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

Abstract

The aim of this paper was to obtain Gauss–Bonnet theorems on the Lorentzian Heisenberg group and the Lorentzian group of rigid motions of the Minkowski plane. At the same time, the sub-Lorentzian limits of Gaussian curvature for surfaces which are $C^2$-smooth in the Lorentzian Heisenberg group away from characteristic points and signed geodesic curvature for curves which are $C^2$-smooth on surfaces are studied. Using a similar method, we also studied the corresponding contents on Lorentzian group of rigid motions of the Minkowski plane.

1. Introduction

The Gauss–Bonnet theorem and the definition of Gaussian curvature for surfaces which are non-horizontal in sub-Riemannian Heisenberg space ${\mathbb{H}}^1$ have been introduced by Diniz-Veloso in [1]. The notion of Gaussian curvature for non-horizontal surfaces in sub-Riemannian Heisenberg space ${\mathbb{H}}^1$ was similar to Gauss curvature of surfaces in ${\mathbb{R}}^3$ with Hausdorff measure of area and particular normal to surface. The image of Gauss map was in the cylinder which has a radius of one. For Euclidean surface, which is $C^2$-smooth in the first Heisenberg group ${\mathbb{H}}^1$, the suitable candidate for the definition of intrinsic Gaussian curvature has been given by Balogh–Tyson–Vecchi in [2]. Taking advantage of these results, the Gauss–Bonnet theorem on Heisenberg group was proved. In M. Diniz and Z. Balogh [1,2], Gaussian curvature of surfaces and normal curvature of curves in surfaces in Heisenberg space ${\mathbb{H}}^1$ have been introduced respectively. In M. Veloso [3], Veloso applied the above definitions respectively to prove that the Gauss–Bonnet theorem on Hesenberg groups ${\mathbb{H}}^1$ were not equal. Veloso then used the same formalism of reference [1] to obtain the curvatures of reference [2]. Using the obtained formulas, Gauss–Bonnet theorem in reference [2] is possible as an application of Stokes theorem. In P. Gilkey [4], P. Gilkey and J. H. Park use analytic continuation to derive the Chern–Gauss–Bonnet theorem for pseudo-Riemannian manifolds with boundary directly from the corresponding result in the Riemannian setting. In this paper, we will use similar methods to get the Gauss–Bonnet theorem in the Lorentzian Heisenberg group and the Lorentzian group of rigid motions of the Minkowski plane.

In Z. Balogh [2], the scheme which is called Riemannian approximation scheme was gave. Moreover, the scheme depends on the choice of the complement to the horizontal distribution. In Z. Balogh [2], the choice is rather natural in ${\mathbb{H}}^1$. The existence of the limit of the intrinsic curvature of a surface is closely related to the cancellation of some divergent quantities in the limit. This cancellation is related to the specific choice of the frame bundle that is adapted on the surface. On the other hand, the cancellation stems from the symmetries that is of the underlying left-invariant group structure on the Heisenberg group.

In Z. Balogh [2], they want to get to what extant similar phenomena hold in other spaces. In this paper, we have solved this question. In Y. Wang [5], we study Gauss–Bonnet theorems in the affine group and the group of rigid motions of the Minkowski plane. The main results of this paper are Gauss–Bonnet type theorems for the spacelike surfaces and Lorentzian surfaces in Lorentzian Heisenberg group and the Lorentzian group of rigid motions of the Minkowski plane (see Theorems 2, 4, 6 and 8).

In Lorentzian Heisenberg group, we investigate the sub-Lorentzian limit of curvature of curves in Section 2. Secondly, we study sub-Lorentzian limits of geodesic curvature of curves on Lorentzian surfaces, and give the Riemannian Gaussian curvature of surfaces in Section 3. Moreover, we obtain sub-Lorentzian limits of geodesic curvature of curves on spacelike surfaces, and get the Riemannian Gaussian curvature of surfaces in Section 4. On the other hand, we also get Gauss–Bonnet theorems in Section 3 and Section 4.

In the Lorentzian group of rigid motions of the Minkowski plane, we investigate the sub-Lorentzian limit of curvature of curves in Section 5. We then give sub-Lorentzian limits of geodesic curvature of curves on Lorentzian surfaces, and obtain the Riemannian Gaussian curvature of surfaces and a Gauss–Bonnet theorem in Section 6. In Section 7, firstly, we study the sub-Lorentzian limit of curvature of curves. Secondly, we compute sub-Lorentzian limits of geodesic curvature of curves on spacelike surfaces and the Riemannian Gaussian curvature of surfaces. Moreover, we also get another Gauss–Bonnet theorem.

2. The Sub-Lorentzian Limit of Curvature of Curves in the Lorentzian Heisenberg Group

In this section, some basic notions in the Lorentzian Heisenberg group will be introduced. Let $\mathbb{H}$ be the first Heisenberg group. The non-commutative group law in $\mathbb{H}$ is given by

$$(b_1,b_2,b_3)🟉(a_1,a_2,a_3)=(a_1+b_1,a_2+b_2,a_3+b_3-\frac{1}{2}(a_1{b}_2-a_2{b}_1)).$$

Let

$$X_1={\partial }_{x_1}-\frac{1}{2}{x}_2{\partial }_{x_3},X_2={\partial }_{x_2}+\frac{1}{2}{x}_1{\partial }_{x_3},X_3=[X_1,X_2]={\partial }_{x_3},$$

(1)

then

$${\partial }_{x_1}=X_1+\frac{1}{2}{X}_3,{\partial }_{x_2}=X_2-\frac{1}{2}{x}_1{X}_3,{\partial }_{x_3}=X_3,$$

(2)

and $\mathrm{span}\{X_1,X_2,X_3\}=T\mathbb{H}.$ Let $H=\mathrm{span}\{X_1,X_2\}$ be the horizontal distribution on $\mathbb{H}$. Let ${\omega }_1=d{x}_1,{\omega }_2=d{x}_2,\omega =d{x}_3+\frac{1}{2}(x_{2}d{x}_1-x_{1}d{x}_2).$ Therefore, $H=\mathrm{Ker}\omega$. For $L>0$, where L is a constant, let $g_L=-{\omega }_1\otimes {\omega }_1+{\omega }_2\otimes {\omega }_2+L\omega \otimes \omega ,g=g_1$ be the Lorentzian metric on $\mathbb{H}$. We call $(\mathbb{H},g_L)$ a Lorentzian Heisenberg group and write ${\mathbb{H}}_L^1$ instead of $(\mathbb{H},g_L)$. Therefore, $X_1,X_2,\widetilde{X_3}:=L^{-\frac{1}{2}}{X}_3$ form an orthonormal basis on $T{\mathbb{H}}_L^1$ with respect to $g_L$. Then

$$[X_1,X_2]=X_3,[X_2,X_3]=0,[X_1,X_3]=0.$$

(3)

For a non-zero vector $\mathbf{x}\in {\mathbb{H}}_L^1$, $\mathbf{x}$ is called to be $spacelike$, $null$ or $timelike$ if $⟨\mathbf{x},\mathbf{x}⟩>0$, $⟨\mathbf{x},\mathbf{x}⟩=0$ or $⟨\mathbf{x},\mathbf{x}⟩<0$ respectively. For $\mathbf{x}\in {\mathbb{H}}_L^1$, $\parallel \mathbf{x}\parallel =\sqrt{|⟨\mathbf{x},\mathbf{x}⟩|}$ is called to be the norm of the vector $\mathbf{x}$. Let $\gamma :I\to {\mathbb{H}}_L^1$ be a regular curve, where I is an open interval in $\mathbb{R}$. The regular curve $\gamma$ is called a spacelike curve, timelike curve or null curve if ${\gamma }^{\prime }\left(t\right)$ is a spacelike vector, timelike vector or null vector at any $t\in I$, respectively.

We assume that ${\nabla }^L$ is the Levi–Civita connection with respect to $g_L$ on ${\mathbb{H}}_L^1$. Using the Koszul formula, we have

$$2{⟨{\nabla }_{X_i}^L{X}_j,X_k⟩}_L={⟨[X_i,X_j],X_k⟩}_L-{⟨[X_j,X_k],X_i⟩}_L+{⟨[X_k,X_i],X_j⟩}_L,$$

(4)

where $i,j,k=1,2,3$. Taking advantage of (3) and (4), we obtain

Lemma 1.

Let${\mathbb{H}}_L^1$be Lorentzian Heisenberg group, then

$$\begin{array}{cc}\hfill & {\nabla }_{X_j}^L{X}_j=0,1\le j\le 3,{\nabla }_{X_1}^L{X}_2=\frac{1}{2}{X}_3,{\nabla }_{X_2}^L{X}_1=-\frac{1}{2}{X}_3,\hfill \\ \hfill & {\nabla }_{X_1}^L{X}_3=-\frac{L}{2}{X}_2,{\nabla }_{X_3}^L{X}_1=-\frac{L}{2}{X}_2,\hfill \\ \hfill & {\nabla }_{X_2}^L{X}_3={\nabla }_{X_3}^L{X}_2=-\frac{L}{2}{X}_1.\hfill \end{array}$$

(5)

Definition 1.

Let$\gamma :I\to {\mathbb{H}}_L^1$be a$C^1$-smooth curve. If$\dot{\gamma }\ne 0$for every$t\in I$, then γ is called regular. Moreover,$\gamma \left(t\right)$is called a horizontal point of γ if

$$\omega \left(\dot{\gamma }\left(t\right)\right)=\frac{{\gamma }_2\left(t\right)}{2}{\dot{\gamma }}_1\left(t\right)-\frac{{\gamma }_1\left(t\right)}{2}{\dot{\gamma }}_2\left(t\right)+{\dot{\gamma }}_3\left(t\right)=0.$$

As is well know, if $\gamma$ is a curve with arc length parametrization, then the standard definition of curvature for $\gamma$ in Riemannian geometry is $k_{\gamma }^L:=\parallel {\nabla }_{\dot{\gamma }}^L\dot{\gamma }\parallel$. If $\gamma$ is a curve with an arbitrary parametrization, then we give the definitions as follows:

Definition 2.

Let$\gamma :I\to {\mathbb{H}}_L^1$be a$C^2$-smooth regular curve.

(1) If${\nabla }_{\dot{\gamma }}^L\dot{\gamma }$is a spacelike vector, we define the curvature$k_{\gamma }^L$of γ at$\gamma \left(t\right)$by

$$k_{\gamma }^L:=\sqrt{\frac{\left|\right|{\nabla }_{\dot{\gamma }}^L\dot{\gamma }{\left|\right|}_L^2}{\left|\right|\dot{\gamma }{\left|\right|}_L^4}-\frac{{⟨{\nabla }_{\dot{\gamma }}^L\dot{\gamma },\dot{\gamma }⟩}_L^2}{{⟨\dot{\gamma },\dot{\gamma }⟩}_L^3}}.$$

(6)

(2) If${\nabla }_{\dot{\gamma }}^L\dot{\gamma }$is a timelike vector, we define the curvature$k_{\gamma }^L$of γ at$\gamma \left(t\right)$by

$$k_{\gamma }^L:=\sqrt{\frac{\left|\right|{\nabla }_{\dot{\gamma }}^L\dot{\gamma }{\left|\right|}_L^2}{\left|\right|\dot{\gamma }{\left|\right|}_L^4}+\frac{{⟨{\nabla }_{\dot{\gamma }}^L\dot{\gamma },\dot{\gamma }⟩}_L^2}{{⟨\dot{\gamma },\dot{\gamma }⟩}_L^3}}.$$

(7)

Lemma 2.

Let$\gamma :I\to {\mathbb{H}}_L^1$be a$C^2$-smooth regular curve. If${\nabla }_{\dot{\gamma }}^L\dot{\gamma }$is a spacelike vector, then

$$\begin{array}{cc}\hfill k_{\gamma }^L& =\left\{\left\{-{\left[{\ddot{\gamma }}_1-L\dot{{\gamma }_2}\omega \left(\dot{\gamma }\left(t\right)\right)\right]}^2+{\left[{\ddot{\gamma }}_2-L\dot{{\gamma }_1}\omega \left(\dot{\gamma }\left(t\right)\right)\right]}^2+L{\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\right]}^2\right\}\right.\hfill \\ \hfill & ·{\left[-{\dot{\gamma }}_1^2+{\dot{\gamma }}_2^2+L{\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)}^2\right]}^{-2}-\left[-{\dot{\gamma }}_1({\ddot{\gamma }}_1-L\dot{{\gamma }_2}\omega \left(\dot{\gamma }\left(t\right)\right))\right.\hfill \\ \hfill & {\left.{\left.+{\dot{\gamma }}_2({\ddot{\gamma }}_2-L\dot{{\gamma }_1}\omega \left(\dot{\gamma }\left(t\right)\right))+L\omega \left(\dot{\gamma }\left(t\right)\right)\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\right]}^2·{\left[-{\dot{\gamma }}_1^2+{\dot{\gamma }}_2^2+L{\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)}^2\right]}^{-3}\right\}}^{\frac{1}{2}}.\hfill \end{array}$$

(8)

In particular, if$\gamma \left(t\right)$is a horizontal point of γ, then

$$\begin{array}{cc}\hfill k_{\gamma }^L& =\{\left\{-{\left({\ddot{\gamma }}_1\right)}^2+{\left({\ddot{\gamma }}_2\right)}^2+L{\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\right]}^2\right\}·{\left(-{\dot{\gamma }}_1^2+{\dot{\gamma }}_2^2\right)}^{-2}.\hfill \\ \hfill & -{\left(-{\dot{\gamma }}_1{\ddot{\gamma }}_1+{\dot{\gamma }}_2{\ddot{\gamma }}_2\right)}^2·{\left(-{\dot{\gamma }}_1^2+{\dot{\gamma }}_2^2\right)}^{-3}{\}}^{\frac{1}{2}}.\hfill \end{array}$$

(9)

Proof. 

Using (2), we get the following equation

$$\dot{\gamma }\left(t\right)={\dot{\gamma }}_1{X}_1+{\dot{\gamma }}_2{X}_2+\omega \left(\dot{\gamma }\left(t\right)\right)X_3.$$

(10)

Combining Lemma 1 and (10), we obtain

$$\begin{array}{cc}\hfill & {\nabla }_{\dot{\gamma }}^L{X}_1=-\frac{L}{2}\left(\frac{{\dot{\gamma }}_1}{2}{\gamma }_2-\frac{{\dot{\gamma }}_2}{2}{\gamma }_1+{\dot{\gamma }}_3\right)X_2-\frac{{\dot{\gamma }}_2}{2}{X}_3,\hfill \\ \hfill & {\nabla }_{\dot{\gamma }}^L{X}_2=-\frac{L}{2}\left(\frac{{\dot{\gamma }}_1}{2}{\gamma }_2-\frac{{\dot{\gamma }}_2}{2}{\gamma }_1+{\dot{\gamma }}_3\right)X_1+\frac{{\dot{\gamma }}_1}{2}{X}_3,\hfill \\ \hfill & {\nabla }_{\dot{\gamma }}^L{X}_3=-\frac{{\dot{\gamma }}_1}{2}L{X}_2-\frac{L}{2}{\dot{\gamma }}_2{X}_1\hfill \end{array}$$

(11)

Using (10) and (11), we get

$$\begin{array}{cc}\hfill {\nabla }_{\dot{\gamma }}^L\dot{\gamma }& =\left[\ddot{{\gamma }_1}-L{\dot{\gamma }}_2\omega \left(\dot{\gamma }\left(t\right)\right)\right]X_1+\left[\ddot{{\gamma }_2}-L{\dot{\gamma }}_1\omega \left(\dot{\gamma }\left(t\right)\right)\right]X_2+\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\right]X_3.\hfill \end{array}$$

(12)

By Definition 2, (10) and (12), we get Lemma 2. □

Lemma 3.

Let$\gamma :I\to {\mathbb{H}}_L^1$be a$C^2$-smooth regular curve. If${\nabla }_{\dot{\gamma }}^L\dot{\gamma }$is a timelike vector, then

$$\begin{array}{cc}\hfill k_{\gamma }^L& =\left\{-\left\{-{\left[{\ddot{\gamma }}_1-L\dot{{\gamma }_2}\omega \left(\dot{\gamma }\left(t\right)\right)\right]}^2+{\left[{\ddot{\gamma }}_2-L\dot{{\gamma }_1}\omega \left(\dot{\gamma }\left(t\right)\right)\right]}^2+L{\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\right]}^2\right\}\right.\hfill \\ \hfill & ·{\left[-{\dot{\gamma }}_1^2+{\dot{\gamma }}_2^2+L{\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)}^2\right]}^{-2}+\left[-{\dot{\gamma }}_1({\ddot{\gamma }}_1-L\dot{{\gamma }_2}\omega \left(\dot{\gamma }\left(t\right)\right))\right.\hfill \\ \hfill & {\left.{\left.+{\dot{\gamma }}_2({\ddot{\gamma }}_2-L\dot{{\gamma }_1}\omega \left(\dot{\gamma }\left(t\right)\right))+L\omega \left(\dot{\gamma }\left(t\right)\right)\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\right]}^2·{\left[-{\dot{\gamma }}_1^2+{\dot{\gamma }}_2^2+L{\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)}^2\right]}^{-3}\right\}}^{\frac{1}{2}}.\hfill \end{array}$$

(13)

In particular, if$\gamma \left(t\right)$is a horizontal point of γ, then

$$\begin{array}{cc}\hfill k_{\gamma }^L& =\{-\left\{-{\left({\ddot{\gamma }}_1\right)}^2+{\left({\ddot{\gamma }}_2\right)}^2+L{\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\right]}^2\right\}·{\left(-{\dot{\gamma }}_1^2+{\dot{\gamma }}_2^2\right)}^{-2}.\hfill \\ \hfill & +{\left(-{\dot{\gamma }}_1{\ddot{\gamma }}_1+{\dot{\gamma }}_2{\ddot{\gamma }}_2\right)}^2·{\left(-{\dot{\gamma }}_1^2+{\dot{\gamma }}_2^2\right)}^{-3}{\}}^{\frac{1}{2}}.\hfill \end{array}$$

(14)

Proof. 

Similar to Lemma 2, combining (2) and Definition 2, (10) and (12), we obtain Lemma 3. □

Definition 3.

Let$\gamma :I\to {\mathbb{H}}_L^1$be a$C^2$-smooth regular curve. The intrinsic curvature$k_{\gamma }^{\infty }$of γ at$\gamma \left(t\right)$is define as

$$k_{\gamma }^{\infty }:={\lim }_{L\to +\infty }{k}_{\gamma }^L,$$

if the limit exists.

The following notation is introduced: for $f_1,f_2:(0,+\infty )\to \mathbb{R}$,

$$f_1\left(L\right)\sim f_2\left(L\right),asL\to +\infty \iff {\lim }_{L\to +\infty }\frac{f_1\left(L\right)}{f_2\left(L\right)}=1,$$

(15)

where $f_1,f_2$ are continuous function.

Lemma 4.

Let$\gamma :I\to {\mathbb{H}}_L^1$be a$C^2$-smooth regular curve. If${\nabla }_{\dot{\gamma }}^L\dot{\gamma }$is a spacelike vector and$-{\dot{\gamma }}_2^2+{\dot{\gamma }}_1^2>0$, then

$$k_{\gamma }^{\infty }=\frac{\sqrt{-{\dot{\gamma }}_2^2+{\dot{\gamma }}_1^2}}{\left|\omega \right(\dot{\gamma }\left(t\right)\left)\right|},if\omega \left(\dot{\gamma }\left(t\right)\right)\ne 0;$$

(16)

$$\begin{array}{cc}\hfill k_{\gamma }^{\infty }& ={\left\{\left\{-{\left[{\ddot{\gamma }}_1\right]}^2+{\left[{\ddot{\gamma }}_2\right]}^2\right\}·{\left[-{\dot{\gamma }}_1^2+{\dot{\gamma }}_2^2\right]}^{-2}-{\left[-{\dot{\gamma }}_1{\ddot{\gamma }}_1+{\dot{\gamma }}_2{\ddot{\gamma }}_2\right]}^2·{\left[-{\dot{\gamma }}_1^2+{\dot{\gamma }}_2^2\right]}^{-3}\right\}}^{\frac{1}{2}},\hfill \\ \hfill & if\omega \left(\dot{\gamma }\left(t\right)\right)=0and\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)=0;\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\lim }_{L\to +\infty }\frac{k_{\gamma }^L}{\sqrt{L}}=\frac{|\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)|}{|-{\dot{\gamma }}_1^2+{\dot{\gamma }}_2^2|},& if\omega \left(\dot{\gamma }\left(t\right)\right)=0and\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\ne 0.\hfill \end{array}$$

Proof. 

By (15), if $\omega \left(\dot{\gamma }\left(t\right)\right)\ne 0$, then we get the following conclusions

$${⟨{\nabla }_{\dot{\gamma }}^L\dot{\gamma },{\nabla }_{\dot{\gamma }}^L\dot{\gamma }⟩}_L\sim {\left[\omega \left(\dot{\gamma }\left(t\right)\right)\right]}^2(-{\dot{\gamma }}_2^2+{\dot{\gamma }}_1^2)L^2,asL\to +\infty ,$$

$${⟨\dot{\gamma },\dot{\gamma }⟩}_L\sim L\omega {\left(\dot{\gamma }\left(t\right)\right)}^2,asL\to +\infty ,$$

$${⟨{\nabla }_{\dot{\gamma }}^L\dot{\gamma },\dot{\gamma }⟩}_L^2\sim O\left(L^2\right)asL\to +\infty .$$

Thus,

$$\frac{{⟨{\nabla }_{\dot{\gamma }}^L\dot{\gamma },{\nabla }_{\dot{\gamma }}^L\dot{\gamma }⟩}_L}{\left|\right|\dot{\gamma }{\left|\right|}_L^4}\to \frac{-{\dot{\gamma }}_2^2+{\dot{\gamma }}_1^2}{{\left[\omega \left(\dot{\gamma }\left(t\right)\right)\right]}^2},asL\to +\infty ,$$

$$\frac{{⟨{\nabla }_{\dot{\gamma }}^L\dot{\gamma },\dot{\gamma }⟩}_L^2}{{⟨\dot{\gamma },\dot{\gamma }⟩}_L^3}\to 0,asL\to +\infty .$$

Using Definition 2, we have (16). By (9) and $\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)=0$, it further concludes that (17). If $\omega \left(\dot{\gamma }\left(t\right)\right)=0$ and $\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\ne 0$, then

$${⟨{\nabla }_{\dot{\gamma }}^L\dot{\gamma },{\nabla }_{\dot{\gamma }}^L\dot{\gamma }⟩}_L\sim L{\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\right]}^2,asL\to +\infty ,$$

$${⟨\dot{\gamma },\dot{\gamma }⟩}_L=-{\dot{\gamma }}_2^2+{\dot{\gamma }}_1^2,$$

$${⟨{\nabla }_{\dot{\gamma }}^L\dot{\gamma },\dot{\gamma }⟩}_L^2=O\left(1\right)asL\to +\infty .$$

By (6), we get (18). □

Lemma 5.

Let$\gamma :I\to {\mathbb{H}}_L^1$be a$C^2$-smooth regular curve. If${\nabla }_{\dot{\gamma }}^L\dot{\gamma }$is a timelike vector and${\dot{\gamma }}_2^2-{\dot{\gamma }}_1^2>0$, then

$$k_{\gamma }^{\infty }=\frac{\sqrt{{\dot{\gamma }}_2^2-{\dot{\gamma }}_1^2}}{\left|\omega \right(\dot{\gamma }\left(t\right)\left)\right|},if\omega \left(\dot{\gamma }\left(t\right)\right)\ne 0;$$

(18)

$$\begin{array}{cc}\hfill k_{\gamma }^{\infty }& ={\left\{-\left\{-{\left[{\ddot{\gamma }}_1\right]}^2+{\left[{\ddot{\gamma }}_2\right]}^2\right\}·{\left[-{\dot{\gamma }}_1^2+{\dot{\gamma }}_2^2\right]}^{-2}+{\left[-{\dot{\gamma }}_1{\ddot{\gamma }}_1+{\dot{\gamma }}_2{\ddot{\gamma }}_2\right]}^2·{\left[-{\dot{\gamma }}_1^2+{\dot{\gamma }}_2^2\right]}^{-3}\right\}}^{\frac{1}{2}},\hfill \\ \hfill & if\omega \left(\dot{\gamma }\left(t\right)\right)=0and\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)=0.\hfill \end{array}$$

(19)

Proof. 

By (15). Similar to Lemma 4. Using Definition 2, we have (19). Combining (14) and $\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)=0$, we have (20). □

Example 1.

Let${\mathbb{H}}_L^1$be Lorentzian Heisenberg group,$\gamma :I\to {\mathbb{R}}^3={\mathbb{H}}_L^1$be a$C^2$-smooth regular curve, where I is a open interval in$\mathbb{R}$and$\gamma \left(t\right)=\left({\gamma }_1\left(t\right),{\gamma }_2\left(t\right),{\gamma }_3\left(t\right)\right)$. Let${\nabla }^L$be the Levi–Civita connection on${\mathbb{H}}_L^1$with respect to$g_L$and

$$\omega \left(\dot{\gamma }\left(t\right)\right)=\frac{{\gamma }_2\left(t\right)}{2}{\dot{\gamma }}_1\left(t\right)-\frac{{\gamma }_1\left(t\right)}{2}{\dot{\gamma }}_2\left(t\right)+{\dot{\gamma }}_3\left(t\right).$$

By (12), we have

$$\begin{array}{cc}\hfill {\nabla }_{\dot{\gamma }}^L\dot{\gamma }& =\left[\ddot{{\gamma }_1}-L{\dot{\gamma }}_2\omega \left(\dot{\gamma }\left(t\right)\right)\right]X_1+\left[\ddot{{\gamma }_2}-L{\dot{\gamma }}_1\omega \left(\dot{\gamma }\left(t\right)\right)\right]X_2+\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\right]X_3\hfill \end{array}$$

and

$$g^L(X_1,X_1)=-1,g^L(X_2,X_2)=1,g^L(X_3,X_3)=L.$$

Now let us assume that$\gamma \left(t\right)=(t,1,0)$be a$C^2$-smooth regular curve and${\gamma }_1\left(t\right)=t,{\gamma }_2\left(t\right)=1,$${\gamma }_3\left(t\right)=0$. Then we have

$$\omega \left(\dot{\gamma }\left(t\right)\right)=\frac{1}{2}\times 1-\frac{t}{2}\times 0+0=\frac{1}{2}\ne 0,$$

and

$$\begin{array}{cc}\hfill {\nabla }_{\dot{\gamma }}^L\dot{\gamma }& =\left[0-L\times 0\times \frac{1}{2}\right]X_1+\left[0-L\times 1\times \frac{1}{2}\right]X_2+\left[\frac{d}{dt}\left(\frac{1}{2}\right)\right]X_3=-\frac{L}{2}{X}_2.\hfill \end{array}$$

Therefore, we have

$$|{\nabla }_{\dot{\gamma }}^L\dot{\gamma }|=\frac{L^2}{4}>0.$$

It implies that${\nabla }_{\dot{\gamma }}^L\dot{\gamma }$is a spacelike vector and$-{\dot{\gamma }}_2^2+{\dot{\gamma }}_1^2=1>0$. By Lemma 2.7 (2.16), we have

$$k_{\gamma }^{\infty }=\frac{\sqrt{1}}{\frac{1}{2}}=2.$$

Definition 3 could use the example above to illustrate what it means, in the case where the limit exists.

3. Lorentzian Surfaces and a Gauss–Bonnet Theorem in the Lorentzian Heisenberg Group

A surface $\Sigma \subset {\mathbb{H}}_L^1$ is called regular if $\Sigma$ is a $C^2$-smooth compact and oriented surface. In particular, we consider that there is a $C^2$-smooth function $u:{\mathbb{H}}_L^1\to \mathbb{R}$ such that

$$\Sigma =\{(x_1,x_2,x_3)\in {\mathbb{H}}_L^1:u(x_1,x_2,x_3)=0\}$$

and ${\nabla }_{{\mathbb{H}}_L^1}u=u_{x_1}{\partial }_{x_1}+u_{x_2}{\partial }_{x_2}+u_{x_3}{\partial }_{x_3}\ne 0.$ We will say that a point $x\in \Sigma$ is characteristic if ${\nabla }_{H}u\left(x\right)=(0,0)$. At local and away from characteristic points of $\Sigma$, we give our all computations.

Firstly, let

$$p:=X_{1}u,q:=X_{2}u,\mathrm{and}r:={\tilde{X}}_{3}u.$$

If $-p^2+q^2>0$, we say that $\Sigma \subset {\mathbb{H}}_L^1$ is a Horizontal spacelike surface. Let $-p^2+q^2>0$. When $L\to +\infty$, then $-p^2+q^2+r^2>0$. Then, we define

$$\begin{array}{cc}\hfill & l:=\sqrt{-p^2+q^2},l_L:=\sqrt{-p^2+q^2+r^2},\overline{p}:=\frac{p}{l},\hfill \\ \hfill & \overline{q}:=\frac{q}{l},\overline{p_L}:=\frac{p}{l_L},\overline{q_L}:=\frac{q}{l_L},\overline{r_L}:=\frac{r}{l_L}.\hfill \end{array}$$

(20)

In particular, $-{\overline{p}}^2+{\overline{q}}^2=1$. At every non-characteristic point, the functions above are well defined. Secondly, let

$$\begin{array}{c}\hfill v_L=-\overline{p_L}{X}_1+\overline{q_L}{X}_2+\overline{r_L}\widetilde{X_3},e_1=\overline{q}{X}_1-\overline{p}{X}_2,e_2=\overline{r_L}\overline{p}{X}_1-\overline{r_L}\overline{q}{X}_2+\frac{l}{l_L}\widetilde{X_3},\end{array}$$

(21)

then $v_L$ is the unit spacelike normal vector to $\Sigma$ and $e_1$ is the unit timelike vector, $e_2$ is the unit spacelike vector. $\{e_1,e_2\}$ are the orthonormal basis of $\Sigma$. $\Sigma$ is called to be a Lorentzian surface in Lorentzian Hersenberg group.

Let $\dot{\gamma }=a{e}_1+b{e}_2$. If $\gamma :I\to {\mathbb{H}}_L^1$ be a $C^2$-smooth timelike curve, then we define $J_L\left(\dot{\gamma }\right):=a{e}_2+b{e}_1$. If $\gamma :I\to {\mathbb{H}}_L^1$ be a $C^2$-smooth spacelike curve, then $J_L\left(\dot{\gamma }\right):=-a{e}_2-b{e}_1$. Then $⟨\dot{\gamma },J_L\left(\dot{\gamma }\right)⟩=0$ and $(\dot{\gamma },J_L\left(\dot{\gamma }\right))$ has the same orientation with $\{e_1,e_2\}$.

We then define the projection by $\pi :T{\mathbb{H}}_L^1\to T\Sigma$. For every $U,V\in T\Sigma$, let ${\nabla }_U^{\Sigma ,L}V=\pi {\nabla }_U^{L}V$. Therefore, the Levi–Civita connection with respect to the metric $g_L$ on $\Sigma$ is define as ${\nabla }^{\Sigma ,L}$. Combining (12), (21) and

$${\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }=-{⟨{\nabla }_{\dot{\gamma }}^L\dot{\gamma },e_1⟩}_L{e}_1+{⟨{\nabla }_{\dot{\gamma }}^L\dot{\gamma },e_2⟩}_L{e}_2,$$

(22)

we have

$$\begin{array}{cc}\hfill {\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }& =-\left\{-\overline{q}\left[{\ddot{\gamma }}_1-L{\gamma }_2\omega \left(\dot{\gamma }\left(t\right)\right)\right]-\overline{p}\left[{\ddot{\gamma }}_2-L{\gamma }_1\omega \left(\dot{\gamma }\left(t\right)\right)\right]\right\}{e}_1\hfill \\ \hfill & +\{-\overline{r_L}\overline{p}\left[{\ddot{\gamma }}_1-L{\gamma }_2\omega \left(\dot{\gamma }\left(t\right)\right)\right]-\overline{r_L}\overline{q}\left[{\ddot{\gamma }}_2-L{\gamma }_1\omega \left(\dot{\gamma }\left(t\right)\right)\right].\hfill \\ \hfill & +\frac{l}{l_L}{L}^{\frac{1}{2}}\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\right]\}{e}_2.\hfill \end{array}$$

(23)

Therefore, when $\omega \left(\dot{\gamma }\left(t\right)\right)=0$, we have

$${\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }=[\overline{q}{\ddot{\gamma }}_1+\overline{p}{\ddot{\gamma }}_2]e_1+\left\{-\overline{r_L}\overline{p}{\ddot{\gamma }}_1-\overline{r_L}\overline{q}{\ddot{\gamma }}_2+\frac{l}{l_L}{L}^{\frac{1}{2}}\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\right]\right\}{e}_2$$

(24)

Definition 4.

Let$\Sigma \subset {\mathbb{H}}_L^1$be a regular Lorentzian surface. Let$\gamma :I\to \Sigma$be a$C^2$-smooth regular curve.

(1) If${\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }$is spacelike vectors, we define the geodesic curvature$k_{\gamma ,\Sigma }^L$of γ at$\gamma \left(t\right)$by

$$k_{\gamma ,\Sigma }^L:=\sqrt{\frac{\left|\right|{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }{\left|\right|}_{\Sigma ,L}^2}{\left|\right|\dot{\gamma }{\left|\right|}_{\Sigma ,L}^4}-\frac{{⟨{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma },\dot{\gamma }⟩}_{\Sigma ,L}^2}{{⟨\dot{\gamma },\dot{\gamma }⟩}_{\Sigma ,L}^3}}.$$

(25)

(2) If${\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }$is timelike vectors, we define the curvature$k_{\gamma }^L$of γ at$\gamma \left(t\right)$by

$$k_{\gamma ,\Sigma }^L:=\sqrt{\frac{\left|\right|{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }{\left|\right|}_{\Sigma ,L}^2}{\left|\right|\dot{\gamma }{\left|\right|}_{\Sigma ,L}^4}+\frac{{⟨{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma },\dot{\gamma }⟩}_{\Sigma ,L}^2}{{⟨\dot{\gamma },\dot{\gamma }⟩}_{\Sigma ,L}^3}}.$$

(26)

Definition 5.

Let$\Sigma \subset {\mathbb{H}}_L^1$be a regular Lorentzian surface. We assume that$\gamma :I\to \Sigma$is a$C^2$-smooth regular curve. The intrinsic geodesic curvature$k_{\gamma ,\Sigma }^{\infty }$of γ at$\gamma \left(t\right)$is defined as

$$k_{\gamma ,\Sigma }^{\infty }:={\lim }_{L\to +\infty }{k}_{\gamma ,\Sigma }^L,$$

if the limit exists.

Lemma 6.

Let$\Sigma \subset {\mathbb{H}}_L^1$be a regular Lorentzian surface. Let$\gamma :I\to \Sigma$be a$C^2$-smooth regular curve.

(1) If${\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }$is a timelike vector, we have

$$k_{\gamma ,\Sigma }^{\infty }=\frac{|\overline{p}{\dot{\gamma }}_1+\overline{q}{\dot{\gamma }}_2|}{\left|\omega \right(\dot{\gamma }\left(t\right)\left)\right|},if\omega \left(\dot{\gamma }\left(t\right)\right)\ne 0;$$

(27)

$$k_{\gamma ,\Sigma }^{\infty }=0,if\omega \left(\dot{\gamma }\left(t\right)\right)=0and\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)=0;$$

(28)

(2) If${\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }$is a spacelike vector, then

$${\lim }_{L\to +\infty }\frac{k_{\gamma ,\Sigma }^L}{\sqrt{L}}=\frac{|\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)|}{{\left(\overline{q}{\dot{\gamma }}_1+\overline{p}{\dot{\gamma }}_2\right)}^2},if\omega \left(\dot{\gamma }\left(t\right)\right)=0and\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\ne 0.$$

(29)

Proof. 

Combining (10) and $\dot{\gamma }\in T\Sigma$, then

$$\dot{\gamma }=(\overline{q}{\dot{\gamma }}_1+\overline{p}{\dot{\gamma }}_2)e_1+\frac{l_L}{l}{L}^{\frac{1}{2}}\omega \left(\dot{\gamma }\left(t\right)\right)e_2.$$

(30)

By (23), we have

$$\begin{array}{cc}\hfill {⟨{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma },{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }⟩}_{L,\Sigma }& =-{\left\{-\overline{q}\left[{\ddot{\gamma }}_1-L{\gamma }_2\omega \left(\dot{\gamma }\left(t\right)\right)\right]-\overline{p}\left[{\ddot{\gamma }}_2-L{\gamma }_1\omega \left(\dot{\gamma }\left(t\right)\right)\right]\right\}}^2\hfill \\ \hfill & +\left\{-\overline{r_L}\overline{p}\left[{\ddot{\gamma }}_1-L{\gamma }_2\omega \left(\dot{\gamma }\left(t\right)\right)\right]-\overline{r_L}\overline{q}\left[{\ddot{\gamma }}_2-L{\gamma }_1\omega \left(\dot{\gamma }\left(t\right)\right)\right]\right.\hfill \\ \hfill & {\left.+\frac{l}{l_L}{L}^{\frac{1}{2}}\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\right]\right\}}^2.\hfill \end{array}$$

(31)

Similarly, if $\omega \left(\dot{\gamma }\left(t\right)\right)\ne 0$, then we have

$${⟨\dot{\gamma },\dot{\gamma }⟩}_{\Sigma ,L}=-{(\overline{q}{\dot{\gamma }}_1+\overline{p}{\dot{\gamma }}_2)}^2+{\left[\frac{l_L}{l}{L}^{\frac{1}{2}}\omega \left(\dot{\gamma }\left(t\right)\right)\right]}^2\sim L{\left[\omega \left(\dot{\gamma }\left(t\right)\right)\right]}^2,\mathrm{as}L\to +\infty .$$

(32)

By (23) and (30), we have

$$\begin{array}{cc}\hfill {⟨{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma },\dot{\gamma }⟩}_{\Sigma ,L}& =(\overline{q}{\dot{\gamma }}_1+\overline{p}{\dot{\gamma }}_2)·\left\{-\overline{q}\left[{\ddot{\gamma }}_1-L{\gamma }_2\omega \left(\dot{\gamma }\left(t\right)\right)\right]\right.\hfill \\ \hfill & \left.-\overline{p}\left[{\ddot{\gamma }}_2-L{\gamma }_1\omega \left(\dot{\gamma }\left(t\right)\right)\right]\right\}+\left[\frac{l_L}{l}{L}^{\frac{1}{2}}\omega \left(\dot{\gamma }\left(t\right)\right)\right]·\left\{-\overline{r_L}\overline{p}\left[{\ddot{\gamma }}_1-L{\gamma }_2\omega \left(\dot{\gamma }\left(t\right)\right)\right]\right.\hfill \\ \hfill & \left.-\overline{r_L}\overline{q}\left[{\ddot{\gamma }}_2-L{\gamma }_1\omega \left(\dot{\gamma }\left(t\right)\right)\right]+\frac{l}{l_L}{L}^{\frac{1}{2}}\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\right]\right\}\sim M_{0}L,\hfill \end{array}$$

(33)

where $M_0$ does not depend on L. By Definition 4, (31)–(33), (27) holds.

If $\omega \left(\dot{\gamma }\left(t\right)\right)=0$ and $\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)=0$, then

$$\begin{array}{cc}\hfill {⟨{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma },{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }⟩}_{L,\Sigma }& =-{\left\{-\overline{q}{\ddot{\gamma }}_1-\overline{p}{\ddot{\gamma }}_2\right\}}^2+{\left\{-\overline{r_L}\overline{p}{\ddot{\gamma }}_1-\overline{r_L}\overline{q}{\ddot{\gamma }}_2\right\}}^2\hfill \\ \hfill & \sim -{\left\{-\overline{q}{\ddot{\gamma }}_1-\overline{p}{\ddot{\gamma }}_2\right\}}^2,\mathrm{as}L\to +\infty ,\hfill \end{array}$$

(34)

$${⟨\dot{\gamma },\dot{\gamma }⟩}_{\Sigma ,L}=-{(\overline{q}{\dot{\gamma }}_1+\overline{p}{\dot{\gamma }}_2)}^2,$$

(35)

$$\begin{array}{cc}\hfill {⟨{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma },\dot{\gamma }⟩}_{\Sigma ,L}& =(\overline{q}{\dot{\gamma }}_1+\overline{p}{\dot{\gamma }}_2)·(-\overline{q}{\ddot{\gamma }}_1-\overline{p}{\ddot{\gamma }}_2)\hfill \end{array}$$

(36)

By (34)–(36) and Definition 4, we get $k_{\gamma ,\Sigma }^{\infty }=0$.

If $\omega \left(\dot{\gamma }\left(t\right)\right)=0$ and $\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\ne 0$, then

$${⟨{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma },{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }⟩}_{L,\Sigma }\sim L{\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\right]}^2,$$

$${⟨{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma },\dot{\gamma }⟩}_{\Sigma ,L}=O\left(1\right),$$

so (26) holds. □

Example 2.

We assume that there exists a$C^2$-smooth function$u=x_2:{\mathbb{H}}_L^1\to \mathbb{R}$such that

$$\Sigma =\{(x_1,x_2,x_3)\in {\mathbb{H}}_L^1:x_2=0\}.$$

Then${\nabla }_{{\mathbb{H}}_L^1}u=u_{x_1}{\partial }_{x_1}+u_{x_2}{\partial }_{x_2}+u_{x_3}{\partial }_{x_3}={\partial }_{x_2}\ne 0.$Let

$$X_1={\partial }_{x_1}-\frac{1}{2}{x}_2{\partial }_{x_3},X_2={\partial }_{x_2}+\frac{1}{2}{x}_1{\partial }_{x_3},X_3=[X_1,X_2]={\partial }_{x_3}.$$

So we have

$$p:=X_{1}u=({\partial }_{x_1}-\frac{1}{2}{x}_2{\partial }_{x_3})\left(x_2\right)=0,q:=X_{2}u=({\partial }_{x_2}+\frac{1}{2}{x}_1{\partial }_{x_3})\left(x_2\right)=1,$$

$$r:={\tilde{X}}_{3}u=\left(L^{-\frac{1}{2}}{\partial }_{x_3}\right)\left(x_2\right)=0.$$

Therefore,$-p^2+q^2=1>0$, so$\Sigma \subset {\mathbb{H}}_L^1$is a Horizontal spacelike surface. By (20), we have

$$\begin{array}{cc}\hfill & l:=\sqrt{-p^2+q^2}=1,l_L:=\sqrt{-p^2+q^2+r^2}=1,\overline{p}:=\frac{p}{l}=0,\hfill \\ \hfill & \overline{q}:=\frac{q}{l}=1,\overline{p_L}:=\frac{p}{l_L}=\frac{0}{1}=0,\overline{q_L}:=\frac{q}{l_L}=1,\overline{r_L}:=\frac{r}{l_L}=0.\hfill \end{array}$$

By (21), we have

$$\begin{array}{c}\hfill v_L=X_2,e_1=\overline{q}{X}_1-\overline{p}{X}_2=X_1,e_2=\overline{r_L}\overline{p}{X}_1-\overline{r_L}\overline{q}{X}_2+\frac{l}{l_L}\widetilde{X_3}=\widetilde{X_3},\end{array}$$

Then,$\{e_1,e_2\}=\{X_1,\widetilde{X_3}\}$, $T\Sigma =span\{{\partial }_{x_1},{\partial }_{x_3}\}$. Thus, it is concluded that Σ is a Lorentzian surface in Lorentzian Hersenberg group.

Let

$$\gamma :[0,2\pi ]\to \Sigma ;\theta \to (cos\theta ,0,sin\theta )$$

be the circle centered at the origin on$x_2=0$. By

$$\omega \left(\dot{\gamma }\left(\theta \right)\right)=\frac{{\gamma }_2\left(\theta \right)}{2}{\dot{\gamma }}_1\left(\theta \right)-\frac{{\gamma }_1\left(\theta \right)}{2}{\dot{\gamma }}_2\left(\theta \right)+{\dot{\gamma }}_3\left(\theta \right)$$

and${\gamma }_1\left(\theta \right)=cos\theta ,{\gamma }_2\left(\theta \right)=0,{\gamma }_3\left(\theta \right)=sin\theta$, we have

$$\omega \left(\dot{\gamma }\left(\theta \right)\right)={\dot{\gamma }}_3\left(\theta \right)=cos\theta .$$

By (23), we have

$${\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }=-cos\theta e_1-L^{\frac{1}{2}}sin\theta e_2.$$

Then

$$|{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }{|}^2={|-cos\theta e_1-L^{\frac{1}{2}}sin\theta e_2|}^2=-co{s}^2\theta +Lsi{n}^2\theta .$$

If$sin\theta =0$, then$cos\theta =\pm 1$. In this case,$\omega \left(\dot{\gamma }\left(t\right)\right)\ne 0$. Then we have${\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }$is a timelike vector. By Lemma 6 (27), we have at the point θ which satisfies$sin\theta =0$

$$k_{\gamma ,\Sigma }^{\infty }=\frac{|\overline{p}{\dot{\gamma }}_1+\overline{q}{\dot{\gamma }}_2|}{\left|\omega \right(\dot{\gamma }\left(t\right)\left)\right|}=\frac{0}{\left|cos\theta \right|}=0.$$

We know that we do not use$k_{\gamma ,\Sigma }^{\infty }$in the Gauss–Bonnet theorem.

Definition 6.

Let$\Sigma \subset {\mathbb{H}}_L^1$be a regular surface. We assume that$\gamma :I\to \Sigma$is a$C^2$-smooth regular curve. We define the signed geodesic curvature$k_{\gamma ,\Sigma }^{L,s}$of γ at$\gamma \left(t\right)$by

$$k_{\gamma ,\Sigma }^{L,s}:=\frac{{⟨{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma },J_L\left(\dot{\gamma }\right)⟩}_{\Sigma ,L}}{\left|\right|\dot{\gamma }{\left|\right|}_{\Sigma ,L}^3}.$$

(37)

Definition 7.

Let$\Sigma \subset {\mathbb{H}}_L^1$be a regular surface. Let$\gamma :[a,b]\to \Sigma$be a$C^2$-smooth regular curve. We define the intrinsic geodesic curvature$k_{\gamma ,\Sigma }^{\infty }$of γ at the non-characteristic point$\gamma \left(t\right)$to be

$$k_{\gamma ,\Sigma }^{\infty ,s}:={\lim }_{L\to +\infty }{k}_{\gamma ,\Sigma }^{L,s},$$

if the limit exists.

Lemma 7.

Let$\Sigma \subset {\mathbb{H}}_L^1$be a regular Lorentzian surface.

(1) If$\gamma :I\to \Sigma$be a timelike$C^2$-smooth regular curve, then$\omega \left(\dot{\gamma }\left(t\right)\right)=0$and

$$k_{\gamma ,\Sigma }^{\infty }=0,if\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)=0;$$

(38)

$${\lim }_{L\to +\infty }\frac{k_{\gamma ,\Sigma }^{L,s}}{\sqrt{L}}=\frac{|\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)|}{{\left(\overline{q}{\dot{\gamma }}_1+\overline{p}{\dot{\gamma }}_2\right)}^2},if\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\ne 0.$$

(39)

(2) Let$\gamma :I\to \Sigma$be a spacelike$C^2$-smooth regular curve, then$\omega \left(\dot{\gamma }\left(t\right)\right)\ne 0$and

$$k_{\gamma ,\Sigma }^{\infty ,s}=-\frac{\overline{p}{\dot{\gamma }}_1+\overline{q}{\dot{\gamma }}_2}{\left|\omega \right(\dot{\gamma }\left(t\right)\left)\right|}.$$

(40)

Proof. 

For (1), by (30), we have

$$J_L\left(\dot{\gamma }\right)=\frac{l_L}{l}{L}^{\frac{1}{2}}\omega \left(\dot{\gamma }\left(t\right)\right)e_1+(\overline{p}{\dot{\gamma }}_2+\overline{q}{\dot{\gamma }}_1)e_2.$$

(41)

By (23) and (41), we have

$$\begin{array}{cc}\hfill {⟨{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma },J_L\left(\dot{\gamma }\right)⟩}_{L,\Sigma }& =-\frac{l_L}{l}{L}^{\frac{1}{2}}\omega \left(\dot{\gamma }\left(t\right)\right)\left\{\overline{q}\left[{\ddot{\gamma }}_1-L{\gamma }_2\omega \left(\dot{\gamma }\left(t\right)\right)\right]+\overline{p}\left[{\ddot{\gamma }}_2-L{\gamma }_1\omega \left(\dot{\gamma }\left(t\right)\right)\right]\right\}\hfill \\ \hfill & +(\overline{p}{\dot{\gamma }}_2+\overline{q}{\dot{\gamma }}_1)·\left\{-\overline{r_L}\overline{p}\left[{\ddot{\gamma }}_1-L{\gamma }_2\omega \left(\dot{\gamma }\left(t\right)\right)\right]-\overline{r_L}\overline{q}\left[{\ddot{\gamma }}_2-L{\gamma }_1\omega \left(\dot{\gamma }\left(t\right)\right)\right]\right.\hfill \\ \hfill & \left.+\frac{l}{l_L}{L}^{\frac{1}{2}}\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\right]\right\}\sim L^{\frac{3}{2}}\omega {\left(\dot{\gamma }\left(t\right)\right)}^2(\overline{p}{\dot{\gamma }}_1+\overline{q}{\dot{\gamma }}_2),\mathrm{as}L\to +\infty .\hfill \end{array}$$

(42)

If $\omega \left(\dot{\gamma }\left(t\right)\right)=0$ and $\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)=0,$ then

$$\begin{array}{cc}\hfill {⟨{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma },J_L\left(\dot{\gamma }\right)⟩}_{L,\Sigma }=& (\overline{p}{\dot{\gamma }}_2+\overline{q}{\dot{\gamma }}_1)·\left\{-\overline{r_L}\overline{p}{\ddot{\gamma }}_1-\overline{r_L}\overline{q}{\ddot{\gamma }}_2\right]\sim O\left(L^{-\frac{1}{2}}\right)\mathrm{as}L\to +\infty .\hfill \end{array}$$

(43)

So $k_{\gamma ,\Sigma }^{\infty ,s}=0.$ If $\omega \left(\dot{\gamma }\left(t\right)\right)=0$ and $\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\ne 0$, then

$$\begin{array}{c}\hfill {⟨{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma },J_L\left(\dot{\gamma }\right)⟩}_{L,\Sigma }\sim L^{\frac{1}{2}}(\overline{p}{\dot{\gamma }}_2+\overline{q}{\dot{\gamma }}_1)\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\mathrm{as}L\to +\infty .\end{array}$$

(44)

So (39) holds. Similarly, we have (2). □

Example 3.

We take$u=x_2:{\mathbb{H}}_L^1\to \mathbb{R}$and$\gamma \left(\theta \right)=(cos\theta ,0,sin\theta )$as in the Example 2. Then

$$\omega \left(\dot{\gamma }\left(\theta \right)\right)={\dot{\gamma }}_3\left(\theta \right)=cos\theta .$$

By (30), we have

$$\dot{\gamma }={\dot{\gamma }}_1{e}_1+L^{\frac{1}{2}}\omega \left(\dot{\gamma }\left(t\right)\right)e_2=-sin\theta e_1+L^{\frac{1}{2}}cos\theta e_2.$$

Then

$$|\dot{\gamma }{|}^2={|-sin\theta e_1+L^{\frac{1}{2}}cos\theta e_2|}^2=-si{n}^2\theta +Lco{s}^2\theta .$$

So we have when$cos\theta \ne 0$, then$|\dot{\gamma }{|}^2>0$for the large L and we have$\dot{\gamma }$is a spacelike vector. If$cos\theta \ne 0$, by Lemma 7 (2), we have

$$k_{\gamma ,\Sigma }^{\infty }=-\frac{\overline{p}{\dot{\gamma }}_1+\overline{q}{\dot{\gamma }}_2}{\left|\omega \right(\dot{\gamma }\left(t\right)\left)\right|}=-\frac{{\dot{\gamma }}_2}{\left|cos\theta \right|}=0.$$

In the Lorentzian Heisenberg space, we then investigate the sub-Lorentzian limit of the Riemannian Gaussian curvature of surfaces. The second fundamental form $I{I}^L$ of the embedding of $\Sigma$ into ${\mathbb{H}}_L^1$ is defined by

$$I{I}^L=\left(\begin{array}{cc}⟨{\nabla }_{e_1}^L{v}_L,e_1){⟩}_L,& ⟨{\nabla }_{e_1}^L{v}_L,e_2){⟩}_L\\ ⟨{\nabla }_{e_2}^L{v}_L,e_1){⟩}_L,& ⟨{\nabla }_{e_2}^L{v}_L,e_2){⟩}_L\end{array}\right).$$

(45)

Similarly to Theorem 4.3 in [6], we have

Theorem 1.

For the embedding of Σ into ${\mathbb{H}}_L^1$, the second fundamental form $I{I}^L$ of Σ is given by

$$I{I}^L=\left(\begin{array}{cc}\frac{l}{l_L}[X_1\left(\overline{p}\right)-X_2\left(\overline{q}\right)],& \frac{l_L}{l}{⟨e_1,{\nabla }_H\left(\overline{r_L}\right)⟩}_L+\frac{\sqrt{L}}{2}\\ \frac{l_L}{l}{⟨e_1,{\nabla }_H\left(\overline{r_L}\right)⟩}_L+\frac{\sqrt{L}}{2},& \frac{l^2}{l_L^2}{⟨e_2,{\nabla }_H\left(\frac{r}{l}\right)⟩}_L+\widetilde{X_3}\left(\overline{r_L}\right)\end{array}\right).$$

(46)

We define the mean curvature ${\mathcal{H}}_L$ of $\Sigma$ by

$${\mathcal{H}}_L:=\mathrm{tr}\left(I{I}^L\right).$$

The curvature of a connection ∇ is defined as

$$R(A,B)C={\nabla }_A{\nabla }_B-{\nabla }_B{\nabla }_A-{\nabla }_{[A,B]}.$$

(47)

Let

$${\mathcal{K}}^{\Sigma ,L}(e_1,e_2)=-{⟨R^{\Sigma ,L}(e_1,e_2)e_1,e_2⟩}_{\Sigma ,L},{\mathcal{K}}^L(e_1,e_2)=-{⟨R^L(e_1,e_2)e_1,e_2⟩}_L.$$

(48)

Taking advantage of the Gauss equation, we obtain

$${\mathcal{K}}^{\Sigma ,L}(e_1,e_2)={\mathcal{K}}^L(e_1,e_2)+\det \left(I{I}^L\right).$$

(49)

Proposition 1.

The horizontal mean curvature${\mathcal{H}}_{\infty }$of$\Sigma \subset \mathbb{H}$away from characteristic points is the following form:

$${\mathcal{H}}_{\infty }={\lim }_{L\to +\infty }{\mathcal{H}}_L=X_1\left(\overline{p}\right)-X_2\left(\overline{q}\right).$$

(50)

Proof. 

By

$$\frac{l^2}{l_L^2}{⟨e_2,{\nabla }_H\left(\frac{r}{l}\right)⟩}_L=\frac{\overline{p}r}{l}{X}_1\left(\overline{r_L}\right)+\frac{\overline{q}r}{l}{X}_2\left(\overline{r_L}\right)=O\left(L^{-1}\right)$$

$$\frac{l}{l_L}[X_1\left(\overline{p}\right)-X_2\left(\overline{q}\right)]\to X_1\left(\overline{p}\right)-X_2\left(\overline{q}\right),\widetilde{X_3}\left(\overline{r_L}\right)\to 0,\overline{p_L}\to \overline{p},$$

we get (50). □

By Lemma 1 and (48), then we get the following lemma.

Lemma 8.

Let${\mathbb{H}}_L^1$be the Lorentzian Heisenberg space, then

$$\begin{array}{cc}\hfill & R^L(X_1,X_2)X_1=\frac{3}{4}L{X}_2,R^L(X_1,X_2)X_2=\frac{3}{4}L{X}_1,R^L(X_1,X_2)X_3=0,\hfill \\ \hfill & R^L(X_1,X_3)X_1=-\frac{1}{4}L{X}_3,R^L(X_1,X_3)X_2=0,R^L(X_1,X_3)X_3=-\frac{L^2}{4}{X}_1,\hfill \\ \hfill & R^L(X_2,X_3)X_1=0,R^L(X_2,X_3)X_2=\frac{L}{4}{X}_3,R^L(X_2,X_3)X_3=-\frac{L^2}{4}{X}_2.\hfill \end{array}$$

(51)

Proposition 2.

Away from characteristic points, we have the following assertion

$${\mathcal{K}}^{\Sigma ,L}(e_1,e_2)\to A+O\left(\frac{1}{L}\right),\mathrm{as}L\to +\infty ,$$

(52)

where

$$A:=-⟨e_1,{\nabla }_H\left(\frac{X_{3}u}{|{\nabla }_{H}u|}\right)⟩-\frac{{\left(X_{3}u\right)}^2}{l^2}.$$

(53)

Proof. 

By (21), we have

$$\begin{array}{cc}\hfill & {⟨R^L(e_1,e_2)e_1,e_2⟩}_L\hfill \\ \hfill & ={\overline{r_L}}^2{⟨R^L(X_1,X_2)X_1,X_2⟩}_L-2\frac{l}{l_L}\overline{q}{L}^{-\frac{1}{2}}\overline{r_L}{⟨R^L(X_1,X_2)X_1,X_3⟩}_L\hfill \\ \hfill & +2\frac{l}{l_L}\overline{p}{L}^{-\frac{1}{2}}\overline{r_L}{⟨R^L(X_1,X_2)X_2,X_3⟩}_L+{\left(\frac{l}{l_L}\overline{q}\right)}^2{L}^{-1}{⟨R^L(X_1,X_3)X_1,X_3⟩}_L\hfill \\ \hfill & -2{\left(\frac{l}{l_L}\right)}^2\overline{p}\overline{q}{L}^{-1}{⟨R^L(X_1,X_3)X_2,X_3⟩}_L+{\left(\overline{p}\frac{l}{l_L}\right)}^2{L}^{-1}{⟨R^L(X_2,X_3)X_2,X_3⟩}_L.\hfill \\ \hfill \end{array}$$

(54)

By Lemma 8, we have

$$\begin{array}{c}\hfill {\mathcal{K}}^L(e_1,e_2)=\frac{L}{4}{\left(\frac{l}{l_L}\right)}^2-\frac{3}{4}L{\overline{r_L}}^2.\end{array}$$

(55)

By (46) and

$${\nabla }_H\left(\overline{r_L}\right)=L^{-\frac{1}{2}}{\nabla }_H\left(\frac{X_{3}u}{|{\nabla }_{H}u|}\right)+O\left(L^{-1}\right)\mathrm{as}L\to +\infty$$

we get

$$\det \left(I{I}^L\right)=-\frac{L}{4}-⟨e_1,{\nabla }_H\left(\frac{X_{3}u}{|{\nabla }_{H}u|}\right)⟩+O\left(L^{-1}\right)\mathrm{as}L\to +\infty .$$

(56)

By (48), (55), (56) we get (52). □

For the case of a spacelike curve $\gamma :I\to {\mathbb{H}}_L^1$, the Riemannian length measure is defined as $d{s}_L=\left|\right|\dot{\gamma }{\left|\right|}_{L}dt.$

Lemma 9.

Let$\gamma :I\to {\mathbb{H}}_L^1$be a$C^2$-smooth spacelike curve. Let

$$ds:=|\omega \left(\dot{\gamma }\left(t\right)\right)|dt,d\overline{s}:=\frac{1}{2}\frac{1}{\left|\omega \right(\dot{\gamma }\left(t\right)\left)\right|}\left(-{\dot{\gamma }}_1^2+{\dot{\gamma }}_2^2\right)dt.$$

(57)

Then,

$${\lim }_{L\to +\infty }\frac{1}{\sqrt{L}}{\int }_{\gamma }d{s}_L={\int }_a^{b}ds.$$

(58)

If$\omega \left(\dot{\gamma }\left(t\right)\right)\ne 0$, then

$$\frac{1}{\sqrt{L}}d{s}_L=ds+d\overline{s}{L}^{-1}+O\left(L^{-2}\right)\mathrm{as}L\to +\infty .$$

(59)

If$\omega \left(\dot{\gamma }\left(t\right)\right)=0$, then

$$\frac{1}{\sqrt{L}}d{s}_L=\frac{1}{\sqrt{L}}\sqrt{-{\dot{\gamma }}_1^2+{\dot{\gamma }}_2^2}dt.$$

(60)

Proof. 

Since $\left|\right|\dot{\gamma }\left(t\right){\left|\right|}_L=\sqrt{-{\dot{\gamma }}_1^2+{\dot{\gamma }}_2^2+L\omega {\left(\dot{\gamma }\left(t\right)\right)}^2}$, similar to the proof of Lemma 6.1 in [2], (58) holds. If $\omega \left(\dot{\gamma }\left(t\right)\right)\ne 0$, then

$$\frac{1}{\sqrt{L}}d{s}_L=\sqrt{L^{-1}\left(-{\dot{\gamma }}_1^2+{\dot{\gamma }}_2^2\right)+\omega {\left(\dot{\gamma }\left(t\right)\right)}^2}dt.$$

By the Taylor expansion, (59) holds. Combining the definition of $d{s}_L$ and $\omega \left(\dot{\gamma }\left(t\right)\right)=0$, (60) holds. □

Proposition 3.

Let$\Sigma \subset {\mathbb{H}}_L^1$be a regular Lorentzian$C^2$-smooth surface. Let$d{\sigma }_{\Sigma ,L}$denote the surface measure on Σ with respect to the Lorentzian metric $g_L$. Let

$$d{\sigma }_{\Sigma }:=-(\overline{p}{\omega }_2-\overline{q}{\omega }_1)\wedge \omega ,d\overline{{\sigma }_{\Sigma }}:=-\frac{X_{3}u}{l}{\omega }_1\wedge {\omega }_2+\frac{{\left(X_{3}u\right)}^2}{2l^2}(\overline{p}{\omega }_2-\overline{q}{\omega }_1)\wedge \omega .$$

(61)

Then

$$\frac{1}{\sqrt{L}}d{\sigma }_{\Sigma ,L}=d{\sigma }_{\Sigma }+d\overline{{\sigma }_{\Sigma }}{L}^{-1}+O\left(L^{-2}\right),\mathrm{as}L\to +\infty .$$

(62)

If $\Sigma =f\left(D\right)$ with

$$f=f(u_1,u_2)=(f_1,f_2,f_3):D\subset {\mathbb{R}}^2\to {\mathbb{H}}_L^1,$$

then

$$\begin{array}{cc}\hfill & {\lim }_{L\to +\infty }\frac{1}{\sqrt{L}}{\int }_{\Sigma }d{\sigma }_{\Sigma ,L}={\int }_D\left\{{\left[\frac{f_1[{\left(f_1\right)}_{u_1}{\left(f_2\right)}_{u_2}-{\left(f_2\right)}_{u_1}{\left(f_1\right)}_{u_2}]}{2}+{\left(f_2\right)}_{u_1}{\left(f_3\right)}_{u_2}-{\left(f_2\right)}_{u_2}{\left(f_3\right)}_{u_1}\right]}^2\right.\hfill \\ \hfill & {\left.-{\left[\frac{f_2[{\left(f_1\right)}_{u_1}{\left(f_2\right)}_{u_2}-{\left(f_1\right)}_{u_2}{\left(f_2\right)}_{u_1}]}{2}+{\left(f_1\right)}_{u_1}{\left(f_3\right)}_{u_2}-{\left(f_1\right)}_{u_2}{\left(f_3\right)}_{u_1}\right]}^2\right\}}^{\frac{1}{2}}d{u}_{1}d{u}_2.\hfill \end{array}$$

(63)

Proof. 

We know that

$$g_L(X_1,·)=-{\omega }_1,g_L(X_2,·)={\omega }_2,g_L(X_3,·)=L\omega ,$$

so

$$e_1^{🟉}=g_L(e_1,·)=\overline{q}{\omega }_1-\overline{p}{\omega }_2,e_2^{🟉}=g_L(e_2,·)=\overline{r_L}\overline{p}{\omega }_1-\overline{r_L}\overline{q}{\omega }_2+\frac{l}{l_L}{L}^{\frac{1}{2}}\omega .$$

(64)

Then,

$$\frac{1}{\sqrt{L}}d{\sigma }_{\Sigma ,L}=\frac{1}{\sqrt{L}}{e}_1^{🟉}\wedge e_2^{🟉}=-\frac{l}{l_L}(\overline{p}{\omega }_2-\overline{q}{\omega }_1)\wedge \omega -\frac{1}{\sqrt{L}}\overline{r_L}{\omega }_1\wedge {\omega }_2.$$

(65)

By

$$\overline{r_L}=\frac{\left(X_{3}u\right)L^{-\frac{1}{2}}}{\sqrt{-p^2+q^2+L^{-1}{\left(X_{3}u\right)}^2}}$$

and the Taylor expansion

$$\frac{1}{l_L}=\frac{1}{l}-\frac{1}{2l^3}{\left(X_{3}u\right)}^2{L}^{-1}+O\left(L^{-2}\right)\mathrm{as}L\to +\infty$$

we obtain (58). Using (2), then we get

$$\begin{array}{cc}\hfill f_{u_1}& ={\left(f_1\right)}_{u_1}{\partial }_{x_1}+{\left(f_2\right)}_{u_1}{\partial }_{x_2}+{\left(f_3\right)}_{u_1}{\partial }_{x_3}\hfill \\ \hfill & ={\left(f_1\right)}_{u_1}{X}_1+{\left(f_2\right)}_{u_1}{X}_2+\sqrt{L}\left[\frac{{\left(f_2\right)}_{u_1}}{2}{f}_2-\frac{{\left(f_1\right)}_{u_1}}{2}{f}_1+{\left(f_3\right)}_{u_1}\right]\widetilde{X_3},\hfill \end{array}$$

(66)

and

$$f_{u_2}={\left(f_1\right)}_{u_2}{X}_1+{\left(f_2\right)}_{u_2}{X}_2+\sqrt{L}\left[\frac{{\left(f_2\right)}_{u_2}}{2}{f}_2-\frac{{\left(f_1\right)}_{u_2}}{2}{f}_1+{\left(f_3\right)}_{u_2}\right]\widetilde{X_3}.$$

(67)

Let

$$\overline{v_L}=\left|\begin{array}{ccc}-X_1,& X_2,& \widetilde{X_3}\\ {\left(f_1\right)}_{u_1},& {\left(f_2\right)}_{u_1},& \sqrt{L}\left[\frac{{\left(f_2\right)}_{u_1}}{2}{f}_2-\frac{{\left(f_1\right)}_{u_1}}{2}{f}_1+{\left(f_3\right)}_{u_1}\right]\\ {\left(f_1\right)}_{u_2},& {\left(f_2\right)}_{u_2},& \sqrt{L}\left[\frac{{\left(f_2\right)}_{u_2}}{2}{f}_2-\frac{{\left(f_1\right)}_{u_2}}{2}{f}_1+{\left(f_3\right)}_{u_2}\right]\end{array}\right|.$$

(68)

We know that

$$d{\sigma }_{\Sigma ,L}=\sqrt{\det \left(g_{ij}\right)}d{u}_{1}d{u}_2,g_{ij}=g_L(f_{u_i},f_{u_j}),\det \left(g_{ij}\right)=\left|\right|\overline{v_L}{\left|\right|}_L^2=-⟨\overline{v_L},\overline{v_L}⟩,$$

by the dominated convergence theorem, (64) holds. □

Theorem 2.

(The Gauss–Bonnet theorem for the Lorentzian surface in${\mathbb{H}}_L^1$) Let$\Sigma \subset {\mathbb{H}}_L^1$be a regular Lorentzian surface. We assume that Σ is with finitely many boundary components ${(\partial \Sigma )}_i,$ $i\in \{1,\cdots ,n\}$, given by $C^2$-smooth regular and closed spacelike curves ${\gamma }_i:[0,2\pi ]\to {(\partial \Sigma )}_i$. Let the characteristic set $C(\Sigma )$ be the empty set. Suppose that A is defined by (53) and $d{\sigma }_{\Sigma }$ is defined by (61) and $k_{{\gamma }_i,\Sigma }^{\infty ,s}$ is the sub-Lorentzian signed geodesic curvature of ${\gamma }_i$ relative to Σ. Thus

$${\int }_{\Sigma }Ad{\sigma }_{\Sigma }+\sum _{i=1}^n{\int }_{{\gamma }_i}{k}_{{\gamma }_i,\Sigma }^{\infty ,s}ds=0.$$

(69)

Proof. 

By the discussions in [2], suppose that all points satisfy $\omega \left(\dot{{\gamma }_i}\left(t\right)\right)\ne 0$ on ${\gamma }_i$. Therefore, using Lemma 7, we obtain

$$k_{{\gamma }_i,\Sigma }^{L,s}=k_{{\gamma }_i,\Sigma }^{\infty ,s}+O\left(L^{-1}\right).$$

(70)

By the Gauss–Bonnet theorem (see [4] page 90 Theorem 1.4), we have

$${\int }_{\Sigma }{\mathcal{K}}^{\Sigma ,L}\frac{1}{\sqrt{L}}d{\sigma }_{\Sigma ,L}+\sum _{i=1}^n{\int }_{{\gamma }_i}{k}_{{\gamma }_i,\Sigma }^{L,s}\frac{1}{\sqrt{L}}d{s}_L=0.$$

(71)

Therefore, by (71), (72), (60), (53), (61), we get

$$\left({\int }_{\Sigma }Ad{\sigma }_{\Sigma }+\sum _{i=1}^n{\int }_{{\gamma }_i}{k}_{{\gamma }_i,\Sigma }^{\infty ,s}ds\right)+O\left(L^{-\frac{1}{2}}\right)=0.$$

(72)

Let L go to the infinity. By the dominated convergence theorem, (69) holds. □

4. Spacelike Surfaces and a Gauss–Bonnet Theorem in the Lorentzian Heisenberg Group

The geodesic curvature of spacelike curves on spacelike surface in the Lorentzian Heisenberg group is investigated in this section.

Let

$$p:=X_{1}u,q:=X_{2}u,\mathrm{and}r:={\tilde{X}}_{3}u.$$

Let $p^2-q^2>0$, when $L\to +\infty$, we have $p^2-q^2-r^2>0$. We then define

$$\begin{array}{cc}\hfill & l:=\sqrt{p^2-q^2},l_L:=\sqrt{p^2-q^2-r^2},\overline{p}:=\frac{p}{l},\hfill \\ \hfill & \overline{q}:=\frac{q}{l},\overline{p_L}:=\frac{p}{l_L},\overline{q_L}:=\frac{q}{l_L},\overline{r_L}:=\frac{r}{l_L}.\hfill \end{array}$$

(73)

In particular, ${\overline{p}}^2-{\overline{q}}^2=1$. At every non-characteristic point, the functions above are well defined. Let

$$\begin{array}{c}\hfill v_L=-\overline{p_L}{X}_1+\overline{q_L}{X}_2+\overline{r_L}\widetilde{X_3},e_1=\overline{q}{X}_1-\overline{p}{X}_2,e_2=\overline{r_L}\overline{p}{X}_1-\overline{r_L}\overline{q}{X}_2-\frac{l}{l_L}\widetilde{X_3},\end{array}$$

(74)

then $v_L$ is the unit timelike normal vector to $\Sigma$, $e_1$ and $e_2$ is the unit spacelike vector. $\{e_1,e_2\}$ are the orthonormal basis of $\Sigma$. We call $\Sigma$ a spacelike surface in Lorentzian Hersenberg group. We define a linear transformation on $T\Sigma$ by $J_L:T\Sigma \to T\Sigma$ and the transformation is well defined.

$$J_L\left(e_1\right)=e_2,J_L\left(e_2\right)=-e_1.$$

(75)

We then define the projection by $\pi :T{\mathbb{H}}_L^1\to T\Sigma$. For every $U,V\in T\Sigma$, let ${\nabla }_U^{\Sigma ,L}V=\pi {\nabla }_U^{L}V$. Therefore, the Levi–Civita connection with respect to the metric $g_L$ on $\Sigma$ is defined as ${\nabla }^{\Sigma ,L}$. Combining (14), (74) and

$${\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }={⟨{\nabla }_{\dot{\gamma }}^L\dot{\gamma },e_1⟩}_L{e}_1+{⟨{\nabla }_{\dot{\gamma }}^L\dot{\gamma },e_2⟩}_L{e}_2,$$

(76)

we have

$$\begin{array}{cc}\hfill {\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }& =\left\{-\overline{q}\left[{\ddot{\gamma }}_1-L{\gamma }_2\omega \left(\dot{\gamma }\left(t\right)\right)\right]-\overline{p}\left[{\ddot{\gamma }}_2-L{\gamma }_1\omega \left(\dot{\gamma }\left(t\right)\right)\right]\right\}{e}_1\hfill \\ \hfill & +\left\{-\overline{r_L}\overline{p}\left[{\ddot{\gamma }}_1-L{\gamma }_2\omega \left(\dot{\gamma }\left(t\right)\right)\right]-\overline{r_L}\overline{q}\left[{\ddot{\gamma }}_2-L{\gamma }_1\omega \left(\dot{\gamma }\left(t\right)\right)\right]\right.\hfill \\ \hfill & \left.-\frac{l}{l_L}{L}^{\frac{1}{2}}\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\right]\right\}{e}_2.\hfill \end{array}$$

(77)

When $\omega \left(\dot{\gamma }\left(t\right)\right)=0$, then

$${\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }=[-\overline{q}{\ddot{\gamma }}_1-\overline{p}{\ddot{\gamma }}_2]e_1+\left\{-\overline{r_L}\overline{p}{\ddot{\gamma }}_1-\overline{r_L}\overline{q}{\ddot{\gamma }}_2-\frac{l}{l_L}{L}^{\frac{1}{2}}\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\right]\right\}{e}_2$$

(78)

Definition 8.

Let$\Sigma \subset {\mathbb{H}}_L^1$be a regular spacelike surface. Let$\gamma :I\to \Sigma$be a$C^2$-smooth regular curve, we define the geodesic curvature$k_{\gamma ,\Sigma }^L$of γ at$\gamma \left(t\right)$by

$$k_{\gamma ,\Sigma }^L:=\sqrt{\frac{\left|\right|{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }{\left|\right|}_{\Sigma ,L}^2}{\left|\right|\dot{\gamma }{\left|\right|}_{\Sigma ,L}^4}-\frac{{⟨{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma },\dot{\gamma }⟩}_{\Sigma ,L}^2}{{⟨\dot{\gamma },\dot{\gamma }⟩}_{\Sigma ,L}^3}}.$$

(79)

Definition 9.

Let$\Sigma \subset {\mathbb{H}}_L^1$be a regular spacelike surface. Let$\gamma :I\to \Sigma$be a$C^2$-smooth regular curve. The intrinsic geodesic curvature$k_{\gamma ,\Sigma }^{\infty }$of γ at$\gamma \left(t\right)$is defined as

$$k_{\gamma ,\Sigma }^{\infty }:={\lim }_{L\to +\infty }{k}_{\gamma ,\Sigma }^L,$$

if the limit exists.

Lemma 10.

Let$\Sigma \subset {\mathbb{H}}_L^1$be a regular spacelike surface. Let$\gamma :I\to \Sigma$be a$C^2$-smooth spacelike curve, then we have the following assertions

$$k_{\gamma ,\Sigma }^{\infty }=\frac{|\overline{p}{\dot{\gamma }}_1+\overline{q}{\dot{\gamma }}_2|}{\left|\omega \right(\dot{\gamma }\left(t\right)\left)\right|},if\omega \left(\dot{\gamma }\left(t\right)\right)\ne 0,$$

(80)

$$k_{\gamma ,\Sigma }^{\infty }=0,if\omega \left(\dot{\gamma }\left(t\right)\right)=0and\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)=0,$$

$${\lim }_{L\to +\infty }\frac{k_{\gamma ,\Sigma }^L}{\sqrt{L}}=\frac{|\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)|}{{\left(\overline{q}{\dot{\gamma }}_1+\overline{p}{\dot{\gamma }}_2\right)}^2},if\omega \left(\dot{\gamma }\left(t\right)\right)=0and\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\ne 0.$$

(81)

Proof. 

By (10) and $\dot{\gamma }\in T\Sigma$, we have

$$\dot{\gamma }=-(\overline{q}{\dot{\gamma }}_1+\overline{p}{\dot{\gamma }}_2)e_1-\frac{l_L}{l}{L}^{\frac{1}{2}}\omega \left(\dot{\gamma }\left(t\right)\right)e_2.$$

(82)

By (78), we have

$$\begin{array}{cc}\hfill {⟨{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma },{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }⟩}_{L,\Sigma }& ={\left\{-\overline{q}\left[{\ddot{\gamma }}_1-L{\gamma }_2\omega \left(\dot{\gamma }\left(t\right)\right)\right]-\overline{p}\left[{\ddot{\gamma }}_2-L{\gamma }_1\omega \left(\dot{\gamma }\left(t\right)\right)\right]\right\}}^2\hfill \\ \hfill & +\left\{-\overline{r_L}\overline{p}\left[{\ddot{\gamma }}_1-L{\gamma }_2\omega \left(\dot{\gamma }\left(t\right)\right)\right]-\overline{r_L}\overline{q}\left[{\ddot{\gamma }}_2-L{\gamma }_1\omega \left(\dot{\gamma }\left(t\right)\right)\right]\right.\hfill \\ \hfill & {\left.-\frac{l}{l_L}{L}^{\frac{1}{2}}\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\right]\right\}}^2.\hfill \end{array}$$

(83)

Similarly, if $\omega \left(\dot{\gamma }\left(t\right)\right)\ne 0$,

$${⟨\dot{\gamma },\dot{\gamma }⟩}_{\Sigma ,L}={(\overline{q}{\dot{\gamma }}_1+\overline{p}{\dot{\gamma }}_2)}^2+{\left[\frac{l_L}{l}{L}^{\frac{1}{2}}\omega \left(\dot{\gamma }\left(t\right)\right)\right]}^2\sim L{\left[\omega \left(\dot{\gamma }\left(t\right)\right)\right]}^2,\mathrm{as}L\to +\infty .$$

(84)

By (78) and (82), we have

$$\begin{array}{cc}\hfill {⟨{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma },\dot{\gamma }⟩}_{\Sigma ,L}& =-(\overline{q}{\dot{\gamma }}_1+\overline{p}{\dot{\gamma }}_2)·\left\{-\overline{q}\left[{\ddot{\gamma }}_1-L{\gamma }_2\omega \left(\dot{\gamma }\left(t\right)\right)\right]\right.\hfill \\ \hfill & \left.-\overline{p}\left[{\ddot{\gamma }}_2-L{\gamma }_1\omega \left(\dot{\gamma }\left(t\right)\right)\right]\right\}+[-\frac{l_L}{l}{L}^{\frac{1}{2}}\omega \left(\dot{\gamma }\left(t\right)\right)]·\left\{-\overline{r_L}\overline{p}\left[{\ddot{\gamma }}_1-L{\gamma }_2\omega \left(\dot{\gamma }\left(t\right)\right)\right]\right.\hfill \\ \hfill & \left.-\overline{r_L}\overline{q}\left[{\ddot{\gamma }}_2-L{\gamma }_1\omega \left(\dot{\gamma }\left(t\right)\right)\right]-\frac{l}{l_L}{L}^{\frac{1}{2}}\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\right]\right\}\sim M_{0}L,\hfill \end{array}$$

(85)

where $M_0$ does not depend on L. Combining (79), (83)–(85), (80) holds.

If $\omega \left(\dot{\gamma }\left(t\right)\right)=0$ and $\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)=0$, then

$$\begin{array}{cc}\hfill {⟨{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma },{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }⟩}_{L,\Sigma }& ={\left\{-\overline{q}{\ddot{\gamma }}_1-\overline{p}{\ddot{\gamma }}_2\right\}}^2+{\left\{-\overline{r_L}\overline{p}{\ddot{\gamma }}_1-\overline{r_L}\overline{q}{\ddot{\gamma }}_2\right\}}^2\hfill \\ \hfill & \sim {\left\{-\overline{q}{\ddot{\gamma }}_1-\overline{p}{\ddot{\gamma }}_2\right\}}^2,\mathrm{as}L\to +\infty ,\hfill \end{array}$$

(86)

$${⟨\dot{\gamma },\dot{\gamma }⟩}_{\Sigma ,L}={(\overline{q}{\dot{\gamma }}_1+\overline{p}{\dot{\gamma }}_2)}^2,$$

(87)

$$\begin{array}{cc}\hfill {⟨{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma },\dot{\gamma }⟩}_{\Sigma ,L}& =-(\overline{q}{\dot{\gamma }}_1+\overline{p}{\dot{\gamma }}_2)·(-\overline{q}{\ddot{\gamma }}_1-\overline{p}{\ddot{\gamma }}_2)\hfill \end{array}$$

(88)

By (86)–(88) and (79), $k_{\gamma ,\Sigma }^{\infty }=0$ holds.

If $\omega \left(\dot{\gamma }\left(t\right)\right)=0$ and $\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\ne 0$, then

$${⟨{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma },{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }⟩}_{L,\Sigma }\sim L{\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\right]}^2,$$

$${⟨{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma },\dot{\gamma }⟩}_{\Sigma ,L}=O\left(1\right),$$

so we get (81). □

Lemma 11.

Let$\Sigma \subset {\mathbb{H}}_L^1$be a regular spacelike surface. Let$\gamma :I\to \Sigma$be a$C^2$-smooth spacelike curve, then

$${\lim }_{L\to +\infty }\frac{k_{\gamma ,\Sigma }^{L,s}}{\sqrt{L}}=\frac{\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)}{{\left(\overline{q}{\dot{\gamma }}_1+\overline{p}{\dot{\gamma }}_2\right)}^2},if\omega \left(\dot{\gamma }\left(t\right)\right)=0and\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\ne 0.$$

(89)

$$k_{\gamma ,\Sigma }^{\infty }=0,if\omega \left(\dot{\gamma }\left(t\right)\right)=0and\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)=0,$$

(90)

$$k_{\gamma ,\Sigma }^{\infty ,s}=\frac{\overline{p}{\dot{\gamma }}_1+\overline{q}{\dot{\gamma }}_2}{\left|\omega \right(\dot{\gamma }\left(t\right)\left)\right|},if\omega \left(\dot{\gamma }\left(t\right)\right)\ne 0,$$

(91)

Proof. 

By (75) and (82), we have

$$J_L\left(\dot{\gamma }\right)=\frac{l_L}{l}{L}^{\frac{1}{2}}\omega \left(\dot{\gamma }\left(t\right)\right)e_1-(\overline{p}{\dot{\gamma }}_2+\overline{q}{\dot{\gamma }}_1)e_2.$$

(92)

By (77) and (92), we have

$$\begin{array}{cc}\hfill {⟨{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma },J_L\left(\dot{\gamma }\right)⟩}_{L,\Sigma }& =\frac{l_L}{l}{L}^{\frac{1}{2}}\omega \left(\dot{\gamma }\left(t\right)\right)\left\{-\overline{q}\left[{\ddot{\gamma }}_1-L{\gamma }_2\omega \left(\dot{\gamma }\left(t\right)\right)\right]-\overline{p}\left[{\ddot{\gamma }}_2-L{\gamma }_1\omega \left(\dot{\gamma }\left(t\right)\right)\right]\right\}\hfill \\ \hfill & -(\overline{p}{\dot{\gamma }}_2+\overline{q}{\dot{\gamma }}_1)·\left\{-\overline{r_L}\overline{p}\left[{\ddot{\gamma }}_1-L{\gamma }_2\omega \left(\dot{\gamma }\left(t\right)\right)\right]-\overline{r_L}\overline{q}\left[{\ddot{\gamma }}_2-L{\gamma }_1\omega \left(\dot{\gamma }\left(t\right)\right)\right]\right.\hfill \\ \hfill & \left.-\frac{l}{l_L}{L}^{\frac{1}{2}}\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\right]\right\}\sim L^{\frac{3}{2}}\omega {\left(\dot{\gamma }\left(t\right)\right)}^2(\overline{p}{\dot{\gamma }}_1+\overline{q}{\dot{\gamma }}_2)\mathrm{as}L\to +\infty .\hfill \end{array}$$

(93)

$${⟨\dot{\gamma },\dot{\gamma }⟩}_{\Sigma ,L}={(\overline{q}{\dot{\gamma }}_1+\overline{p}{\dot{\gamma }}_2)}^2+{\left[\frac{l_L}{l}{L}^{\frac{1}{2}}\omega \left(\dot{\gamma }\left(t\right)\right)\right]}^2\sim L{\left[\omega \left(\dot{\gamma }\left(t\right)\right)\right]}^2,\mathrm{as}L\to +\infty .$$

(94)

Therefore, (91) holds. If $\omega \left(\dot{\gamma }\left(t\right)\right)=0$ and $\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)=0,$ then

$$\begin{array}{cc}\hfill {⟨{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma },J_L\left(\dot{\gamma }\right)⟩}_{L,\Sigma }=& -(\overline{p}{\dot{\gamma }}_2+\overline{q}{\dot{\gamma }}_1)·\left\{-\overline{r_L}\overline{p}{\ddot{\gamma }}_1-\overline{r_L}\overline{q}{\ddot{\gamma }}_2\right]\sim O\left(L^{-\frac{1}{2}}\right)\mathrm{as}L\to +\infty .\hfill \end{array}$$

(95)

So $k_{\gamma ,\Sigma }^{\infty ,s}=0.$ If $\omega \left(\dot{\gamma }\left(t\right)\right)=0$ and $\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\ne 0$, then

$$\begin{array}{c}\hfill {⟨{\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma },J_L\left(\dot{\gamma }\right)⟩}_{L,\Sigma }\sim L^{\frac{1}{2}}(\overline{p}{\dot{\gamma }}_2+\overline{q}{\dot{\gamma }}_1)\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\mathrm{as}L\to +\infty .\end{array}$$

(96)

So (89) holds. □

In the Lorentzian Heisenberg group, the sub-Lorentzian limit of the Riemannian Gaussian curvature of spacelike surfaces is investigated next. The second fundamental form $I{I}^L$ of the embedding of $\Sigma$ into ${\mathbb{H}}_L^1$ is defined as:

$$I{I}^L=\left(\begin{array}{cc}{⟨{\nabla }_{e_1}^L{v}_L,e_1⟩}_L,& {⟨{\nabla }_{e_1}^L{v}_L,e_2⟩}_L\\ {⟨{\nabla }_{e_2}^L{v}_L,e_1⟩}_L,& {⟨{\nabla }_{e_2}^L{v}_L,e_2⟩}_L\end{array}\right).$$

(97)

Similarly to Theorem 4.3 in [6], we have

Theorem 3.

For the embedding of Σ into ${\mathbb{H}}_L^1$, the second fundamental form $I{I}^L$ of Σ is

$$I{I}^L=\left(\begin{array}{cc}-\frac{l}{l_L}[X_1\left(\overline{p}\right)-X_2\left(\overline{q}\right)],& -\frac{l_L}{l}{⟨e_1,{\nabla }_H\left(\overline{r_L}\right)⟩}_L+\frac{\sqrt{L}}{2}\\ -\frac{l_L}{l}{⟨e_1,{\nabla }_H\left(\overline{r_L}\right)⟩}_L+\frac{\sqrt{L}}{2},& -\frac{l^2}{l_L^2}{⟨e_2,{\nabla }_H\left(\frac{r}{l}\right)⟩}_L+\widetilde{X_3}\left(\overline{r_L}\right)\end{array}\right).$$

(98)

Proof. 

Combining

$$\begin{array}{c}\hfill v_L=-\overline{p_L}{X}_1+\overline{q_L}{X}_2+\overline{r_L}\widetilde{X_3},e_1=\overline{q}{X}_1-\overline{p}{X}_2,e_2=\overline{r_L}\overline{p}{X}_1-\overline{r_L}\overline{q}{X}_2-\frac{l}{l_L}\widetilde{X_3},\end{array}$$

and ${⟨{\nabla }_{e_i}^L{v}_L,e_j⟩}_L=-{⟨{\nabla }_{e_i}^L{e}_j,v_L⟩}_L,i=1,2;j=1,2.$ By direct calculation, we obtain

$${\nabla }_{e_1}^L{e}_1=[\overline{q}{X}_1\left(\overline{q}\right)-\overline{p}{X}_2\left(\overline{q}\right)]X_1-[\overline{q}{X}_1\left(\overline{p}\right)-\overline{p}{X}_2\left(\overline{p}\right)]X_2.$$

Since ${\overline{p}}^2-{\overline{q}}^2=1$, we get $\overline{p}{X}_i\left(\overline{p}\right)-\overline{q}{X}_i\left(\overline{q}\right)=0,i=1,2$. Then

$${⟨{\nabla }_{e_1}^L{e}_1,v_L⟩}_L={⟨{\nabla }_{(\overline{q}{X}_1-\overline{p}{X}_2)}^L(\overline{q}{X}_1-\overline{p}{X}_2),-\overline{p_L}{X}_1+\overline{q_L}{X}_2+\overline{r_L}\widetilde{X_3}⟩}_L.$$

Then,

$${⟨{\nabla }_{e_1}^L{v}_L,e_1⟩}_L=-{⟨{\nabla }_{e_1}^L{e}_1,v_L⟩}_L=-\frac{l}{l_L}[X_1\left(\overline{p}\right)-X_2\left(\overline{q}\right)].$$

Similarly, we have

$${⟨{\nabla }_{e_1}^L{v}_L,e_2⟩}_L=-{⟨{\nabla }_{e_1}^L{e}_2,v_L⟩}_L=-\frac{l_L}{l}{⟨e_1,{\nabla }_H\left(\overline{r_L}\right)⟩}_L+\frac{\sqrt{L}}{2};$$

$${⟨{\nabla }_{e_2}^L{v}_L,e_1⟩}_L=-{⟨{\nabla }_{e_2}^L{e}_1,v_L⟩}_L=-\frac{l_L}{l}{⟨e_1,{\nabla }_H\left(\overline{r_L}\right)⟩}_L+\frac{\sqrt{L}}{2};$$

$${⟨{\nabla }_{e_2}^L{v}_L,e_2⟩}_L=-{⟨{\nabla }_{e_2}^L{e}_2,v_L⟩}_L=-\frac{l^2}{l_L^2}{⟨e_2,{\nabla }_H\left(\frac{r}{l}\right)⟩}_L+\widetilde{X_3}\left(\overline{r_L}\right).$$

Thus, (98) holds. □

By the Gauss equation, we have

$${\mathcal{K}}^{\Sigma ,L}(e_1,e_2)={\mathcal{K}}^L(e_1,e_2)-\det \left(I{I}^L\right).$$

(99)

Proposition 4.

The horizontal mean curvature${\mathcal{H}}_{\infty }$of$\Sigma \subset \mathbb{H}$away from characteristic points is

$${\mathcal{H}}_{\infty }={\lim }_{L\to +\infty }{\mathcal{H}}_L=-X_1\left(\overline{p}\right)+X_2\left(\overline{q}\right).$$

(100)

Proof. 

By

$$\frac{l^2}{l_L^2}{⟨e_2,{\nabla }_H\left(\frac{r}{l}\right)⟩}_L=\frac{\overline{p}r}{l}{X}_1\left(\overline{r_L}\right)+\frac{\overline{q}r}{l}{X}_2\left(\overline{r_L}\right)=O\left(L^{-1}\right)$$

$$\frac{l}{l_L}[X_1\left(\overline{p}\right)-X_2\left(\overline{q}\right)]\to X_1\left(\overline{p}\right)-X_2\left(\overline{q}\right),\widetilde{X_3}\left(\overline{r_L}\right)\to 0,\overline{p_L}\to \overline{p},$$

we get (100). □

Proposition 5.

Away from characteristic points, we get the following assertion

$${\mathcal{K}}^{\Sigma ,L}(e_1,e_2)\to A+O\left(\frac{1}{L}\right),\mathrm{as}L\to +\infty ,$$

(101)

where

$$A:=-{⟨e_1,{\nabla }_H\left(\frac{X_{3}u}{|{\nabla }_{H}u|}\right)⟩}_L-\frac{{\left(X_{3}u\right)}^2}{l^2}.$$

(102)

Proof. 

By (21), we have

$$\begin{array}{cc}\hfill & {⟨R^L(e_1,e_2)e_1,e_2⟩}_L\hfill \\ \hfill & ={\overline{r_L}}^2{⟨R^L(X_1,X_2)X_1,X_2⟩}_L-2\frac{l}{l_L}\overline{q}{L}^{-\frac{1}{2}}\overline{r_L}{⟨R^L(X_1,X_2)X_1,X_3⟩}_L\hfill \\ \hfill & +2\frac{l}{l_L}\overline{p}{L}^{-\frac{1}{2}}\overline{r_L}{⟨R^L(X_1,X_2)X_2,X_3⟩}_L+{\left(\frac{l}{l_L}\overline{q}\right)}^2{L}^{-1}{⟨R^L(X_1,X_3)X_1,X_3⟩}_L\hfill \\ \hfill & -2{\left(\frac{l}{l_L}\right)}^2\overline{p}\overline{q}{L}^{-1}{⟨R^L(X_1,X_3)X_2,X_3⟩}_L+{\left(\overline{p}\frac{l}{l_L}\right)}^2{L}^{-1}{⟨R^L(X_2,X_3)X_2,X_3⟩}_L.\hfill \\ \hfill \end{array}$$

(103)

By Lemma 8, we have

$$\begin{array}{c}\hfill {\mathcal{K}}^L(e_1,e_2)=-\frac{L}{4}{\left(\frac{l}{l_L}\right)}^2-\frac{3}{4}L{\overline{r_L}}^2.\end{array}$$

(104)

By (98) and

$${\nabla }_H\left(\overline{r_L}\right)=L^{-\frac{1}{2}}{\nabla }_H\left(\frac{X_{3}u}{|{\nabla }_{H}u|}\right)+O\left(L^{-1}\right)\mathrm{as}L\to +\infty$$

we get

$$\det \left(I{I}^L\right)=-\frac{L}{4}+⟨e_1,{\nabla }_H\left(\frac{X_{3}u}{|{\nabla }_{H}u|}\right)⟩+O\left(L^{-1}\right)\mathrm{as}L\to +\infty .$$

(105)

By (99), (104) and (105), we get (101). □

Proposition 6.

Let$\Sigma \subset {\mathbb{H}}_L^1$be a spacelike$C^2$-smooth surface. We assume that$d{\sigma }_L$is the surface measure on Σ with respect to the metric $g_L$. Suppose that

$$d{\sigma }_{\Sigma }:=(\overline{p}{\omega }_2-\overline{q}{\omega }_1)\wedge \omega ,d\overline{{\sigma }_{\Sigma }}:=\frac{X_{3}u}{l}{\omega }_1\wedge {\omega }_2+\frac{{\left(X_{3}u\right)}^2}{2l^2}(\overline{p}{\omega }_2-\overline{q}{\omega }_1)\wedge \omega .$$

(106)

Then,

$$\frac{1}{\sqrt{L}}d{\sigma }_{\Sigma ,L}=d{\sigma }_{\Sigma }+d\overline{{\sigma }_{\Sigma }}{L}^{-1}+O\left(L^{-2}\right),\mathrm{as}L\to +\infty .$$

(107)

Proof. 

We know that

$$g_L(X_1,·)=-{\omega }_1,g_L(X_2,·)={\omega }_2,g_L(X_3,·)=L\omega ,$$

so

$$e_1^{🟉}=g_L(e_1,·)=\overline{q}{\omega }_1-\overline{p}{\omega }_2,e_2^{🟉}=g_L(e_2,·)=\overline{r_L}\overline{p}{\omega }_1-\overline{r_L}\overline{q}{\omega }_2-\frac{l}{l_L}{L}^{\frac{1}{2}}\omega .$$

(108)

Then,

$$\frac{1}{\sqrt{L}}d{\sigma }_{\Sigma ,L}=\frac{1}{\sqrt{L}}{e}_1^{🟉}\wedge e_2^{🟉}=\frac{l}{l_L}(\overline{p}{\omega }_2-\overline{q}{\omega }_1)\wedge \omega +\frac{1}{\sqrt{L}}\overline{r_L}{\omega }_1\wedge {\omega }_2.$$

(109)

By

$$\overline{r_L}=\frac{\left(X_{3}u\right)L^{-\frac{1}{2}}}{\sqrt{p^2-q^2-L^{-1}{\left(X_{3}u\right)}^2}}$$

and the Taylor expansion

$$\frac{1}{l_L}=\frac{1}{l}+\frac{1}{2l^3}{\left(X_{3}u\right)}^2{L}^{-1}+O\left(L^{-2}\right)\mathrm{as}L\to +\infty ,$$

so (109) holds. □

Theorem 4.

(The Gauss–Bonnet theorem for the spacelike surface in the Lorentzian Heisenberg group) Let$\Sigma \subset {\mathbb{H}}_L^1$be a regular spacelike surface. We assume that Σ is with finitely many boundary components ${(\partial \Sigma )}_i,$ $i\in \{1,\cdots ,n\}$, given by $C^2$-smooth regular and closed curves ${\gamma }_i:[0,2\pi ]\to {(\partial \Sigma )}_i$. Let the characteristic set $C(\Sigma )$ be the empty set. Suppose that A is defined by (102) and $d{\sigma }_{\Sigma }$ is defined by (106) and $k_{{\gamma }_i,\Sigma }^{\infty ,s}$ is the sub-Lorentzian signed geodesic curvature of ${\gamma }_i$ relative to Σ. Thus, we get

$${\int }_{\Sigma }Ad{\sigma }_{\Sigma }+\sum _{i=1}^n{\int }_{{\gamma }_i}{k}_{{\gamma }_i,\Sigma }^{\infty ,s}ds=0.$$

(110)

Proof. 

By the discussions in [2], we may assume that there is no points satisfying $\omega \left(\dot{{\gamma }_i}\left(t\right)\right)=0$ and $\frac{d}{dt}\left(\omega \left(\dot{{\gamma }_i}\left(t\right)\right)\right)\ne 0$ on ${\gamma }_i$. Using Lemma 11, we get

$$k_{{\gamma }_i,\Sigma }^{L,s}=k_{{\gamma }_i,\Sigma }^{\infty ,s}+O\left(L^{-\frac{1}{2}}\right).$$

(111)

By the Gauss–Bonnet theorem, we obtain

$${\int }_{\Sigma }{\mathcal{K}}^{\Sigma ,L}\frac{1}{\sqrt{L}}d{\sigma }_{\Sigma ,L}+\sum _{i=1}^n{\int }_{{\gamma }_i}{k}_{{\gamma }_i,\Sigma }^{L,s}\frac{1}{\sqrt{L}}d{s}_L=2\pi \frac{\chi (\Sigma )}{\sqrt{L}}.$$

(112)

So by (59), (60), (101), (109), (112) and (113), we get

$${\int }_{\Sigma }Ad{\sigma }_{\Sigma }+\sum _{i=1}^n{\int }_{{\gamma }_i}{k}_{{\gamma }_i,\Sigma }^{\infty ,s}d\overline{s}+O\left(L^{-\frac{1}{2}}\right)=2\pi \frac{\chi (\Sigma )}{\sqrt{L}}.$$

(113)

Let L go to the infinity. By the dominated convergence theorem, (110) holds. □

5. The Sub-Lorentzian Limit of Curvature of Curves in the Lorentzian Group of Rigid Motions of the Minkowski Plane

Let $E(1,1)$ be the group of rigid motions of the Minkowski plane. $E(1,1)$ is a unimodular Lie group with a natural sub-Lorentzian structure. For the reason of studying $E(1,1)$, we choose the underlying manifold ${\mathbb{R}}^3$ as a model of $E(1,1)$. On ${\mathbb{R}}^3$, let

$$X_1={\partial }_{x_3},X_2=\frac{1}{\sqrt{2}}(-e^{x_3}{\partial }_{x_1}+e^{-x_3}{\partial }_{x_2}),X_3=-\frac{1}{\sqrt{2}}(e^{x_3}{\partial }_{x_1}+e^{-x_3}{\partial }_{x_2}).$$

(114)

Then,

$${\partial }_{x_1}=-\frac{\sqrt{2}}{2}{e}^{-x_3}(X_2+X_3),{\partial }_{x_2}=\frac{\sqrt{2}}{2}{e}^{x_3}(X_2-X_3),{\partial }_{x_3}=X_1,$$

(115)

and $\mathrm{span}\{X_1,X_2,X_3\}=T\left(E(1,1)\right).$ Let $H=\mathrm{span}\{X_1,X_2\}$ be the horizontal distribution on $E(1,1)$. Let ${\omega }_1=d{x}_3,{\omega }_2=\frac{1}{\sqrt{2}}(-e^{-x_3}d{x}_1+e^{x_3}d{x}_2),\omega =-\frac{1}{\sqrt{2}}(e^{-x_3}d{x}_1+e^{x_3}d{x}_2).$ Then $H=\mathrm{Ker}\omega$. For $L>0$, let $g_L=-{\omega }_1\otimes {\omega }_1+{\omega }_2\otimes {\omega }_2+L\omega \otimes \omega ,g=g_1$ be the Riemannian metric on $E(1,1)$, where L is a constant. We call $(E(1,1),g_L)$the Lorentzian group of rigid motions of the Minkowski plane and write $E_L^1(1,1)$ instead of $(E(1,1),g_L)$. Then $X_1,X_2,\widetilde{X_3}:=L^{-\frac{1}{2}}{X}_3$ are orthonormal basis on $T\left(E_L^1(1,1)\right)$ with respect to $g_L$. Therefore,

$$[X_1,X_2]=X_3,[X_2,X_3]=0,[X_1,X_3]=X_2.$$

(116)

A non-zero vector $\mathbf{x}\in E_L^1(1,1)$ is called to be $spacelike$, $null$ or $timelike$ if $⟨\mathbf{x},\mathbf{x}⟩>0$, $⟨\mathbf{x},\mathbf{x}⟩=0$ or $⟨\mathbf{x},\mathbf{x}⟩<0$ respectively. We define the norm of the vector $\mathbf{x}\in E_L^1(1,1)$ by $\parallel \mathbf{x}\parallel =\sqrt{|⟨\mathbf{x},\mathbf{x}⟩|}$.

Let $\gamma :I\to E_L^1(1,1)$ be a regular curve, where I is an open interval in $\mathbb{R}$. The regular curve $\gamma$ is called a spacelike curve, timelike curve or null curve if ${\gamma }^{\prime }\left(t\right)$ is a spacelike vector, timelike vector or null vector at any $t\in I$, respectively.

We assume that ${\nabla }^L$ is the Levi–Civita connection on $E_L^1(1,1)$ with respect to $g_L$. Using the Koszul formula and (116), we have

Lemma 12.

We assume that$E_L^1(1,1)$is the Lorentzian group of rigid motions of the Minkowski plane. Thus,

$$\begin{array}{cc}\hfill & {\nabla }_{X_j}^L{X}_j=0,1\le j\le 3,{\nabla }_{X_1}^L{X}_2=\frac{L-1}{2L}{X}_3,{\nabla }_{X_2}^L{X}_1=\frac{-L-1}{2L}{X}_3,\hfill \\ \hfill & {\nabla }_{X_1}^L{X}_3=\frac{1-L}{2}{X}_2,{\nabla }_{X_3}^L{X}_1=\frac{-1-L}{2}{X}_2,{\nabla }_{X_2}^L{X}_3={\nabla }_{X_3}^L{X}_2=-\frac{1+L}{2}{X}_1.\hfill \end{array}$$

(117)

Definition 10.

Let$\gamma :I\to E_L^1(1,1)$be a$C^1$-smooth curve.$\gamma \left(t\right)$is called a horizontal point of γ when

$$\omega \left(\dot{\gamma }\left(t\right)\right)=-\frac{\sqrt{2}}{2}\left(e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)=0.$$

Similar to the Definitions 2 and 3, $k_{\gamma }^L$ and $k_{\gamma }^{\infty }$ for the Lorentzian group of rigid motions of the Minkowski plane can be defined, then we obtain

Lemma 13.

Suppose that$\gamma :I\to E_L^1(1,1)$is a$C^2$-smooth regular curve in the Riemannian manifold$E_L^1(1,1)$.

(1) If${\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }$is a spacelike vector, then

$$k_{\gamma }^{\infty }=\frac{\sqrt{-\frac{1}{2}{\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)}^2+{\dot{\gamma }}_3^2}}{\left|\omega \right(\dot{\gamma }\left(t\right)\left)\right|},if\omega \left(\dot{\gamma }\left(t\right)\right)\ne 0,$$

(118)

$$\begin{array}{cc}\hfill & k_{\gamma }^{\infty }=\left\{\frac{-{\ddot{\gamma }}_3^2+\frac{1}{2}{({\ddot{\gamma }}_2{e}^{{\gamma }_3}+{\dot{\gamma }}_2{\dot{\gamma }}_3{e}^{{\gamma }_3}-{\ddot{\gamma }}_1{e}^{-{\gamma }_3}+{\dot{\gamma }}_1{\dot{\gamma }}_3{e}^{-{\gamma }_3})}^2}{{\left[\frac{1}{2}{\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)}^2-{\dot{\gamma }}_3^2\right]}^2}\right.\hfill \\ \hfill & {\left.-\frac{{\left[-{\dot{\gamma }}_3{\ddot{\gamma }}_3+\frac{1}{2}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)({\ddot{\gamma }}_2{e}^{{\gamma }_3}+{\dot{\gamma }}_2{\dot{\gamma }}_3{e}^{{\gamma }_3}-{\ddot{\gamma }}_1{e}^{-{\gamma }_3}+{\dot{\gamma }}_1{\dot{\gamma }}_3{e}^{-{\gamma }_3})\right]}^2}{{\left[\frac{1}{2}{\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)}^2-{\dot{\gamma }}_3^2\right]}^3}\right\}}^{\frac{1}{2}}\hfill \\ \hfill & if\omega \left(\dot{\gamma }\left(t\right)\right)=0and\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)=0,\hfill \end{array}$$

(119)

$${\lim }_{L\to +\infty }\frac{k_{\gamma }^L}{\sqrt{L}}=\frac{|\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)|}{|\frac{1}{2}{\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)}^2-{\dot{\gamma }}_3^2|},if\omega \left(\dot{\gamma }\left(t\right)\right)=0and\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\ne 0.$$

(120)

(2) If${\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }$is a timelike vector, then

$$k_{\gamma }^{\infty }=\frac{\sqrt{\frac{1}{2}{\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)}^2-{\dot{\gamma }}_3^2}}{\left|\omega \right(\dot{\gamma }\left(t\right)\left)\right|},if\omega \left(\dot{\gamma }\left(t\right)\right)\ne 0,$$

(121)

$$\begin{array}{cc}\hfill & k_{\gamma }^{\infty }=\left\{-\frac{-{\ddot{\gamma }}_3^2+\frac{1}{2}{({\ddot{\gamma }}_2{e}^{{\gamma }_3}+{\dot{\gamma }}_2{\dot{\gamma }}_3{e}^{{\gamma }_3}-{\ddot{\gamma }}_1{e}^{-{\gamma }_3}+{\dot{\gamma }}_1{\dot{\gamma }}_3{e}^{-{\gamma }_3})}^2}{{\left[\frac{1}{2}{\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)}^2-{\dot{\gamma }}_3^2\right]}^2}\right.\hfill \\ \hfill & {\left.+\frac{{\left[-{\dot{\gamma }}_3{\ddot{\gamma }}_3+\frac{1}{2}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)({\ddot{\gamma }}_2{e}^{{\gamma }_3}+{\dot{\gamma }}_2{\dot{\gamma }}_3{e}^{{\gamma }_3}-{\ddot{\gamma }}_1{e}^{-{\gamma }_3}+{\dot{\gamma }}_1{\dot{\gamma }}_3{e}^{-{\gamma }_3})\right]}^2}{{\left[\frac{1}{2}{\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)}^2-{\dot{\gamma }}_3^2\right]}^3}\right\}}^{\frac{1}{2}}\hfill \\ \hfill & if\omega \left(\dot{\gamma }\left(t\right)\right)=0and\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)=0,\hfill \end{array}$$

(122)

Proof. 

Using (115), we get

$$\dot{\gamma }\left(t\right)={\dot{\gamma }}_3{X}_1+\frac{\sqrt{2}}{2}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)X_2+\omega \left(\dot{\gamma }\left(t\right)\right)X_3.$$

(123)

Combining Lemma 12 and (123), we obtain

$$\begin{array}{cc}\hfill & {\nabla }_{\dot{\gamma }}^L{X}_1=-\frac{L+1}{2}\omega \left(\dot{\gamma }\left(t\right)\right)X_2-\frac{\sqrt{2}}{2}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)\frac{L+1}{2L}{X}_3,\hfill \\ \hfill & {\nabla }_{\dot{\gamma }}^L{X}_2=-\frac{L+1}{2}\omega \left(\dot{\gamma }\left(t\right)\right)X_1+\frac{L-1}{2L}{\dot{\gamma }}_3{X}_3,\hfill \\ \hfill & {\nabla }_{\dot{\gamma }}^L{X}_3=-\frac{\sqrt{2}}{4}(L+1)\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)X_1+\frac{1-L}{2}{\dot{\gamma }}_3{X}_2.\hfill \end{array}$$

(124)

Combining (123) and (124), we have

$$\begin{array}{cc}\hfill {\nabla }_{\dot{\gamma }}^L\dot{\gamma }& =-\left[{\ddot{\gamma }}_3-\frac{\sqrt{2}}{2}(L+1)\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)\omega \left(\dot{\gamma }\left(t\right)\right)\right]X_1\hfill \\ \hfill & +\left[\frac{\sqrt{2}}{2}({\ddot{\gamma }}_2{e}^{{\gamma }_3}+{\dot{\gamma }}_2{\dot{\gamma }}_3{e}^{{\gamma }_3}-{\ddot{\gamma }}_1{e}^{-{\gamma }_3}+{\dot{\gamma }}_1{\dot{\gamma }}_3{e}^{-{\gamma }_3})-L\omega \left(\dot{\gamma }\left(t\right)\right){\dot{\gamma }}_3\right]X_2\hfill \\ \hfill & +\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)-\frac{\sqrt{2}}{2L}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right){\dot{\gamma }}_3\right]X_3\hfill \end{array}$$

(125)

By (123) and (125), if $\omega \left(\dot{\gamma }\left(t\right)\right)\ne 0$, then

$${⟨{\nabla }_{\dot{\gamma }}^L\dot{\gamma },{\nabla }_{\dot{\gamma }}^L\dot{\gamma }⟩}_L\sim \left[-\frac{1}{2}{\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)}^2+{\dot{\gamma }}_3^2\right]\omega {\left(\dot{\gamma }\left(t\right)\right)}^2{L}^2,asL\to +\infty ,$$

$${⟨\dot{\gamma },\dot{\gamma }⟩}_L\sim L\omega {\left(\dot{\gamma }\left(t\right)\right)}^2,asL\to +\infty ,$$

$${⟨{\nabla }_{\dot{\gamma }}^L\dot{\gamma },\dot{\gamma }⟩}_L^2\sim O\left(L^2\right)asL\to +\infty .$$

Thus,

$$\frac{{⟨{\nabla }_{\dot{\gamma }}^L\dot{\gamma },{\nabla }_{\dot{\gamma }}^L\dot{\gamma }⟩}_L}{\left|\right|\dot{\gamma }{\left|\right|}_L^4}\to \frac{-\frac{1}{2}{\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)}^2+{\dot{\gamma }}_3^2}{\omega {\left(\dot{\gamma }\left(t\right)\right)}^2},asL\to +\infty ,$$

$$\frac{{⟨{\nabla }_{\dot{\gamma }}^L\dot{\gamma },\dot{\gamma }⟩}_L^2}{\left|\right|\dot{\gamma }{\left|\right|}_L^6}\to 0,asL\to +\infty .$$

So using (6), we get (118) and (121). Combining (123), (125), (6) and $\omega \left(\dot{\gamma }\left(t\right)\right)=0$ and $\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)=0$, (119) and (122) hold. If $\omega \left(\dot{\gamma }\left(t\right)\right)=0$ and $\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\ne 0$, then

$${⟨{\nabla }_{\dot{\gamma }}^L\dot{\gamma },{\nabla }_{\dot{\gamma }}^L\dot{\gamma }⟩}_L\sim L{\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\right]}^2,asL\to +\infty ,$$

$${⟨\dot{\gamma },\dot{\gamma }⟩}_L=\frac{1}{2}{\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)}^2-{\dot{\gamma }}_3^2,$$

$${⟨{\nabla }_{\dot{\gamma }}^L\dot{\gamma },\dot{\gamma }⟩}_L^2=O\left(1\right)asL\to +\infty .$$

By (6), we get (120). □

6. Lorentzian Surfaces and a Gauss–Bonnet Theorem in the Lorentzian Group of Rigid Motions of the Minkowski Plane

For a regular surface ${\Sigma }_1\subset E_L^1(1,1)$ and regular curve $\gamma \subset {\Sigma }_1$, suppose that there is a $C^2$-smooth function $u:E_L^1(1,1)\to \mathbb{R}$ such that

$${\Sigma }_1=\{(x_1,x_2,x_3)\in E_L^1(1,1):u(x_1,x_2,x_3)=0\}.$$

Similar to Section 3, we define $p,q,r,l,l_L,\overline{p},\overline{q},\overline{p_L},\overline{q_L},\overline{r_L},v_L,e_1,e_2,J_L,k_{\gamma ,{\Sigma }_1}^L,k_{\gamma ,{\Sigma }_1}^{\infty }$, $k_{\gamma ,{\Sigma }_1}^{L,s},k_{\gamma ,{\Sigma }_1}^{\infty ,s}$. We call $\Sigma$ a Lorentzian surface in the Lorentzian group of rigid motions of the Minkowski plane.

By (22) and (125), we have

$$\begin{array}{cc}\hfill {\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma }& =-\left\{-\overline{q}\left[{\ddot{\gamma }}_3-\frac{\sqrt{2}}{2}(L+1)\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)\omega \left(\dot{\gamma }\left(t\right)\right)\right]\right.\hfill \\ \hfill & \left.-\overline{p}\left[\frac{\sqrt{2}}{2}({\ddot{\gamma }}_2{e}^{{\gamma }_3}+{\dot{\gamma }}_2{\dot{\gamma }}_3{e}^{{\gamma }_3}-{\ddot{\gamma }}_1{e}^{-{\gamma }_3}+{\dot{\gamma }}_1{\dot{\gamma }}_3{e}^{-{\gamma }_3})-L\omega \left(\dot{\gamma }\left(t\right)\right){\dot{\gamma }}_3\right]\right\}{e}_1\hfill \\ \hfill & +\left\{-\overline{r_L}\overline{p}\left[{\ddot{\gamma }}_3-\frac{\sqrt{2}}{2}(L+1)\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)\omega \left(\dot{\gamma }\left(t\right)\right)\right]\right.\hfill \\ \hfill & -\overline{r_L}\overline{q}\left[\frac{\sqrt{2}}{2}({\ddot{\gamma }}_2{e}^{{\gamma }_3}+{\dot{\gamma }}_2{\dot{\gamma }}_3{e}^{{\gamma }_3}-{\ddot{\gamma }}_1{e}^{-{\gamma }_3}+{\dot{\gamma }}_1{\dot{\gamma }}_3{e}^{-{\gamma }_3})-L\omega \left(\dot{\gamma }\left(t\right)\right){\dot{\gamma }}_3\right]\hfill \\ \hfill & \left.+\frac{l}{l_L}{L}^{\frac{1}{2}}\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)-\frac{\sqrt{2}}{2L}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right){\dot{\gamma }}_3\right]\right\}{e}_2\hfill \end{array}$$

(126)

Combining (123) and $\dot{\gamma }\left(t\right)\in T{\Sigma }_1$, we obtain

$$\dot{\gamma }\left(t\right)=\left[\overline{q}{\dot{\gamma }}_3+\frac{\sqrt{2}}{2}\overline{p}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)\right]e_1+\frac{l_L}{l}{L}^{\frac{1}{2}}\omega \left(\dot{\gamma }\left(t\right)\right)e_2.$$

(127)

We have

Lemma 14.

Let${\Sigma }_1\subset E_L^1(1,1)$be a regular Lorentzian surface. Let$\gamma :I\to {\Sigma }_1$be a$C^2$-smooth regular curve.

(1) If${\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }$is a timelike vector, then

$$k_{\gamma ,{\Sigma }_1}^{\infty }=\frac{\sqrt{\frac{1}{2}{\overline{q}}^2{\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)}^2+{\overline{p}}^2{\dot{{\gamma }_3}}^2}}{\left|\omega \right(\dot{\gamma }\left(t\right)\left)\right|},if\omega \left(\dot{\gamma }\left(t\right)\right)\ne 0,$$

(128)

$$k_{\gamma ,{\Sigma }_1}^{\infty }=0,if\omega \left(\dot{\gamma }\left(t\right)\right)=0,and\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)=0,$$

(2) If${\nabla }_{\dot{\gamma }}^{\Sigma ,L}\dot{\gamma }$is a spacelike vector, then

$${\lim }_{L\to +\infty }\frac{k_{\gamma ,{\Sigma }_1}^L}{\sqrt{L}}=\frac{|\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)|}{{\left[\overline{q}{\dot{\gamma }}_3+\frac{\sqrt{2}}{2}\overline{p}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)\right]}^2},if\omega \left(\dot{\gamma }\left(t\right)\right)=0and\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\ne 0.$$

(129)

Proof. 

Using (126), we get

$$\begin{array}{cc}\hfill {⟨{\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma },{\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma }⟩}_{L,{\Sigma }_1}& \sim -L^2\omega {\left(\dot{\gamma }\left(t\right)\right)}^2\left[\frac{1}{2}{\overline{q}}^2{\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)}^2+{\overline{p}}^2{\dot{{\gamma }_3}}^2\right],\hfill \\ \hfill & \mathrm{as}L\to +\infty .\hfill \end{array}$$

(130)

Using (127), if $\omega \left(\dot{\gamma }\left(t\right)\right)\ne 0$,

$${⟨\dot{\gamma },\dot{\gamma }⟩}_{{\Sigma }_1,L}\sim L{\left[\omega \left(\dot{\gamma }\left(t\right)\right)\right]}^2,\mathrm{as}L\to +\infty .$$

(131)

Combining (126) and (127), we get

$$\begin{array}{c}\hfill {⟨{\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma },\dot{\gamma }⟩}_{{\Sigma }_1,L}\sim M_{0}L,\end{array}$$

(132)

where $M_0$ does not depend on L. Using Definition 4 and (130)–(132), (128) holds. If $\omega \left(\dot{\gamma }\left(t\right)\right)=0$ and $\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)=0$, we obtain

$$\begin{array}{cc}\hfill {⟨{\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma },{\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma }⟩}_{L,{\Sigma }_1}& \sim -{\left[\overline{q}{\ddot{\gamma }}_3+\frac{\sqrt{2}}{2}\overline{p}({\ddot{\gamma }}_2{e}^{{\gamma }_3}+{\dot{\gamma }}_2{\dot{\gamma }}_3{e}^{{\gamma }_3}-{\ddot{\gamma }}_1{e}^{-{\gamma }_3}+{\dot{\gamma }}_1{\dot{\gamma }}_3{e}^{-{\gamma }_3})\right]}^2,\hfill \\ \hfill & \mathrm{as}L\to +\infty ,\hfill \end{array}$$

(133)

$${⟨\dot{\gamma },\dot{\gamma }⟩}_{{\Sigma }_1,L}=-{\left[\overline{q}{\dot{\gamma }}_3+\frac{\sqrt{2}}{2}\overline{p}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)\right]}^2,$$

(134)

$$\begin{array}{cc}\hfill {⟨{\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma },\dot{\gamma }⟩}_{{\Sigma }_1,L}=-& \left[\overline{q}{\ddot{\gamma }}_3+\frac{\sqrt{2}}{2}\overline{p}({\ddot{\gamma }}_2{e}^{{\gamma }_3}+{\dot{\gamma }}_2{\dot{\gamma }}_3{e}^{{\gamma }_3}-{\ddot{\gamma }}_1{e}^{-{\gamma }_3}+{\dot{\gamma }}_1{\dot{\gamma }}_3{e}^{-{\gamma }_3})\right]\hfill \\ \hfill & \times \left[\overline{q}{\dot{\gamma }}_3+\frac{\sqrt{2}}{2}\overline{p}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)\right].\hfill \end{array}$$

(135)

By (133)–(135) and Definition 4, we obtain $k_{\gamma ,{\Sigma }_1}^{\infty }=0$. If $\omega \left(\dot{\gamma }\left(t\right)\right)=0$ and $\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\ne 0$, then

$${⟨{\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma },{\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma }⟩}_{L,{\Sigma }_1}\sim L{\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\right]}^2,$$

$${⟨{\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma },\dot{\gamma }⟩}_{{\Sigma }_1,L}=O\left(1\right),$$

so we get (129). □

Lemma 15.

Let${\Sigma }_1\subset E_L^1(1,1)$be a regular Lorntzian surface.

(1) If$\gamma :I\to {\Sigma }_1$be a timelike$C^2$-smooth curve, then$\omega \left(\dot{\gamma }\left(t\right)\right)=0$and

$$\begin{array}{c}\hfill k_{\gamma ,{\Sigma }_1}^{\infty ,s}=0if\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)=0;\end{array}$$

(136)

$$\begin{array}{cc}\hfill {\lim }_{L\to +\infty }\frac{k_{\gamma ,{\Sigma }_1}^{L,s}}{\sqrt{L}}=& \frac{\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)}{{\left|\overline{q}{\dot{\gamma }}_3+\frac{\sqrt{2}}{2}\overline{p}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)\right|}^2},if\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\ne 0.\hfill \end{array}$$

(137)

(2) If$\gamma :[a,b]\to {\Sigma }_1$be a spacelike$C^2$-smooth curve, then$\omega \left(\dot{\gamma }\left(t\right)\right)\ne 0$and

$$k_{\gamma ,{\Sigma }_1}^{\infty ,s}=-\frac{\overline{p}{\dot{\gamma }}_3+\frac{\sqrt{2}}{2}\overline{q}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)}{\left|\omega \right(\dot{\gamma }\left(t\right)\left)\right|}.$$

(138)

Proof. 

For (1), by (22) and (127), we have

$$J_L\left(\dot{\gamma }\right)=\frac{l_L}{l}{L}^{\frac{1}{2}}\omega \left(\dot{\gamma }\left(t\right)\right)e_1+\left[\overline{q}{\dot{\gamma }}_3+\frac{\sqrt{2}}{2}\overline{p}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)\right]e_2.$$

(139)

Combining (126) and (139), we have

$$\begin{array}{c}\hfill {⟨{\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma },J_L\left(\dot{\gamma }\right)⟩}_{L,{\Sigma }_1}\sim L^{\frac{3}{2}}\omega {\left(\dot{\gamma }\left(t\right)\right)}^2\left[\overline{p}{\dot{\gamma }}_3+\frac{\sqrt{2}}{2}\overline{q}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)\right],\mathrm{as}L\to +\infty .\end{array}$$

(140)

If $\omega \left(\dot{\gamma }\left(t\right)\right)=0$ and $\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)=0,$ then

$$\begin{array}{c}\hfill {⟨{\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma },J_L\left(\dot{\gamma }\right)⟩}_{L,{\Sigma }_1}\sim M_0{L}^{-\frac{1}{2}}\mathrm{as}L\to +\infty .\end{array}$$

(141)

So $k_{\gamma ,{\Sigma }_1}^{\infty ,s}=0.$ If $\omega \left(\dot{\gamma }\left(t\right)\right)=0$ and $\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\ne 0$, then

$$\begin{array}{c}\hfill {⟨{\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma },J_L\left(\dot{\gamma }\right)⟩}_{L,{\Sigma }_1}\sim L^{\frac{1}{2}}\left[\overline{q}{\dot{\gamma }}_3+\frac{\sqrt{2}}{2}\overline{p}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)\right]\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right),\mathrm{as}L\to +\infty .\end{array}$$

(142)

Therefore, (137) holds. □

In the Lorentzian group of rigid motions of the Minkowski plane, the sub-Lorentzian limit of the Riemannian Gaussian curvature of surfaces is computed nextly. Similarly to Theorem 4.3 in [6], we obtain

Theorem 5.

The second fundamental form$I{I}_1^L$of the embedding of${\Sigma }_1$into$E_L^1(1,1)$is given by

$$I{I}_1^L=\left(\begin{array}{cc}{h}_{11},& h_{12}\\ h_{21},& h_{22}\end{array}\right),$$

(143)

where

$$h_{11}=\frac{l}{l_L}[X_1\left(\overline{p}\right)-X_2\left(\overline{q}\right)]-\overline{p}\overline{q}\overline{r_L}{L}^{-\frac{1}{2}},$$

$$h_{12}=h_{21}=\frac{l_L}{l}{⟨e_1,{\nabla }_H\left(\overline{r_L}\right)⟩}_L+\frac{\sqrt{L}}{2}+\frac{1}{2\sqrt{L}}({\overline{q_L}}^2-{\overline{p_L}}^2)-\frac{1}{2\sqrt{L}}{\overline{r_L}}^2,$$

$$h_{22}=\frac{l^2}{l_L^2}{⟨e_2,{\nabla }_H\left(\frac{r}{l}\right)⟩}_L+\widetilde{X_3}\left(\overline{r_L}\right)-\overline{p_L}\overline{q_L}\overline{r_L}{L}^{-\frac{1}{2}}-\overline{p}\overline{q}{\overline{r_L}}^3{L}^{-\frac{1}{2}}.$$

Proof. 

Combining

$$\begin{array}{c}\hfill v_L=-\overline{p_L}{X}_1+\overline{q_L}{X}_2+\overline{r_L}\widetilde{X_3},e_1=\overline{q}{X}_1-\overline{p}{X}_2,e_2=\overline{r_L}\overline{p}{X}_1-\overline{r_L}\overline{q}{X}_2+\frac{l}{l_L}\widetilde{X_3},\end{array}$$

and ${⟨{\nabla }_{e_i}^L{v}_L,e_j⟩}_L=-{⟨{\nabla }_{e_i}^L{e}_j,v_L⟩}_L,i=1,2;j=1,2.$ By direct calculation, we obtain

$${\nabla }_{e_1}^L{e}_1=[\overline{q}{X}_1\left(\overline{q}\right)-\overline{p}{X}_2\left(\overline{q}\right)]X_1-[\overline{q}{X}_1\left(\overline{p}\right)-\overline{p}{X}_2\left(\overline{p}\right)]X_2+\frac{\overline{p}\overline{q}}{L}{X}_3.$$

Since ${\overline{p}}^2-{\overline{q}}^2=-1$, we get $\overline{p}{X}_i\left(\overline{p}\right)-\overline{q}{X}_i\left(\overline{q}\right)=0,i=1,2$. Then

$${⟨{\nabla }_{e_1}^L{e}_1,v_L⟩}_L={⟨{\nabla }_{(\overline{q}{X}_1-\overline{p}{X}_2)}^L(\overline{q}{X}_1-\overline{p}{X}_2),-\overline{p_L}{X}_1+\overline{q_L}{X}_2+\overline{r_L}\widetilde{X_3}⟩}_L.$$

Then,

$${⟨{\nabla }_{e_1}^L{v}_L,e_1⟩}_L=\frac{l}{l_L}[X_1\left(\overline{p}\right)-X_2\left(\overline{q}\right)]-\overline{p}\overline{q}\overline{r_L}{L}^{-\frac{1}{2}}.$$

Similarly, we have

$${⟨{\nabla }_{e_1}^L{v}_L,e_2⟩}_L=-{⟨{\nabla }_{e_1}^L{e}_2,v_L⟩}_L=h_{21}=\frac{l_L}{l}{⟨e_1,{\nabla }_H\left(\overline{r_L}\right)⟩}_L+\frac{\sqrt{L}}{2}+\frac{1}{2\sqrt{L}}({\overline{q_L}}^2-{\overline{p_L}}^2)-\frac{1}{2\sqrt{L}}{\overline{r_L}}^2;$$

$${⟨{\nabla }_{e_2}^L{v}_L,e_1⟩}_L=-{⟨{\nabla }_{e_2}^L{e}_1,v_L⟩}_L=h_{21}=\frac{l_L}{l}{⟨e_1,{\nabla }_H\left(\overline{r_L}\right)⟩}_L+\frac{\sqrt{L}}{2}+\frac{1}{2\sqrt{L}}({\overline{q_L}}^2-{\overline{p_L}}^2)-\frac{1}{2\sqrt{L}}{\overline{r_L}}^2;$$

$${⟨{\nabla }_{e_2}^L{v}_L,e_2⟩}_L=-{⟨{\nabla }_{e_2}^L{e}_2,v_L⟩}_L=\frac{l^2}{l_L^2}{⟨e_2,{\nabla }_H\left(\frac{r}{l}\right)⟩}_L+\widetilde{X_3}\left(\overline{r_L}\right)-\overline{p_L}\overline{q_L}\overline{r_L}{L}^{-\frac{1}{2}}-\overline{p}\overline{q}{\overline{r_L}}^3{L}^{-\frac{1}{2}}.$$

Thus, (143) holds. □

Similar to Proposition 1, we have

Proposition 7.

The horizontal mean curvature${\mathcal{H}}_{\infty }^1$of${\Sigma }_1\subset E(1,1)$away from characteristic points is

$${\mathcal{H}}_{\infty }^1=X_1\left(\overline{p}\right)-X_2\left(\overline{q}\right).$$

(144)

By Lemma 12, we have

Lemma 16.

We assume that$E_L^1(1,1)$is the Lorentzian group of rigid motions of the Minkowski plane. Then

$$\begin{array}{cc}\hfill & R^L(X_1,X_2)X_1=(\frac{1}{2}-\frac{1}{4L}+\frac{3L}{4})X_2,R^L(X_1,X_2)X_2=(\frac{1}{2}-\frac{1}{4L}+\frac{3L}{4})X_1,\hfill \\ \hfill & R^L(X_1,X_2)X_3=0,R^L(X_1,X_3)X_1=(\frac{1}{2}-\frac{L}{4}+\frac{3}{4L})X_3,\hfill \\ \hfill & R^L(X_1,X_3)X_2=0,R^L(X_1,X_3)X_3=(-\frac{L^2}{4}+\frac{L}{2}+\frac{3}{4})X_1,\hfill \\ \hfill & R^L(X_2,X_3)X_1=0,R^L(X_2,X_3)X_2=(\frac{1}{2}+\frac{1}{4L}+\frac{L}{4})X_3,\hfill \\ \hfill & R^L(X_2,X_3)X_3=-(\frac{1}{4}+\frac{L^2}{4}+\frac{L}{2})X_2.\hfill \end{array}$$

(145)

Proposition 8.

Away from characteristic points, we have

$${\mathcal{K}}^{{\Sigma }_1,\infty }(e_1,e_2)=-⟨e_1,{\nabla }_H\left(\frac{X_{3}u}{|{\nabla }_{H}u|}\right)⟩-\frac{{\left(X_{3}u\right)}^2}{l^2}-\frac{{\left(X_{2}u\right)}^2}{l^2}.$$

(146)

Proof. 

Combining (54) and Lemma 16, we have

$$\begin{array}{cc}\hfill {\mathcal{K}}^{E_L^1(1,1)}(e_1,e_2)=& -{\overline{r_L}}^2(\frac{1}{2}-\frac{1}{4L}+\frac{3L}{4})-{\left(\frac{l}{l_L}\overline{q}\right)}^2(\frac{1}{2}-\frac{L}{4}+\frac{3}{4L})\hfill \\ \hfill & -{\left(\frac{l}{l_L}\overline{p}\right)}^2(\frac{1}{2}+\frac{1}{4L}+\frac{L}{4})\hfill \\ \hfill & \sim {\left(\frac{l}{l_L}\right)}^2\frac{L}{4}-\frac{3}{4}\frac{{\left(X_{3}u\right)}^2}{l^2}-{\overline{q}}^2+\frac{1}{2},\mathrm{as}L\to +\infty .\hfill \end{array}$$

(147)

Similar to (58), we obtain

$$\det \left(I{I}_1^L\right)=-\frac{L}{4}-⟨e_1,{\nabla }_H\left(\frac{X_{3}u}{|{\nabla }_{H}u|}\right)⟩-\frac{1}{2}+O\left(L^{-\frac{1}{2}}\right)\mathrm{as}L\to +\infty .$$

(148)

Combining (147) and (148), (146) holds. □

Similar to (59) and (62), for the Lorentzian group of rigid motions of the Minkowski plane. We assume that $\gamma :I\to E_L^1(1,1)$ is a spacelike $C^2$-smooth curve, then we get

$${\lim }_{L\to +\infty }\frac{1}{\sqrt{L}}d{s}_L=ds,{\lim }_{L\to +\infty }\frac{1}{\sqrt{L}}d{\sigma }_{{\Sigma }_1,L}=d{\sigma }_{{\Sigma }_1}.$$

(149)

Combining (146), (149) and Lemma 15, similar to the proof of Theorem 2, we have

Theorem 6.

(Gauss–Bonnet theorem for the Lorentzian surface in$E_L^1(1,1)$) We assume that${\Sigma }_1\subset E_L^1(1,1)$is a regular Lorentzian surface. Let${\Sigma }_1$be with finitely many boundary components${(\partial {\Sigma }_1)}_i,$$i\in \{1,\cdots ,n\}$, given by$C^2$-smooth regular and closed spacelike curves${\gamma }_i:[0,2\pi ]\to {(\partial {\Sigma }_1)}_i$. Let the characteristic set$C\left({\Sigma }_1\right)$satisfy${\mathcal{H}}^1\left(C\left({\Sigma }_1\right)\right)=0$and$\left|\right|{\nabla }_{H}u{\left|\right|}_H^{-1}$be locally summable with respect to the 2-dimensional Hausdorff measure near the characteristic set$C\left({\Sigma }_1\right)$, then we get

$${\int }_{{\Sigma }_1}{\mathcal{K}}^{{\Sigma }_1,\infty }d{\sigma }_{{\Sigma }_1}+\sum _{i=1}^n{\int }_{{\gamma }_i}{k}_{{\gamma }_i,{\Sigma }_1}^{\infty ,s}ds=0.$$

(150)

7. Spacelike Surfaces and a Gauss–Bonnet Theorem in the Lorentzian Group of Rigid Motions of the Minkowski Plane

For a regular surface ${\Sigma }_1\subset E_L^1(1,1)$ and regular curve $\gamma \subset {\Sigma }_1$, suppose that there is a $C^2$-smooth function $u:E_L^1(1,1)\to \mathbb{R}$ such that

$${\Sigma }_1=\{(x_1,x_2,x_3)\in E_L^1(1,1):u(x_1,x_2,x_3)=0\}.$$

(151)

Similar to Section 4, we give $p,q,r,l,l_L,\overline{p},\overline{q},\overline{p_L},\overline{q_L},\overline{r_L},v_L,e_1,e_2,J_L,k_{\gamma ,{\Sigma }_1}^L,k_{\gamma ,{\Sigma }_1}^{\infty }$, $k_{\gamma ,{\Sigma }_1}^{L,s},k_{\gamma ,{\Sigma }_1}^{\infty ,s}$. We call ${\Sigma }_1$ a spacelike surface in the Lorentzian group of rigid motions of the Minkowski plane. By (125), we have

$$\begin{array}{cc}\hfill {\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma }& =\left\{-\overline{q}\left[{\ddot{\gamma }}_3-\frac{\sqrt{2}}{2}(L+1)\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)\omega \left(\dot{\gamma }\left(t\right)\right)\right]\right.\hfill \\ \hfill & \left.-\overline{p}\left[\frac{\sqrt{2}}{2}({\ddot{\gamma }}_2{e}^{{\gamma }_3}+{\dot{\gamma }}_2{\dot{\gamma }}_3{e}^{{\gamma }_3}-{\ddot{\gamma }}_1{e}^{-{\gamma }_3}+{\dot{\gamma }}_1{\dot{\gamma }}_3{e}^{-{\gamma }_3})-L\omega \left(\dot{\gamma }\left(t\right)\right){\dot{\gamma }}_3\right]\right\}{e}_1\hfill \\ \hfill & +\left\{-\overline{r_L}\overline{p}\left[{\ddot{\gamma }}_3-\frac{\sqrt{2}}{2}(L+1)\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)\omega \left(\dot{\gamma }\left(t\right)\right)\right]\right.\hfill \\ \hfill & -\overline{r_L}\overline{q}\left[\frac{\sqrt{2}}{2}({\ddot{\gamma }}_2{e}^{{\gamma }_3}+{\dot{\gamma }}_2{\dot{\gamma }}_3{e}^{{\gamma }_3}-{\ddot{\gamma }}_1{e}^{-{\gamma }_3}+{\dot{\gamma }}_1{\dot{\gamma }}_3{e}^{-{\gamma }_3})-L\omega \left(\dot{\gamma }\left(t\right)\right){\dot{\gamma }}_3\right]\hfill \\ \hfill & \left.-\frac{l}{l_L}{L}^{\frac{1}{2}}\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)-\frac{\sqrt{2}}{2L}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right){\dot{\gamma }}_3\right]\right\}{e}_2\hfill \end{array}$$

(152)

By (121) and $\dot{\gamma }\left(t\right)\in T{\Sigma }_1$, we have

$$\dot{\gamma }\left(t\right)=-\left[\overline{q}{\dot{\gamma }}_3+\frac{\sqrt{2}}{2}\overline{p}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)\right]e_1-\frac{l_L}{l}{L}^{\frac{1}{2}}\omega \left(\dot{\gamma }\left(t\right)\right)e_2.$$

(153)

We have

Lemma 17.

Let${\Sigma }_1\subset E_L^1(1,1)$be a regular Lorentzian surface. Let$\gamma :I\to {\Sigma }_1$be a$C^2$-smooth spacelike curve, then

$$k_{\gamma ,{\Sigma }_1}^{\infty }=\frac{\sqrt{\frac{1}{2}{\overline{q}}^2{\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)}^2+{\overline{p}}^2{\dot{{\gamma }_3}}^2}}{\left|\omega \right(\dot{\gamma }\left(t\right)\left)\right|},if\omega \left(\dot{\gamma }\left(t\right)\right)\ne 0,$$

(154)

$$k_{\gamma ,{\Sigma }_1}^{\infty }=0,if\omega \left(\dot{\gamma }\left(t\right)\right)=0,and\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)=0,$$

$${\lim }_{L\to +\infty }\frac{k_{\gamma ,{\Sigma }_1}^L}{\sqrt{L}}=\frac{|\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)|}{{\left[\overline{q}{\dot{\gamma }}_3+\frac{\sqrt{2}}{2}\overline{p}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)\right]}^2},if\omega \left(\dot{\gamma }\left(t\right)\right)=0and\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\ne 0.$$

(155)

Proof. 

Using (152), we obtain

$$\begin{array}{c}\hfill {⟨{\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma },{\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma }⟩}_{L,{\Sigma }_1}\sim L^2\omega {\left(\dot{\gamma }\left(t\right)\right)}^2\left[\frac{1}{2}{\overline{q}}^2{\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)}^2+{\overline{p}}^2{\dot{{\gamma }_3}}^2\right],\mathrm{as}L\to +\infty .\end{array}$$

(156)

Using (153), if $\omega \left(\dot{\gamma }\left(t\right)\right)\ne 0$,

$${⟨\dot{\gamma },\dot{\gamma }⟩}_{{\Sigma }_1,L}\sim L{\left[\omega \left(\dot{\gamma }\left(t\right)\right)\right]}^2,\mathrm{as}L\to +\infty .$$

(157)

By (152) and (153), we have

$$\begin{array}{c}\hfill {⟨{\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma },\dot{\gamma }⟩}_{{\Sigma }_1,L}\sim M_{0}L,\end{array}$$

(158)

where $M_0$ does not depend on L. By Definition 4, (156)–(158), we have (154). If $\omega \left(\dot{\gamma }\left(t\right)\right)=0$ and $\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)=0$, then

$$\begin{array}{c}\hfill {⟨{\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma },{\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma }⟩}_{L,{\Sigma }_1}\sim {\left[-\overline{q}{\ddot{\gamma }}_3-\frac{\sqrt{2}}{2}\overline{p}({\ddot{\gamma }}_2{e}^{{\gamma }_3}+{\dot{\gamma }}_2{\dot{\gamma }}_3{e}^{{\gamma }_3}-{\ddot{\gamma }}_1{e}^{-{\gamma }_3}+{\dot{\gamma }}_1{\dot{\gamma }}_3{e}^{-{\gamma }_3})\right]}^2,\mathrm{as}L\to +\infty ,\end{array}$$

(159)

and

$${⟨\dot{\gamma },\dot{\gamma }⟩}_{{\Sigma }_1,L}={\left[\overline{q}{\dot{\gamma }}_3+\frac{\sqrt{2}}{2}\overline{p}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)\right]}^2,$$

(160)

$$\begin{array}{cc}\hfill {⟨{\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma },\dot{\gamma }⟩}_{{\Sigma }_1,L}=& \left[\overline{q}{\ddot{\gamma }}_3+\frac{\sqrt{2}}{2}\overline{p}({\ddot{\gamma }}_2{e}^{{\gamma }_3}+{\dot{\gamma }}_2{\dot{\gamma }}_3{e}^{{\gamma }_3}-{\ddot{\gamma }}_1{e}^{-{\gamma }_3}+{\dot{\gamma }}_1{\dot{\gamma }}_3{e}^{-{\gamma }_3})\right]\hfill \\ \hfill & ·\left[\overline{q}{\dot{\gamma }}_3+\frac{\sqrt{2}}{2}\overline{p}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)\right].\hfill \end{array}$$

(161)

By (159)–(161) and Definition 4, we obtain $k_{\gamma ,{\Sigma }_1}^{\infty }=0$. If $\omega \left(\dot{\gamma }\left(t\right)\right)=0$ and $\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\ne 0$, then

$${⟨{\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma },{\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma }⟩}_{L,{\Sigma }_1}\sim L{\left[\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\right]}^2,$$

$${⟨{\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma },\dot{\gamma }⟩}_{{\Sigma }_1,L}=O\left(1\right),$$

so (155) holds. □

Lemma 18.

Let${\Sigma }_1\subset E_L^1(1,1)$be a regular Lorentzian surface. Let$\gamma :I\to {\Sigma }_1$be a spacelike$C^2$-smooth regular curve, then

$$k_{\gamma ,{\Sigma }_1}^{\infty ,s}=0if\omega \left(\dot{\gamma }\left(t\right)\right)=0,and\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)=0,$$

$$\begin{array}{cc}\hfill {\lim }_{L\to +\infty }\frac{k_{\gamma ,{\Sigma }_1}^{L,s}}{\sqrt{L}}=& \frac{\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)}{{\left|\overline{q}{\dot{\gamma }}_3+\frac{\sqrt{2}}{2}\overline{p}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)\right|}^2},if\omega \left(\dot{\gamma }\left(t\right)\right)=0and\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\ne 0.\hfill \end{array}$$

(162)

$$k_{\gamma ,{\Sigma }_1}^{\infty ,s}=\frac{\overline{p}{\dot{\gamma }}_3+\frac{\sqrt{2}}{2}\overline{q}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)}{\left|\omega \right(\dot{\gamma }\left(t\right)\left)\right|},if\omega \left(\dot{\gamma }\left(t\right)\right)\ne 0,$$

(163)

Proof. 

By (75) and (153), we have

$$J_L\left(\dot{\gamma }\right)=\frac{l_L}{l}{L}^{\frac{1}{2}}\omega \left(\dot{\gamma }\left(t\right)\right)e_1-\left[\overline{q}{\dot{\gamma }}_3+\frac{\sqrt{2}}{2}\overline{p}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)\right]e_2.$$

(164)

Combining (152) and (164), we have

$$\begin{array}{c}\hfill {⟨{\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma },J_L\left(\dot{\gamma }\right)⟩}_{L,{\Sigma }_1}\sim L^{\frac{3}{2}}\omega {\left(\dot{\gamma }\left(t\right)\right)}^2\left[\overline{p}{\dot{\gamma }}_3+\frac{\sqrt{2}}{2}\overline{q}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)\right],\mathrm{as}L\to +\infty .\end{array}$$

(165)

Therefore, we get (163). If $\omega \left(\dot{\gamma }\left(t\right)\right)=0$ and $\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)=0,$ then

$$\begin{array}{c}\hfill {⟨{\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma },J_L\left(\dot{\gamma }\right)⟩}_{L,{\Sigma }_1}\sim M_0{L}^{-\frac{1}{2}}\mathrm{as}L\to +\infty .\end{array}$$

(166)

So $k_{\gamma ,{\Sigma }_1}^{\infty ,s}=0.$ If $\omega \left(\dot{\gamma }\left(t\right)\right)=0$ and $\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right)\ne 0$, then

$$\begin{array}{c}\hfill {⟨{\nabla }_{\dot{\gamma }}^{{\Sigma }_1,L}\dot{\gamma },J_L\left(\dot{\gamma }\right)⟩}_{L,{\Sigma }_1}\sim L^{\frac{1}{2}}\left[\overline{q}{\dot{\gamma }}_3+\frac{\sqrt{2}}{2}\overline{p}\left(-e^{-{\gamma }_3}\dot{{\gamma }_1}+e^{{\gamma }_3}\dot{{\gamma }_2}\right)\right]\frac{d}{dt}\left(\omega \left(\dot{\gamma }\left(t\right)\right)\right),\mathrm{as}L\to +\infty .\end{array}$$

(167)

So (162) holds. □

In the Lorentzian group of rigid motions of the Minkowski plane, the sub-Lorentzian limit of the Riemannian Gaussian curvature of surfaces is computed. Similarly to Theorem 4.3 in [6], we have

Theorem 7.

For the embedding of${\Sigma }_1$into$(E(1,1),g_L)$, the second fundamental form$I{I}_1^L$of${\Sigma }_1$is

$$I{I}_1^L=\left(\begin{array}{cc}{h}_{11},& h_{12}\\ h_{21},& h_{22}\end{array}\right),$$

(168)

where

$$h_{11}=-\frac{l}{l_L}[X_1\left(\overline{p}\right)-X_2\left(\overline{q}\right)]-\overline{p}\overline{q}\overline{r_L}{L}^{-\frac{1}{2}},$$

$$h_{12}=h_{21}=-\frac{l_L}{l}{⟨e_1,{\nabla }_H\left(\overline{r_L}\right)⟩}_L+\frac{\sqrt{L}}{2}-\frac{1}{2\sqrt{L}}({\overline{q_L}}^2-{\overline{p_L}}^2)+\frac{1}{2\sqrt{L}}{\overline{r_L}}^2,$$

$$h_{22}=-\frac{l^2}{l_L^2}{⟨e_2,{\nabla }_H\left(\frac{r}{l}\right)⟩}_L+\widetilde{X_3}\left(\overline{r_L}\right)+\overline{p_L}\overline{q_L}\overline{r_L}{L}^{-\frac{1}{2}}-\overline{p}\overline{q}{\overline{r_L}}^3{L}^{-\frac{1}{2}}.$$

Similar to Proposition 1, we have

Proposition 9.

The horizontal mean curvature${\mathcal{H}}_{\infty }^1$of${\Sigma }_1\subset E(1,1)$away from characteristic points is

$${\mathcal{H}}_{\infty }^1=-X_1\left(\overline{p}\right)+X_2\left(\overline{q}\right).$$

(169)

Proposition 10.

Away from characteristic points, we have

$${\mathcal{K}}^{{\Sigma }_1,\infty }(e_1,e_2)=-⟨e_1,{\nabla }_H\left(\frac{X_{3}u}{|{\nabla }_{H}u|}\right)⟩-\frac{{\left(X_{3}u\right)}^2}{l^2}-{\overline{q}}^2.$$

(170)

Proof. 

By (103) and Lemma 16, we have

$$\begin{array}{cc}\hfill {\mathcal{K}}^{E(1,1),L}(e_1,e_2)=& -{\overline{r_L}}^2(\frac{1}{2}-\frac{1}{4L}+\frac{3L}{4})-{\left(\frac{l}{l_L}\overline{q}\right)}^2(\frac{1}{2}-\frac{L}{4}+\frac{3}{4L})\hfill \\ \hfill & -{\left(\frac{l}{l_L}\overline{p}\right)}^2(\frac{1}{2}+\frac{1}{4L}+\frac{L}{4})\hfill \\ \hfill & \sim -{\left(\frac{l}{l_L}\right)}^2\frac{L}{4}-\frac{3L}{4}\frac{{\left(X_{3}u\right)}^2}{l^2}-{\overline{q}}^2-\frac{1}{2},\mathrm{as}L\to +\infty .\hfill \end{array}$$

(171)

Similar to (148), we have

$$\det \left(I{I}_1^L\right)=-\frac{L}{4}+⟨e_1,{\nabla }_H\left(\frac{X_{3}u}{|{\nabla }_{H}u|}\right)⟩-\frac{1}{2}+O\left(L^{-\frac{1}{2}}\right)\mathrm{as}L\to +\infty .$$

(172)

Combining (171) and (172), (170) holds. □

Similar to (61) and (111), for the Lorentzian group of rigid motions of the Minkowski plane, we assume that $\gamma :I\to E_L^1(1,1)$ is a spacelike $C^2$-smooth curve, then we obtain

$${\lim }_{L\to +\infty }\frac{1}{\sqrt{L}}d{s}_L=ds,{\lim }_{L\to +\infty }\frac{1}{\sqrt{L}}d{\sigma }_{{\Sigma }_1,L}=d{\sigma }_{{\Sigma }_1}.$$

(173)

Combining (170), (173) and Lemma 18, similar to the proof of Theorem 4, we have

Theorem 8.

(Gauss–Bonnet theorem for the spacelike surface in$E_L^1(1,1)$) We assume that${\Sigma }_1\subset {\mathbb{H}}_L^1$is a regular spacelike surface. Let${\Sigma }_1$be with finitely many boundary components${(\partial {\Sigma }_1)}_i,$$i\in \{1,\cdots ,n\}$, given by$C^2$-smooth regular and closed spacelike curves${\gamma }_i:[0,2\pi ]\to {(\partial {\Sigma }_1)}_i$. Let the characteristic set$C\left({\Sigma }_1\right)$be the empty set. Suppose that$d{\sigma }_{{\Sigma }_1}$is defined by (106) and$ds$is defined by (57) and$k_{{\gamma }_i,{\Sigma }_1}^{\infty ,s}$is the sub-Lorentzian signed geodesic curvature of${\gamma }_i$relative to${\Sigma }_1$.

$${\int }_{\Sigma }Ad{\sigma }_{{\Sigma }_1}+\sum _{i=1}^n{\int }_{{\gamma }_i}{k}_{{\gamma }_i,{\Sigma }_1}^{\infty ,s}ds=0.$$

(174)

where

$$A=-{\overline{q}}^2-⟨e_1,{\nabla }_H\left(\frac{X_{3}u}{|{\nabla }_{H}u|}\right)⟩-\frac{{\left(X_{3}u\right)}^2}{l^2}.$$

8. Conclusions

This work has addressed the interesting question of Gauss–Bonnet type theorems in other spaces, especially in Lorentzian Heisenberg group and the Lorentzian group of rigid motions of the Minkowski plane. In Lorentzian Heisenberg group, the main results of this paper are Theorem 2, Theorem 4, which are Gauss–Bonnet type theorem for the Lorentzian surfaces and Gauss–Bonnet type theorem for the spacelike surfaces, respectively.

On the other hand, in the Lorentzian group of rigid motions of the Minkowski plane, the main results of this paper are Theorem 6 and Theorem 8, which are Gauss–Bonnet type theorem for the Lorentzian surfaces and Gauss–Bonnet type theorem for the spacelike surfaces, respectively.