dx2.Similar to the proof of Theorem 4.3 in [3], we get a Gauss-Bonnet theorem in Eλ1,λ2L as follows:Theorem 22.
LetS⊂Eλ1,λ2L be a regular Lorentzian surface with finitely many boundary components ∂Si,i∈1,⋯,n, given by C2-smooth regular and closed spacelike curve αi:0,2π⟶∂Si. Let KS,E,∞ be intrinsic Gaussian curvature of S in Proposition 19 and καi,SE,∞,s the intrinsic signed geodesic curvature of αi relative to S in Proposition 14. Suppose that the characteristic set CS satisfies H1CS=0 and that ∇HFH−1 is locally summable with respect to the 2-dimensional Hausdorff measure near the characteristic set CS. Then,
(169)∫SKS,E,∞dσS,E+∑i=1n∫αiκαi,SE,∞,sdsE=0.Proof.
Using the discussions in [1, 2], we may assume that all points satisfy the following:
(170)ωα̇it=0andddtωα̇it≠0.
Then, by Lemma20, we obtain the following:
(171)καi,SE,L,s=καi,SE,∞,s+OL−1/2.
By the Gauss-Bonnet theorem, we have the following:(172)∫SKS,E,L1LdσS,E,L+∑i=1n∫αiκαi,SE,∞,s1LdsE,L=2πχSL.
So, by Equations (171), (172), (145), (159), and (160) and Lemma 20, we get the following:
(173)∫SAdσS,E+∑i=1n∫αiκαi,SE,∞,sdsE,L+OL−1/2=2πχSL,where A=−T1,∇HE3F/∇HFL−λ24E3F2/l2. Let L go to the infinity, and by using the dominated convergence theorem, we get the desired result.


## 6. Curvatures for Spacelike Surfaces and the Second Gauss-Bonnet Theorem

The geodesic curvature of spacelike curves on spacelike surface and intrinsic Gaussian curvature of spacelike surfaces inEλ1,λ2L will be investigated in this section. For a regular surface S⊂Eλ1,λ2L and regular curves α⊂S, suppose that there is a C2-smooth function F:Eλ1,λ2L⟶ℝ such that
(174)S=u1,u2,u3∈Eλ1,λ2L:Fu1,u2,u3=0.Similar to Section4, we give p1,q1,r1,l,lL,p¯1,q¯1,p¯1,L,q¯1,L,r¯1,L,NL,T1,T2,JL,κα,SE,L,κα,S∞,κα,SE,L,s,κα,SE,∞,s. We consider the case that S is a spacelike surface in Eλ1,λ2L. In particular, let p1≔E1F, q1≔E2F, andr1≔E3~F. Let p12−q12>0, when L⟶+∞, we have p12−q12−r12>0. We then define the following:
(175)l≔p12−q12,lL≔p12−q12−r12,p¯1≔p1l,q¯1≔q1l,p¯1,L≔p1lL,q¯1,L≔q1lL,r¯1,L≔r1lL,l≔p12−q12.In particular,p¯12−q¯12=1. These functions are well defined at every noncharacteristic point. Let
(176)NL=−p¯1,LE1+q¯1,LE2+r¯1,LE3~,(177)T1=q¯1E1−p¯1E2,(178)T2=r¯1,Lp¯1E1−r¯1,Lq¯1E2−llLE3~,then NL is the unit timelike normal vector to S and T1, T2 is the unit spacelike vector. T1,T2 are the orthonormal basis of S. We call S a spacelike surface in the Lorentzian group of rigid motions of the Minkowski plane. We define a linear transformation on TS by JL:TS⟶TS,
(179)JLT1≔T2,JLT2≔−T1.For everyX,Y∈TS, we define ∇XS,E,LY=π∇UE,LV, where π:TG⟶TS is the projection. Then, ∇S,E,L is the Levi-Civita connection on S with respect to the metric gL. In particular,
(180)∇α̇S,E,Lα̇=∇α̇E,Lα̇,T1LT1+∇α̇E,Lα̇,T2LT2.A simple computation shows that(181)∇α̇S,E,Lα̇=−q¯11λ1λ2α¨3t−λ132+λ1λ222L2−e−α3α̇1t+eα3α̇2tωα̇t−p¯1λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3−Lα̇3tωα̇tT1+−r¯1,Lp¯11λ1λ2α¨3t−λ132+λ1λ222L2−e−α3α̇1t+eα3α̇2tωα̇t−r¯1,Lq¯1λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3−Lα̇3tωα̇t−llLL1/2ddtωα̇t−λ1222λ2L−e−α3α̇1t+eα3α̇2tα̇3tT2.Moreover, ifωα̇t=0, then
(182)∇α̇S,E,Lα̇=−q¯11λ1λ2α¨3t−p¯1λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3T1+−r¯1,Lp¯11λ1λ2α¨3t−r¯1,Lq¯1λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3−llLL1/2ddtωα̇t−λ1222λ2L−e−α3α̇1t+eα3α̇2tα̇3tT2.Definition 23.
LetS⊂Eλ1,λ2L be a regular spacelike surface, α:I⟶S be a regular C2-smooth spacelike curve. The geodesic curvature κα,SE,L of α at αt is defined as follows:
(183)κα,SE,L≔∇α̇S,E,Lα̇S,L2α̇S,L4−∇α̇S,E,Lα̇,α̇S,L2α̇S,L6.Definition 24.
LetS⊂Eλ1,λ2L be a regular spacelike surface and α:I⟶S be a regular C2-smooth spacelike curve. The intrinsic geodesic curvature κα,SE,∞,s of α at αt is defined to be
(184)κα,SE,∞,s≔limL⟶+∞κα,SE,L,if the limit exists.Proposition 25.
LetS⊂Eλ1,λ2L be a regular spacelike surface and α:I⟶S be a regular C2-smooth spacelike curve. Then,
(185)κα,SE,∞,s=λ12λ24/2q¯12−e−α3α̇1t+eα3α̇2t2+p¯12α̇3t2ωα̇tifωα̇t≠0,(186)κα,SE,∞,s=0,ifωα̇t=0,(187)ddtωα̇t=0,(188)limL⟶+∞κα,SE,LL=d/dtωα̇t1/λ1λ2α̇3tq¯1+λ12/2−e−α3α̇1t+eα3α̇2tp¯12,ifωα̇t=0,ddtωα̇t≠0.Proof.
By Equation (25) and α̇∈TS, we get the following:
(189)α̇=−1λ1λ2α̇3tq¯1+λ122−e−α3α̇1t+eα3α̇2tp¯1T1−lLlL1/2ωα̇tT2.
By Equation (181), we have the following:
(190)∇α̇S,E,Lα̇,∇α̇S,E,Lα̇S,L~λ12λ242q¯12−e−α3α̇1t+eα3α̇2t2+p¯12α̇3t2L2ωα̇t2asL⟶+∞.
Similarly, we have that whenωα̇t≠0,
(191)α̇,α̇S,L~Lωα̇t2asL⟶+∞.
By Equations (181) and (189), we have the following:
(192)∇α̇S,E,Lα̇,α̇S,L~M0LasL⟶+∞,where M0 does not depend on L. By Equation (183), we have the following:
(193)κα,SE,∞=limL⟶+∞κα,SE,L=λ12λ24/2q¯12−e−α3α̇1t+eα3α̇2t2+p¯12α̇3t2ωα̇tifωα̇t≠0.
Whenωα̇t=0, and d/dtωα̇t=0, we have the following:
(194)∇α̇S,E,Lα̇,∇α̇S,E,Lα̇S,L~−q¯11λ1λ2α¨3t−p¯1λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α32asL⟶+∞,(195)α̇,α̇S,L=1λ1λ2α̇3tq¯1+λ122−e−α3α̇1t+eα3α̇2tp¯12,(196)∇α̇S,E,Lα̇,α̇S,L=1λ1λ2α̇3tq¯1+λ122−e−α3α̇1t+eα3α̇2tp¯1·q¯11λ1λ2α¨3t+p¯1λ122α¨2eα3+α̇2α̇3eα3−α¨1e−α3+α̇1α̇3e−α3≔B1B2.
By Equations (194)–(196) and Equation (182), we get the following:
(197)κα,SE,∞,s=B22B14−B12B22B16=0.
Whenωα̇t=0 and d/dtωα̇t≠0, we have the following:
(198)∇α̇S,E,Lα̇,∇α̇S,E,Lα̇S,L~Lddtωα̇t2asL⟶+∞,α̇,α̇S,L=1λ1λ2α̇3tq¯1+λ122−e−α3α̇1t+eα3α̇2tp¯12,∇α̇S,E,Lα̇,α̇S,L=O1,limL⟶+∞κα,SE,LL=d/dtωα̇t1/λ1λ2α̇3tq¯1+λ12/2−e−α3α̇1t+eα3α̇2tp¯12,if ωα̇t=0, and d/dtωα̇t≠0, so we get Equation (188).Proposition 26.
LetS⊂Eλ1,λ2L be a regular spacelike surface. Let α:I⟶S be a spacelike C2-smooth regular curve. Then,
(199)κα,SE,∞,s=q¯1λ1λ222/2−e−α3α̇1t+eα3α̇2t+p¯1α̇3tωα̇t,ifωα̇t≠0,κα,SE,∞,s=0,ifωα̇t=0,ddtωα̇t=0,limL⟶+∞κα,SE,L,sL=d/dtωα̇t1/λ1λ2α̇3tq¯1+λ12/2−e−α3α̇1t+eα3α̇2tp¯12,if ωα̇t=0, and d/dtωα̇t≠0.Proof.
By Equations (176) and (188), we get the following:
(200)JLα̇=lLlL1/2ωα̇tT1−1λ1λ2α̇3tq¯1+λ122−e−α3α̇1t+eα3α̇2tp¯1T2.
By Equation (180) and the above equation, we have the following:
(201)∇α̇S,E,Lα̇,JLα̇S,L~q¯1λ1λ2222−e−α3α̇1t+eα3α̇2t+p¯1α̇3tωα̇t2L3/2asL⟶+∞.
So, we get the following:(202)κα,SE,L,s=∇α̇S,E,Lα̇,JLα̇S,Lα̇S,L3=L3/2q¯1λ1λ222/2−e−α3α̇1t+eα3α̇2t+p¯1α̇3tωα̇t2L3/2ωα̇t3.
Furthermore,(203)κα,SE,∞,s=limL⟶+∞κα,SE,L,s=q¯1λ1λ222/2−e−α3α̇1t+eα3α̇2t+p¯1α̇3tωα̇t.
Whenωα̇t=0, and d/dtωα̇t=0, we get the following:
(204)∇α̇S,E,Lα̇,JLα̇S,L~M1L−1/2asL⟶+∞,where M1 does not depend on L. So, κα,SE,∞,s=0. When ωα̇t=0, and d/dtωα̇t≠0, we have the following:
(205)∇α̇S,E,Lα̇,JLα̇S,L~L1/21λ1λ2α̇3tq¯1+λ122−e−α3α̇1t+eα3α̇2tp¯1ddtωα̇tas L⟶+∞. We get the following:
(206)κα,SE,∞,s=limL⟶+∞κα,SE,LL=d/dtωα̇t1/λ1λ2α̇3tq¯1+λ12/2−e−α3α̇1t+eα3α̇2tp¯12.In the following, we compute the intrinsic Gaussian curvature of spacelike surfaces inEλ1,λ2L. Similar to Theorem 4.3 in [16], we have the following:Theorem 27.
The second fundamental formIIE,L of the embedding of S into Eλ1,λ2L is given by the following:
(207)IIE,L=h11h12h21h22,where
(208)h11=−llLE1p¯1−E2q¯1−λ12r¯1,Lp¯1q¯1L−1/2,h12=h21=−llLT1,∇Hr¯1,LL+L2λ22+λ122Lq¯1,L2+p¯1,L2−r¯1,L2λ122Lp¯12+q¯12,h22=−l2lL2T2,∇HrlL+E3~r¯1,L+r¯1,Lp¯1,Lq¯1,LLλ12−r¯1,L3p¯1q¯1Lλ12.Proof.
SinceT1,NLL=0,T2,NLL=0, we have the following:
(209)∇T1E,LNL,T1L=−∇T1E,LT1,NLL,∇T2E,LNL,T2L=−∇T2E,LT2,NLL.
Using the definition of the connection, the identities in Equation (11), and grouping terms, we have the following:
(210)∇T1E,LT1=∇q¯1E1−p¯1E2E,Lq¯1E1−p¯1E2=q¯1E1q¯1E1−E1p¯1E2−p¯1∇E1E,LE2−p¯1E2q¯1E1−E2p¯1E2+q¯1∇E2E,LE1=q¯1E1q¯1−p¯1E2q¯1E1−q¯1E1p¯1−p¯1E2p¯1E2+p¯1q¯1λ12LE3.
Sincep¯12−q¯12=1, we have p¯1Eip¯1−q¯1Eiq¯1=0,i=1,2,3. We have the following:
(211)h11=−∇T1E,LT1,NLL=−p¯1,Lq¯1E1q¯1−p¯1E2q¯1−q¯1,Lq¯1E1p¯1−p¯1E2p¯1+r¯1,Lp¯1q¯1λ12L=−llLE1p¯1−E2q¯1−λ12r¯1,Lp¯1q¯1L−1/2.
Similarly, we have the following:(212)h12=−llLT1,∇Hr¯1,LL+L2λ22+λ122Lq¯1,L2+p¯1,L2−r¯1,L2λ122Lp¯12+q¯12,h21=−llLT1,∇Hr¯1,LL+L2λ22+λ122Lq¯1,L2+p¯1,L2−r¯1,L2λ122Lp¯12+q¯12,h22=−l2lL2T2,∇HrlL+E3~r¯1,L+r¯1,Lp¯1,Lq¯1,LLλ12−r¯1,L3p¯1q¯1Lλ12.Similar to Proposition17, we get the expression of the horizontal mean curvature of the spacelike surface.Proposition 28.
Away from characteristic point, the horizontal mean curvatureHE,∞ of S⊂Eλ1,λ2L is given by the following:
(213)HE,∞=−E1p¯1+E2q¯1.Proposition 29.
Away from characteristic points, we have the following:(214)KS,E,∞T1,T2=−T1,∇HE3F∇HFL−λ24E3F2l2.Proof.
By Equation (141) and Lemma 18, we have the following:
(215)KE,LT1,T2=−RE,LT1,T2T1,T2L=−r¯1,L2λ12λ222−λ144L+3λ24L4−llLq¯12λ12λ222−λ24L4+3λ144L−llLp¯12λ12λ222+λ144L+λ24L4~−llL2λ24L4−3λ244E3u2l2−λ12λ22q¯12−λ12λ222asL⟶∞.
Similar to Equation (148), we obtain the following:
(216)detIIE,L=h11h22−h122=−λ24L4+T1,∇HE3F∇HFL−λ12λ222p¯12+q¯12+OL−1/2asL⟶∞.
By Equations (215) and (216) and KS,E,L=KE,L−detIIE,L, we get the desired equation.
Similar to Lemma20 and Proposition 21, for spacelike curve and spacelike surface, we obtain the following:
(217)limL⟶∞1LdsE,L=dsE,(218)limL⟶∞1LdσS,E,L=dσS,E.
Combining Equations (214) and (217) and Proposition 26, similar to the proof of Theorem 22, we have the second Gauss-Bonnet theorem.Theorem 30.
LetS⊂Eλ1,λ2L be a regular spacelike surface with finitely many boundary components ∂Si,i∈1,⋯,n, given by C2-smooth closed and regular spacelike curves αi:0,2π⟶∂Si. Let KS,E,∞ be intrinsic Gaussian curvature of S in Proposition 29 and καi,SE,∞,s the intrinsic signed geodesic curvature of αi relative to S in Proposition 26. Suppose that the characteristic set CS satisfies H1CS=0 and that ∇HFH−1 is locally summable with respect to the 2-dimensional Hausdorff measure near the characteristic set CS. Then,
(219)∫SKS,E,∞dσS,E+∑i=1n∫αiκαi,SE,∞,sdsE=0.



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