# Lorentzian Approximations and Gauss–Bonnet Theorem forE1,1 with the Second Lorentzian Metric

Authors: Haiming Liu; Xiawei Chen
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we consider the Lorentzian approximations of rigid motions of the Minkowski planeEL21,1. By using the method of Lorentzian approximations, we define the notions of the intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on Lorentzian surface, and the intrinsic Gaussian curvature of Lorentzian surface in E1,1 with the second Lorentzian metric away from characteristic points. Furthermore, we derive the expressions of those curvatures and prove Gauss–Bonnet theorem for the Lorentzian surface in E1,1 with the second left-invariant Lorentzian metric g2.


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## Body

## 1. Introduction

In 2021, Wang and Wei studied Gauss–Bonnet theorems in the affine group, the group of rigid motions of the Minkowski plane, BCV spaces, and the twisted Heisenberg group by using the method of Riemannian approximations in References [1, 2], which were first took by Balogh, Tyson, and Vecchi to prove a Heisenberg version of the Gauss–Bonnet theorem in References [3, 4]. Riemannian approximations can be extended to the case for any Lie group equipped with left-invariant Lorentzian metric g named Lorentzian approximations. Some typical work of Lorentzian approximations in Lorentzian Heisenberg group is obtained in Reference [5, 6]. For Lorentzian Sasakian space forms, we refer to Reference [7], and for semiaffine transformations of the Euclidean plane and Borel subgroup of the group SL2,R, we refer to References [8–10]. In Reference [11], the authors classified parallel surfaces in the groups of rigid motions of Minkowski plane. Inspired by the abovementioned work, we proved Gauss–Bonnet theorems in E1,1 with the general left-invariant metric in References [12, 13], where E1,1 is one of the three-dimensional unimodular Lie groups classified by Milnor in Reference [14].One motivation for this paper was our observation in Reference [15] that there are two classes of left-invariant Lorentzian metrics g1 and g2 on the group of rigid motions of Minkowski 2-space E1,1, where g1 satisfies a Ricci soliton equation and g2 is geodesically complete and flat. A natural question is how to prove Gauss–Bonnet theorem for Lorentzian surface in E1,1 with the second left-invariant Lorentzian metric g2. In this paper, we try to solve this question by employing the method of the Lorentzian approximation scheme. To do this, we define the notions of the intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on Lorentzian surface, and the intrinsic Gaussian curvature of Lorentzian surface away from characteristic points. Furthermore, we derive the expressions of those curvatures and prove Gauss–Bonnet theorem for the Lorentzian surface in E1,1 with the second left-invariant Lorentzian metric g2.The paper is organized in the following way: In Section2, we introduce some notations on the Lorentzian approximations of E1,1. Furthermore, we define the notions of curvature and intrinsic curvature of curves in the Lorentzian approximations of E1,1. We then get the expressions of those curvatures. In Section 3, we compute intrinsic geodesic curvatures of regular curves on Lorentzian surfaces and the intrinsic Gaussian curvature of Lorentzian surface in the Lorentzian approximations of E1,1. Finally, we prove the Gauss–Bonnet theorem for the Lorentzian surface in E1,1. In Section 4, we summarize the main results.


## 2. Lorentzian Approximations forE1,1

LetE1,1 be the group of rigid motions of Minkowski 2-space in Reference [15]. This consists of all matrices of the following form:(1)coshx1sinhx1x2sinhx1coshx1x3001.Topologically,E1,1 is diffeomorphic to R3 under the map(2)E1,1∋coshx1sinhx1x2sinhx1coshx1x3001↦x1,x2,x3.It is Lie algebra that has a basis consisting of(3)X1=∂∂x1,X2=coshx1∂∂x2+sinhx1∂∂x3,X3=sinhx1∂∂x2+coshx1∂∂x3,for which(4)X1,X2=X3,X2,X3=0,X1,X3=X2.Putx=x1, y=1/2x2+x3, and z=1/2x2−x3. Then, we get the following equation:(5)X1=∂∂x,X2=12ex∂∂y+e−x∂∂z,X3=12ex∂∂y−e−x∂∂z.We consider left-invariant Lorentzian metric which has a pseudo-orthonormal basisXii=13, given by(6)g2=dx2+e−xdy+exdz2−e−xdy−exdz2.Let our frame be defined byF1=X1, F2=X2, and F3=X3, then(7)∂∂x=F1,∂∂y=e−xF2+F3,∂∂z=exF2−F3,and spanF1,F2,F3=TE1,1. Let H=spanF1,F2 be the horizontal distribution on E1,1. Let(8)θ1=dx,θ2=e−xdy+exdz,θ=e−xdy−exdz.Then,H=kerθ. For the constant L>0, let g2L=θ1⊗θ1+θ2⊗θ2−Lθ⊗θ, and g2=g21 be the Lorentzian metric on E1,1 in Reference [15]. We call E1,1,g2L the Lorentzian approximations of rigid motions of the Minkowski plane and write EL21,1 instead of E1,1,g2L. Then, F1,F2,F3˜≔L−1/2F3 are pseudo-orthonormal bases on TEL21,1 with respect to g2L, with brackets(9)F1,F2=F3,F2,F3=0,F1,F3=F2.A nonzero vectorx∈EL21,1 is called to be space-like, null, or time-like if x,x>0, x,x=0, and x,x<0, respectively. We define the norm of the vector x∈EL21,1 by x=x,x. Let β:I⟶EL21,1 be a regular curve, where I is an open interval in R. The regular curve β is called a space-like curve, time-like curve, or null curve if β˙t is a space-like vector, time-like vector, or null vector at any t∈I, respectively.We assume that∇L is the Levi-Civita connection on EL21,1 with respect to g2L. By the Koszul formula, we have the following equation:(10)2∇FiLFj,FkL=Fi,Fj,FkL−Fj,Fk,FiL+Fk,Fi,FjL,where i,j,k=1,2,3. By (9) and (10), we have the following:Lemma 1.
The Levi-Civita connection onEL21,1 relative to the coordinate frame F1,F2,F3˜ is given by(11)∇F1LF1=0,∇F1LF2=1+L2LF3,∇F1LF3=1+L2F2,∇F2LF1=1−L2LF3,∇F2LF2=0,∇F2LF3=1−L2F1,∇F3LF1=L−12F2,∇F3LF2=1−L2F1,∇F3LF3=0.
Define the curvature of the connection∇X,YL by(12)RLX,YZ=∇XL∇YLZ−∇YL∇XLZ−∇X,YLZ.
We get the following proposition.Proposition 1.
The curvatures of the connection∇L for EL21,1 are given by(13)RLF1,F2F1=1−L1+3L4LF2,RLF1,F2F2=−1−L1+3L4LF1,RLF1,F2F3=0,RLF1,F3F1=L+3L−14LF3,RLF1,F3F2=0,RLF1,F3F3=L+3L−14F1,RLF2,F3F1=0,RLF2,F3F2=1−L24LF3,RLF2,F3F3=1−L24F2.Proof.
It is a direct computation using(14)RLX,YZ=∇XL∇YLZ−∇YL∇XLZ−∇X,YLZ.
Taking(15)RLF1,F2F1=∇F1L∇F2LF1−∇F2L∇F1LF1−∇F1,F2LF1,for example, we compute(16)∇F1L∇F2LF1=1−L24LF2,∇F2L∇F1LF1=0,∇F1,F2LF1=L−12F2.
Hence,(17)RLF1,F2F1=1−L1+3L4LF2.Definition 1.
Letβ:I⟶EL21,1 be a C1-smooth curve, we say that β is regular if β˙t≠0 for every t∈I. Moreover, we say that βt is a horizontal point of β if(18)θβ˙t=e−β1β˙2t−eβ1β˙3t=0,where βt=β1t,β2t,β3t.
The curvature for a curve with an arbitrary parametrization is given as follows:Definition 2.
Letβ:I⟶EL21,1 be a C2-smooth regular curve.(1)
If∇β˙Lβ˙ is a space-like vector, we define the curvature κβL of β at βt by(19)κβL≔∇β˙Lβ˙L2β˙L4−∇β˙Lβ˙,β˙L2β˙,β˙L3.(2)
If∇β˙Lβ˙ is a time-like vector, we define the curvature κβL of β at βt by(20)κβL≔∇β˙Lβ˙L2β˙L4+∇β˙Lβ˙,β˙L2β˙,β˙L3.
One can get the expression of curvature for curve as follows:Lemma 2.
Suppose thatβ:I⟶EL21,1 is a C2-smooth regular curve.(1)
If∇β˙Lβ˙ is a space-like vector, then(21)κβL=β¨1+1−LAθβ˙t2+B+Lβ˙1θβ˙t2−LL−1Aβ˙1+ddtθβ˙t2×β˙12+A2−Lθβ˙t2−2−β˙1β¨1+1−LAθβ˙t+AB+Lβ˙1θβ˙t−Lθβ˙tL−1Aβ˙1+ddtθβ˙t2×β˙12+A2−Lθβ˙t2−31/2,whichA=e−β1β˙2+eβ1β˙3 and B=−β˙1e−β1β˙2+β¨2e−β1+β˙1eβ1β˙3+β¨3eβ1.In particular, ifβt is a horizontal point of β,(22)κβL=β¨12+B2−LL−1Aβ˙1+ddtθβ˙t2×β˙12+A2−2−β˙1β¨1+AB2×β˙12+A2−31/2.(2)
If∇β˙Lβ˙ is a time-like vector, then(23)κβL=−β¨1+1−LAθβ˙t2+B+Lβ˙1θβ˙t2−LL−1Aβ1.+ddtθβ˙t2×β˙12+A2−Lθβ˙t2−2+β˙1β¨1+1−LAθβ˙t+AB+Lβ˙1θβ˙t−Lθβ˙tL−1Aβ˙1+ddtθβ˙t2×β˙12+A2−Lθβ˙t2−31/2.
In particular, ifβt is a horizontal point of β,(24)κβL=−β¨12+B2−LL−1Aβ˙1+ddtθβt.2×β˙12+A2−2+β˙1β¨1+AB2×β˙12+A2−31/2.Proof.
By using equation (7), we have the following equation:(25)β˙t=β˙1F1+AF2+θβ˙tF3,where A=e−β1β˙2+eβ1β˙3 and B=−β˙1e−β1β˙2+β¨2e−β1+β˙1eβ1β˙3+β¨3eβ1. By Lemma 1 and (25), we obtain the following equation:(26)∇β˙LF1=1−L2LAF3−1−L2θβ˙tF2,∇β˙LF2=1+L2Lβ˙1F3+1−L2θβ˙tF1,∇β˙LF3=1+L2β˙1F2+1−L2AF1.
By (25) and (26), we have the following equation:(27)∇β˙Lβ˙=β¨1+1−LAθβ˙tF1+B+Lβ˙1θβ˙tF2+L−1Aβ˙1+ddtθβ˙tF3.
By Definition2, (25), and (27), we get Lemma 2.Definition 3.
Letβ:I⟶EL21,1 be a C2-smooth regular curve. We define the intrinsic curvature κβ∞ of β at βt to be(28)κβ∞≔limL⟶∞κβLif the limit exists.
We introduce the following notation: for continuous functionsf1,f2:0,+∞⟶ℝ,(29)f1L∼f2L,asL⟶+∞⇔limL⟶∞f1Lf2L=1.Lemma 3.
Letβ:I⟶EL21,1 be a C2-smooth regular curve. Then(1)
if∇β˙Lβ˙ is a space-like vector, then(30)κβ∞=A2+β˙12θβ˙t,ifθβ˙t≠0,(31)κβ∞=β¨12+B2β˙12+A2−2−β˙1β¨1+AB2β˙12+A2−31/2,ifθβ˙t=0 and d/dtθβ˙t=0. If θβ˙t=0, and d/dtθβ˙t≠0,(32)limL⟶∞κβLL=−d/dtθβ˙t2β˙12+A2.Therefore, this situation does not exist.(2)
If∇β˙Lβ˙ is a time-like vector, then(33)κβ∞=−A2+β˙12θβ˙t,ifθβ˙t≠0.
Therefore, this situation does not exist.(34)κβ∞=−β¨12+B2β˙12+A2−2+β˙1β¨1+AB2β˙12+A2−31/2,if θβ˙t=0 and d/dtθβ˙t=0. If θβ˙t=0, and d/dtθβ˙t≠0,(35)limL⟶∞κβLL=d/dtθβ˙tβ˙12+A2.Proof.
(1) If∇β˙Lβ˙ is a space-like vector.
Using the notation introduced in (29), when θβ˙t≠0, we have the following equations:(36)∇β˙Lβ˙,∇β˙Lβ˙L∼L2θβ˙t2A2+β˙12asL⟶+∞,(37)β˙,β˙L∼−Lθβ˙t2,∇β˙Lβ˙,β˙L2=OL2asL⟶+∞.
Thus(38)∇β˙Lβ˙,∇β˙Lβ˙Lβ˙L4⟶A2+β˙12θβ˙tasL⟶+∞,∇β˙Lβ˙,β˙L2β˙,β˙L3⟶0asL⟶+∞.
So by Definition2, we have (30). If θβ˙t=0 and d/dtθβ˙t=0, we get (31). If θβ˙t=0 and d/dtθβ˙t≠0, then(39)∇β˙Lβ˙,∇β˙Lβ˙L∼−Lddtθβ˙t2,asL⟶+∞,β˙,β˙L=β˙12+A2,∇β˙Lβ˙,β˙L2=O1asL⟶+∞.
Then, this situation does not exist.(2) If∇β˙Lβ˙ is a time-like vector, by similar calculation, we get (2).


## 3. Lorentzian Surface and Gauss–Bonnet Theorem inEL21,1

In this section, we will compute intrinsic geodesic curvatures of regular curves on Lorentzian surfaces and the intrinsic Gaussian curvature of Lorentzian surface in the Lorentzian approximations ofE1,1. Furthermore, we will prove the Gauss–Bonnet theorem for the Lorentzian surface in EL21,1.We say that a surfaceS⊂EL21,1 is regular if S is a C2-smooth compact and oriented surface. In particular, we will assume that there exists a C2-smooth function h:EL21,1⟶ℝ such that(40)S=x1,x2,x3∈EL21,1:hx1,x2,x3=0,and ∇EL21,1h=hx1∂x1+hx2∂x2+hx3∂x3≠0. Let ∇Hh=F1hF1+F2hF2. A point x∈S is called characteristic if ∇Hhx=0,0.Our computations will be local and away from characteristic points ofS.We definep:=F1h,q:=F2h,andr:=F3˜h. Since p2+q2>0, we say S⊂EL21,1 is a horizontal space-like surface. When L⟶+∞, then p2+q2−r2>0.Then we define(41)l:=p2+q2,lL:=p2+q2−r2,p¯:=pl,q¯:=ql,pL¯:=plL,qL¯:=qlL,rL¯:=rlL.In particular,p¯2+q¯2=1. These functions are well defined at every noncharacteristic point. Let(42)NL=pL¯F1+qL¯F2−rL¯F3˜,E1=q¯F1−p¯F2,E2=rL¯p¯F1+rL¯q¯F2−llLF3˜,then NL is the unit space-like normal vector to S, E1 is a unit space-like vector, and E2 is a unit time-like vector of S. E1,E2 is the pseudo-orthonormal basis of S. We call S a Lorentzian surface in EL21,1.Letβ˙=λ1E1+λ2E2, we define JLβ˙=λ1E2+λ2E1, if β is a C2-smooth space-like curve. We define JLβ˙=−λ1E2−λ2E1, if β is a C2-smooth time-like curve. Then, gLβ˙,JLβ˙=0 and β˙,JLβ˙ have the same orientation with E1,E2.For everyU,V∈TS, we define ∇US,LV=π∇ULV where π:TEL21,1⟶TS is the projection. Then, ∇S,L is the Levi-Civita connection on S with respect to the metric gL. By (26), (42), and the following(43)∇β˙S,Lβ˙=∇β˙Lβ˙,E1LE1−∇β˙Lβ˙,E2LE2,we have(44)∇β˙S,Lβ˙=q¯β¨1+1−LAθβ˙t−p¯B+Lβ˙1θβ˙tE1−r¯Lp¯β¨1+1−LAθβ˙t+r¯Lq¯B+Lβ1.θβ˙t+llLL1/2L−1Aβ˙1+ddtθβ˙tE2.Therefore, whenθβ˙t=0, we have the following equation:(45)∇β˙S,Lβ˙=q¯β¨1−p¯BE1−rL¯p¯β¨1+rL¯q¯B+llLL1/2L−1Aβ˙1+ddtθβ˙tE2.Definition 4.
LetS⊂EL21,1 be a Lorentzian regular surface and β:I⟶S be a C2-smooth regular curve.(1)
If∇β˙S,Lβ˙ is a space-like vector, the geodesic curvature κβ,SL of β at βt is defined as follows:(46)κβ,SL:=∇β˙S,Lβ˙S,L2β˙S,L4−∇β˙S,Lβ˙,β˙S,L2β˙,β˙S,L3.(2)
If∇β˙S,Lβ˙ is a time-like vector, the geodesic curvature κβ,SL of β at βt is defined as follows:(47)κβ,SL:=∇β˙S,Lβ˙S,L2β˙S,L4+∇β˙S,Lβ˙,β˙S,L2β˙,β˙S,L3.Definition 5.
LetS⊂EL21,1 be a Lorentzian regular surface and β:I⟶S be a C2-smooth regular curve. We define the intrinsic geodesic curvature κβ,S∞ of β at βt to be(48)κβ,S∞:=limL⟶+∞κβ,SL,if the limit exists.Lemma 4.
LetS⊂EL21,1 be a regular Lorentzian surface. Let β:I⟶S be a C2-smooth regular curve.(1)
If∇β˙S,Lβ˙ is a space-like vector, then(49)κβ,S∞=Aq¯+β˙1p¯θβ˙t,ifθβ˙t≠0,κβ,S∞=0,ifθβ˙t=0,andddtθβ˙t=0,(50)limL⟶+∞κβ,SLL=−d/dtθβ˙t2q¯β˙1−Ap¯2,fθβ˙t=0,andddtθβ˙t≠0.Therefore, this situation does not exist.(2)
If∇β˙S,Lβ˙ is a time-like vector, then(51)κβ,S∞=−Aq¯+β˙1p¯2θβ˙t,ifθβ˙t≠0.
Therefore, this situation does not exist.(52)κβ,S∞=0,ifθβ˙t=0,andddtθβ˙t=0,(53)limL⟶+∞κβ,SLL=d/dtθβ˙tq¯β˙1−Ap¯2,ifθβ˙t=0,andddtθβ˙t≠0.Proof.
(1) If∇β˙S,Lβ˙ is a space-like vector. By (25) and β˙∈TS, we have the following equation:(54)β˙t=q¯β˙1−Ap¯E1−lLlL1/2θβ˙tE2.
By (44), we have the following equation:(55)∇β˙S,Lβ˙,∇β˙S,Lβ˙S,L=q¯β¨1+1−LAθβ˙t−p¯B+Lβ˙1θβ˙t2−rL¯p¯β¨1+1−LAθβ˙t+rL¯q¯B+Lβ˙1θβ˙t+llLL1/2L−1Aβ˙1+ddtθβ˙t2.
Similarly, we have that whenθβ˙t≠0,(56)β˙,β˙S,L=q¯β˙1−Ap¯2−lLl2Lθβ˙t2∼−Lθβ˙t2asL⟶+∞.
By (44) and (54), we have the following equation:(57)∇β˙S,Lβ˙,β˙S,L∼M0L,where M0 does not depend on L. By Definition 4 and (55)-(57), we get (49).
Whenθβ˙t=0 and d/dtθβ˙t=0, then(58)∇β˙S,Lβ˙,∇β˙S,Lβ˙S,L=q¯β¨1−p¯B2−llLL−1/2Aβ˙12∼q¯β¨1−p¯B2,(59)β˙,β˙S,L=q¯β˙1−Ap¯2,asL⟶+∞,(60)∇β˙S,Lβ˙,β˙S,L=q¯β˙1−Ap¯q¯β¨1−p¯B.
By (58)-(60) and Definition 4, we get κβ,S∞=0.
Whenθβ˙t=0 and d/dtθβ˙t≠0, we have the following equations:(61)∇β˙S,Lβ˙,∇β˙S,Lβ˙S,L∼−Lddtθβ˙t2,(62)∇β˙S,Lβ˙,β˙S,L=O1,so we get (50).
(2) If∇β˙S,Lβ˙ is a time-like vector, by similar calculation, we get (2).Definition 6.
LetS⊂EL21,1 be a Lorentzian regular surface and β:I⟶S be a C2-smooth regular curve. The signed geodesic curvature κβ,SL,c of β at βt is defined as follows:(63)κβ,SL,c≔∇β˙S,Lβ˙,JLβ˙S,Lβ˙S,L3.Definition 7.
LetS⊂EL21,1 be a regular surface. Let β:a,b⟶S be a C2-smooth regular curve. We define the intrinsic geodesic curvature κβ,S∞ of β at the noncharacteristic point βt to be(64)κβ,S∞,c:=limL⟶+∞κβ,SL,c,if the limit exists.Lemma 5.
LetS⊂EL21,1 be a Lorentzian regular surface.(1)
Ifβ:I⟶S be a C2-smooth regular space-like curve, then(65)κβ,S∞,c=q¯A+p¯β1.θβ˙t.ifθβ˙t≠0.(66)κβ,S∞,c=0,ifθβ˙t=0,andddtθβ˙t=0,(67)limL⟶+∞kβ,SL,cL=d/dtθβ˙tβ˙1q¯−Ap¯2,ifθβ˙t=0,andddtθβ˙t≠0.(2)
Ifβ:I⟶S be a C2-smooth regular time-like curve, then(68)κβ,S∞,c=−q¯A+p¯β˙1θβ˙t.ifθβ˙t≠0,(69)κβ,S∞,c=0,ifθβ˙t=0,andddtθβ˙t=0,(70)limL⟶+∞kβ,SL,cL=−d/dtθβ˙tβ˙1q¯−Ap¯2,ifθβ˙t=0,andddtθβ˙t≠0.Proof.
For (1), by (54), we have the following equation:(71)JLβ˙=−lLlL1/2θβ˙tE1+q¯β˙1−Ap¯E2.
By (44) and (71), we have the following equations:(72)∇β˙S,Lβ˙,JLβ˙S,L=−lLlL1/2θβ˙tq¯β¨1+1−LAθβ˙t−p¯B+Lβ˙1θβ˙t+q¯β˙1−Ap¯rL¯p¯β¨1+1−LAθβ˙t+rL¯q¯B+Lβ˙1θβ˙t+llLL1/2L−1Aβ˙1+ddtθβ˙t∼−L3/2θβ˙t2q¯A+p¯β˙1,asL⟶+∞,(73)β˙S,L2=q¯β˙1−Ap¯2−lLlL1/2θβ˙t2∼−Lθβ˙t2,asL⟶+∞.
Whenθβ˙t=0 and d/dtθβ˙t=0, we get the following equation:(74)∇β˙S,Lβ˙,JLβ˙L,S=OL−1/2,asL⟶+∞.
Soκβ,S∞,c=0. When θβ˙t=0 and d/dtθβ˙t≠0, we have the following equation:(75)∇β˙S,Lβ˙,JLβ˙L,S∼q¯β˙1−Ap¯llLL1/2L−1Aβ˙1+ddtθβ˙t∼Lq¯β˙1−Ap¯ddtθβ˙t,asL⟶+∞.
So we get (68).
(2) If β:I⟶S is a C2-smooth regular time-like curve, by similar calculation, we get (2).
Next, we compute the sub-Riemannian limit of the Gaussian curvature of surfaces in theEL21,1. The second fundamental form IIL of the embedding of S into EL21,1 is defined by(76)IIL=∇F1LNL,F1L∇F1LNL,F2L∇F2LNL,F1L∇F2LNL,F2L.
Similar to Theorem 4.3 in Reference [16], we have the following theorem:Theorem 1.
For the embedding ofS into EL21,1, the second fundamental form IIL of the embedding of S is given by(77)IIL=h11h12h21h22,where(78)h11=llLF1p¯+F2q¯+rL¯p¯q¯L−1/2,(79)h12=−lLlE1,∇HrL¯L−L−1/21−L2pL¯2−1+L2qL¯2+rL¯L1/21−L2L−L−1p2¯,(80)h21=−llLrL¯F1q¯−F2p¯+rL¯L−1/21lLp¯F3q¯−q¯F3p¯−1−L2−rL¯2L1/21+L2L+q¯2,(81)h22=lLl−2llLE2,∇HrL¯L+llL2F3˜rL¯+rL¯p¯q¯L−1/2rL¯+1+llLrL¯F3˜llL.
We define the mean curvatureℋL of S by(82)ℋL≔trIIL=h11+h22.
Let(83)KS,LE1,E2=−RS,LE1,E2E1,E2S,L,KLE1,E2=−RLE1,E2E1,E2L.
By the Gauss equation, we have(84)KS,LE1,E2=KLE1,E2+detIIL.Proposition 2.
The horizontal mean curvatureℋ∞ of S⊂EL21,1 away from characteristic point is the following form:(85)ℋ∞=limL⟶+∞ℋL=F1p¯+F2q¯.Proof.
By(86)lLlF2,∇HrL¯=OL−1/2,(87)F3˜rL¯⟶0,pL¯⟶p¯,(88)llLF1p¯+F2q¯⟶F1p¯+F2q¯,we get (85).Proposition 3.
Away from characteristic points, we have(89)KS,LE1,E2=C+OL−1,asL⟶+∞,where(90)C≔−1−L4LllL2L+3q¯2+L−1p¯2.Proof.
We compute(91)RLE1,E2E1=1−L4LrL¯p¯1+3Lq¯2+1−Lp¯2F1+1−L1+3L4LrL¯q¯F2+1−L4LllLL+3q¯2+L−1p¯2F3˜,then(92)RLF1,F2F1,F2L=1−L4LrL¯2p¯21+3Lq¯2+1−Lp¯2F1+1−L1+3L4LrL¯2q¯2F2+1−L4LllL2L+3q¯2+L−1p¯2F3˜.
By Proposition1, we find(93)KLF1,F2=−1−L4LllL2L+3q¯2+L−1p¯2.
By the second fundamental form and∇Hr¯L=L−1/2∇HF3h/∇Hh+OL−1 as L⟶+∞, we get(94)detIIL=h11h22−h12h21=0,asL⟶+∞.
By Theorem1 and equations (93) and (94), we get the desired equation.
We consider the case of a regular time-like curveβ:I⟶EL21,1 and define the length measure dsL=β˙Ldt.Lemma 6.
Letβ:I⟶EL21,1 be a C2-smooth time-like curve. Let(95)ds≔θβ˙tdt,ds¯≔121θβ˙t−1α2β˙32+e2β3β˙12dt.
Then(96)limL⟶+∞1L∫βdsL=∫abds.
Whenθβ˙t≠0, we have(97)1LdsL=ds+ds¯L−1+OL−2asL⟶+∞.
The situation ofθβ˙t=0, we have(98)1LdsL=1L−β˙12−A2dt.Proof.
We know that(99)β˙tL=−β˙12−A2+Lθβ˙t2,similar to the proof of Lemma 6.1 in Reference [3], we can prove (97). When θβ˙t≠0, we have(100)1LdsL=L−1−β˙12−A2+θβ˙t2dt.
Using the Taylor expansion, we can prove (96). From the definition of dsL and θβ˙t=0, we get (98).Proposition 4.
LetS⊂EL21,1 be a regular Lorentzian C2-smooth surface. Let dσS,L denote the surface measure on S with respect to the Lorentzian metric g2L. Let(101)dσS≔p¯θ2−q¯θ1∧θ,dσ¯S≔F3hlθ1∧θ2+F3h22l2p¯θ2−q¯θ1∧θ.
Then,(102)1LdσS,L=dσS+dσ¯SL−1+OL−2,asL⟶+∞.
IfS=fD with f=fu1,u2=f1,f2,f3:D⊂ℝ2⟶EL21,1, then(103)limL⟶+∞1L∫SdσS,L=∫D−4f3u1f2u2−f2u1f3u22−f1u1f2u2e−x−f3u2ex−f1u2f2u1e−x−f3u1ex21/2du1du2.Proof.
It is well known that(104)g2LF1,⋅=θ1,g2LF2,⋅=θ2,g2LE3,⋅=−Lθ.
We defineE1∗≔g2LE1,⋅,E2∗≔g2LE2,⋅, then(105)E1∗=q¯θ1−p¯θ2,E2∗=r¯Lp¯θ1+r¯Lq¯θ2−llLL1/2θ.
Therefore,(106)1LdσS,L=1LE1∗∧E2∗=llLp¯θ2−q¯θ1∧θ+1Lr¯Lθ1∧θ2.
Recalling(107)r¯L=F3hL−1/2p2+q2−L−1F3h2,and the Taylor expansion(108)1lL=1l−12l3F3h2L−1+OL−2asL⟶+∞,we get (102). By (4), we have(109)fu1=f1u1∂x+f2u1∂y+f3u1∂z=f1u1F1+f2u1e−xF2+F3+f3u1exF2−F3=f1u1F1+f2u1e−x+f3u1exF2+Lf2u1e−x−f3u1exF3˜,and(110)fu2=f1u2F1+f2u2e−x+f3u2exF2+Lf2u2e−x−f3u2exF3˜.
Let(111)N¯L=F1F2−F3˜f1u1f2u1e−x+f3u1exLf2u1e−x−f3u1exf1u2f2u2e−x+f3u2exLf2u2e−x−f3u2ex.
We know that(112)dσS,L=−detgijdu1du2,gij=g2Lfui,fuj,detgij=−N¯L,N¯LL,so by the dominated convergence theorem, we get E1∗,E2∗.
We get the Gauss–Bonnet theorem inEL21,1 as follows:Theorem 2.
LetS⊂EL21,1 be a regular Lorentzian surface with finitely many boundary components ∂Si,i∈1,…,n, given by C2-smooth regular and closed time-like curves βi:0,2π⟶∂Si. Suppose that the characteristic set CS satisfies ℋ1CS=0 and that ∇HhH−1 is locally summable with respect to the 2-dimensional Hausdorff measure near the characteristic set CS. Let C be defined by (90) and dσS be defined by (101), and κβi,S∞,c is the intrinsic geodesic curvature of βi relative to S.(113)∫SCdσS+∑i=1n∫βiκβi,S∞,cds=0.Proof.
By the discussions in References [3, 4], one can get the number of points satisfying(114)θβ˙it=0,ddtθβ˙it≠0,on γi is finite. We remark that the proof of Theorem 2 is based on an approximation argument by using the well-known Lebesgue dominated convergence theorem. According to the conditions of Lebesgue dominated convergence theorem, a set of finitely many points can be ignored as a null set. Therefore, by Lemma 5, we have(115)κβi,SL,c=κβi,S∞,c+OL−1.
By the classical Gauss–Bonnet theorem, we get(116)∫SKS,L1LdσS,L+∑i=1n∫βiκβi,SL,c1LdsL=2πχΣL.
We assume firstly thatCS is the empty set. By using the Lebesgue dominated convergence theorem and let L go to the infinity, then we obtain(117)∫SKS,∞dσS+∑i=1n∫βiκβi,S∞,cds=0.
Therefore, by (115), (116), (90), (101), and (102), we get(118)∫SCdσS+∑i=1n∫βiκβi,S∞,cds+OL−1/2=0.
LetL go to the infinity and use the dominated convergence theorem, we get the desired result.


## 4. Conclusion

This paper proved a Gauss–Bonnet theorem inE1,1 with the second left-invariant Lorentzian metric from the Lorentzian approximation scheme. The main result of this paper is Theorem 2, which is Gauss–Bonnet type theorem in the Lie group E1,1 with the second left-invariant Lorentzian metric. To prove Theorem 2, we derived the expressions of the intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on Lorentzian surface, and the intrinsic Gaussian curvature of Lorentzian surface in E1,1 with the second Lorentzian metric away from characteristic points.Gauss–Bonnet theorem connects the intrinsic differential geometry of a surface with its topology and has many applications. Therefore, it is interesting to study Gauss–Bonnet theorem for Lie groups.



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