Lorentzian approximations for a Lorentzian $\alpha$-Sasakian manifold and Gauss-Bonnet theorems

1.   Introduction

Lorentzian $\alpha$-Sasakian manifolds were introduced by Yildiz and Murathan in ^[1]. Then, many researchers began to study properties of $\alpha$-Sasakian manifolds, such as second order parallel tensors ^[2], pseudosymmetric Lorentzian $\alpha$-Sasakian manifolds ^[3], some special classes of Lorentzian $\alpha$-Sasakian manifolds ^[4,5,6], certain derivations ^[7], Ricci solitons ^[8,9], Lorentzian $\alpha$-Sasakian manifolds admitting a quarter-symmetric metric connection ^[10], semi-symmetry type $\alpha$-Sasakian manifolds ^[11] and ${\mathcal {M}}$-projectively semi-symmetric Lorentzian $\alpha$-Sasakian manifolds ^[12]. Recently, Wang studied Gauss-Bonnet theorems in the BCV spaces and the twisted Heisenberg group ^[13], the affine group and the group of rigid motions of the Minkowski plane ^[14] by using the method of Riemannian approximations which first took by Balogh, Tyson and Vecchi to prove a Heisenberg version of the Gauss-Bonnet theorem^[15,16]. Riemannian approximations can be extended to the case for any Lie group equipped with left-invariant Lorentzian metric $g$, named Lorentzian approximations. Some typical works of Lorentzian approximations in a Lorentzian Heisenberg group are obtained in ^[17,18]. Inspired by the above work, we proved Gauss-Bonnet theorems in the rototranslation group ^[19,20], Lorentzian Sasakian space forms ^[21] and the group of rigid motions of the Minkowski plane with the general left-invariant metric ^[22]. However, very little is known about the Gauss-Bonnet theorem in $3$-dimensional Lorentzian $\alpha$-Sasakian manifolds. This paper attempts to solve this question by employing the method of the Lorentzian approximation scheme.

We restrict our attention to Lorentzian $\alpha$-Sasakian manifolds. As we know, in ^[8], a differentiable manifold of dimension $(2n+1)$ is called a Lorentzian $\alpha$-Sasakian manifold if it admits a $(1, 1)$ tensor field $\phi$, a vector field $\xi$, $1$-form $\eta$ and Lorentzian metric $g$ which satisfy on $M$, respectively, that

where $\nabla$ denotes the operator of covariant differentiation with respect to the Lorentzian metric $g$ on $M$. Meanwhile, a Lorentzian $\alpha$-Sasakian model of $3$-dimensional Lorentzian $\alpha$-Sasakian manifolds was constructed in ^[8]. In this paper, we focus on Gauss-Bonnet theorems for the Lorentzian surfaces and spacelike surfaces in the Lorentzian $\alpha$-Sasakian model. We define the notions of the intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on Lorentzian surfaces and spacelike surfaces and the intrinsic Gaussian curvature of Lorentzian surfaces and spacelike surfaces away from characteristic points. Furthermore, we derive the expressions of those curvatures and prove Gauss-Bonnet theorems for the Lorentzian surfaces and spacelike surfaces in the $3$-dimensional Lorentzian $\alpha$-Sasakian manifold.

The paper is organized in the following way. Basic notions on $(S_\alpha, g)$ and the Lorentzian approximants $(S_\alpha, g_L)$ of the $\alpha$-Sasakian manifold are given in Section 2. The sub-Lorentzian limit of curvature of curves in $(S_\alpha, g_L)$ will be computed. In Sections 3 and 4, we compute sub-Lorentzian limits of geodesic curvature of curves on Lorentzian surfaces and the intrinsic Gaussian curvature of Lorentzian surfaces in $(S_\alpha, g_L)$. In Section 5, we prove the Gauss-Bonnet theorem for Lorentzian surfaces. In Section 6, we prove the Gauss-Bonnet theorem for spacelike surfaces. Finally, we summarize the conclusions and add an appendix section on length measure and surface measure.

2.   Lorentzian approximants and curvature tensor

In this section, some basic notions on a Lorentzian $\alpha$-Sasakian manifold will be introduced. First, we recall the Lorentzian $\alpha$-Sasakian model of $3$-dimensional Lorentzian $\alpha$-Sasakian manifolds constructed in ^[8]. Let $\alpha$ be some constant, and set $S_\alpha = \{(x, y, z)\in \mathbb{R}^{3}|z > 0\}$ equipped with a Lorentzian metric

Then, $(S_\alpha, g)$ was called the Lorentzian $\alpha$-Sasakian model of $3$-dimensional Lorentzian $\alpha$-Sasakian manifold, where $(x, y, z)$ are the standard coordinates of $\mathbb{R}^{3}$. Let $E_1, E_2$ and $E_3$ be the vector fields on $S_\alpha$ given by

which are linearly independent at each point $p$ of $S_\alpha$. Then,

and $span\left\{ {{E_1}, {E_2}, {E_3}} \right\} = T\left(S_\alpha \right)$. One can check the following brackets

Let $H = span\left\{ {{E_1}, {E_2}} \right\}$ be the horizontal distribution on $S_\alpha$. If we let

then $H = \ker \theta$. To describe the Lorentzian approximants of $S_\alpha$, let $L > 0$ and define a metric

so that ${E_1}, {E_2}, \widetilde{{E_3}}: = {L^{-\frac{1}{2}}}{E_3}$ are a pseudo orthonormal basis on $T\left(S_\alpha \right)$ with respect to $g_{L}.$ Hereafter, we denote the Lorentzian approximants to $S_\alpha$ by $(S_\alpha, g_L)$ and write $S^L_\alpha$ instead of $(S_\alpha, g_L)$. Note that $g = g_{1}$ is the Lorentzian metric on $S_\alpha$. A non-zero vector $x \in S^L_\alpha$ is called spacelike, null or timelike if $\langle x, x \rangle > 0$, $\langle x, x \rangle = 0$ or $\langle x, x \rangle < 0$, respectively. We define the norm of the vector $x \in S^L_\alpha$ by $\parallel x \parallel = \sqrt{\mid \langle x, x \rangle\mid}$. We assume that $\nabla ^L$ is the Levi-civita connection on $S^L_\alpha$ with respect to $g_L$. Using the Koszul formula and $(2.3)$, we have

Proposition 2.1. The Levi-civita connection on $S^L_\alpha$ relative to the coordinate frame ${E_1}, {E_2}, \widetilde{{E_3}}$ is given by

Proof. It follows from a direct application of the Koszul identity, which here simplifies

where $i, j, k = 1, 2, 3.$

For a Lorentzian $\alpha$-Sasakian manifold $M$, one can compute the curvature tensor of the connection $\nabla^{L}$ by the formula $R^{L}(X, Y)Z = \nabla^{L}_X\nabla^{L}_YZ-{\nabla^{L}_Y\nabla^{L}_X}Z-\nabla^{L}_{[X, Y]}Z$ or $R^{L}(X, Y)Z = \alpha^2[g(Y, Z)X-g(X, Z)Y].$ Then, we get the following proposition.

Proposition 2.2. The curvature tensor of $S^L_\alpha$ is given by

Proof. We take $R^{L}(X, Y)Z = \alpha^2[g(Y, Z)X-g(X, Z)Y]$ to compute curvature tensor of $S^L_\alpha$. Taking

for example, we compute

and hence

3.   Curvature for curves in $S^L_\alpha$ and sub-Lorentzian limit

In this section, we will compute the sub-Lorentzian limit of curvature for curves in $S^L_\alpha$. Our approach is to define sub-Lorentzian objects as limits of horizontal objects in $S^L_\alpha$, where a family of metrics $g_{L}$ is essentially obtained as an anisotropic blow-up of the Lorentzian metric $g.$ At the heart of this approach is the fact that the intrinsic horizontal geometry does not change with $L.$ Let $\beta:I \rightarrow S^L_\alpha$ be a regular curve, where $I$ is an open interval in $R$. The regular curve $\beta$ is called a spacelike curve, timelike curve or null curve if $\dot{\beta}(t)$ is a spacelike vector, timelike vector or null vector at any $t \in I$, respectively.

Definition 3.1. Let $\beta :I \to S^L_\alpha$ be a $C^1$ smooth curve, and we say that $\beta$ is regular if $\dot \beta\neq0$ for every $t\in I$. Moreover we say that $\beta(t)$ is a horizontal point of $\beta$ if

where $\beta(t) = (\beta_{1}(t), \beta_{2}(t), \beta_{3}(t))$.

As is well known, if $\beta$ is a curve with arc length parametrization, then the standard definition of curvature for $\beta$ in Riemannian geometry is $\kappa_\beta ^L: = \left\| {\nabla _{\dot \beta }^L\dot \beta } \right\|_L$. If $\beta$ is a curve with an arbitrary parametrization, then we give the definitions as follows:

Definition 3.2. Let $\beta :I \to S^L_\alpha$ be a $C^2$-smooth regular curve.

(1) If $\nabla _{\dot \beta }^L\dot \beta$ is a spacelike vector, we define the curvature $\kappa_\beta ^L$ of $\beta$ at $\beta(t)$ by

(2) If $\nabla _{\dot \beta }^L\dot \beta$ is a timelike vector, we define the curvature $\kappa_\beta ^L$ of $\beta$ at $\beta(t)$ by

Proposition 3.3. Suppose that $\beta :I \to S^L_\alpha$ is a $C^2$-smooth regular curve.

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

In particular, if $\beta(t)$ is a horizontal point of $\beta$,

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

In particular, if $\beta(t)$ is a horizontal point of $\beta$,

Proof. By (2.2), we have

By Proposition 2.1 and (3.7), we obtain

Coming by (3.7), we have

By (3.7), (3.9), and the definition of $\kappa_\beta ^{L},$ we get Proposition 3.3.

Definition 3.4. Let $\beta :I \to S^L_{\alpha}$ be a $C^2$-smooth regular curve. We define the intrinsic curvature $\kappa_{\beta}^\infty$ of $\beta$ at $\beta(t)$ to be

if the limit exists.

We introduce the following notation : For continuous functions $f_1, f_2:(0, +\infty)\rightarrow \mathbb{R}$,

Proposition 3.5. Suppose that $\beta :I \to S^L_{\alpha}$ is a $C^2$-smooth regular curve in the Lorentzian $\alpha$-Sasakian manifold.

(1) If $\nabla _{\dot \beta }^L\dot \beta$ is a spacelike vector, then $\kappa_\beta ^\infty$ does not exist, if $\theta(\dot{\beta}(t))\neq 0.$

if $\theta(\dot{\beta}(t)) = 0$ and $\frac{d}{dt}(\theta(\dot{\beta}(t))) = 0.$

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

if $\theta(\dot{\beta}(t)) = 0$ and $\frac{d}{dt}(\theta(\dot{\beta}(t))) = 0.$

Therefore, this situation does not exist.

Proof. (1) If $\nabla _{\dot \beta }^L\dot \beta$ is a spacelike vector, we have

Thus,

So, using (3.1), we know $\kappa_\beta ^\infty$ does not exist, if $\theta(\dot{\beta}(t))\neq 0$. If $\theta(\dot{\beta}(t)) = 0$ and $\frac{d}{dt}(\theta(\dot{\beta}(t))) = 0$, we get (3.10). If $\theta(\dot{\beta}(t)) = 0$ and $\frac{d}{dt}(\theta(\dot{\beta}(t)))\ne 0$, then

By (3.1), we get (3.11).

(2) If $\nabla _{\dot \beta }^L\dot \beta$ is a timelike vector, we have

We get (3.12). If $\theta(\dot{\beta}(t)) = 0$ and $\frac{d}{dt}(\theta(\dot{\beta}(t))) = 0$, we get (3.13). If $\theta(\dot{\beta}(t)) = 0$ and $\frac{d}{dt}(\theta(\dot{\beta}(t)))\neq 0,$

and then, the situation does not exist.

4.   Geodesic curvatures of curves on Lorentzian surfaces in $S^L_\alpha$

In this section, we will compute the expressions of intrinsic geodesic curvatures of curves on Lorentzian surfaces in $S^L_\alpha$. We will say that a surface $S^L_\alpha$ is regular if $S$ is a $C^{2}$-smooth compact and oriented surface. In particular we will assume that there exists a $C^{2}$-smooth function $h:S^L_\alpha \rightarrow \mathbb{R}$ such that

and $\nabla_{S^L_\alpha}h = h_{x_{1}}\partial_{x_{1}}+h_{x_{2}}\partial_{x_{2}}+h_{x_3}\partial_{x_3}\neq 0$. Let $\nabla_{H}h = E_{1}(h)E_{1}+E_{2}(h)E_{2}$. A point $x\in S$ is called characteristic if $\nabla_{H}h(x) = (0, 0).$ Our computations will be local and away from characteristic points of $S$. Let us define first $p: = E_{1}h, q: = E_{2}h, \ \mbox{and} \ r: = \widetilde{E_{3}}h.$ Since $-p^2+q^2 > 0,$ we say $S \subset S^L_\alpha$ is a horizontal spacelike surface. When $L\rightarrow +\infty,$ $-p^2+q^2+r^2 > 0.$ Then, we define

In particular, $-\bar{p}^{2}+\bar{q}^{2} = 1$. These functions are well defined at every non-characteristic point. Let

Then, $N_{L}$ is the unit spacelike normal vector to $S$, and $F_{1}$ is a unit timelike vector, while $F_{2}$ is a unit spacelike vector of $S$. $\{F_{1}, F_{2}\}$ is the orthonormal basis of $S.$ Let $\dot{\beta} = a F_1+b F_2.$ We define $J_{L}(\dot{\beta}) = aF_2+bF_1$ if $\beta$ is a $C^2$-smooth spacelike curve, and we define $J_{L}(\dot{\beta}) = -aF_2-bF_1$ if $\beta$ is a $C^2$-smooth timelike curve. Then, $g_L(\dot{\beta}, J_L(\dot{\beta})) = 0$ and $(\dot{\beta}, J_L(\dot{\beta}))$ have the same orientation with $\{F_{1}, F_{2}\}$.

For every $U, V \in TS$, we define $\nabla^{S, L}_{U}V = \pi\nabla^{L}_{U}V$ where $\pi:T S^L_\alpha \rightarrow TS$ is the projection. Then, $\nabla^{S, L}$ is the Levi-Civita connection on $S$ with respect to the metric $g_{L}$. By (3.9), (4.2) and

we have

Therefore, when $\theta(\dot{\beta}(t)) = 0$, we have

Definition 4.1. Let $S \subset S^L_{\alpha}$ be a Lorentzian regular surface, $\beta:I\rightarrow S$ be a $C^{2}$-smooth regular curve.

(1) If $\nabla^{S, L}_{\dot{\beta}}\dot{\beta}$ is a spacelike vector, the geodesic curvature $\kappa^{L}_{\beta, S}$ of $\beta$ at $\beta(t)$ is defined as

(2) If $\nabla^{S, L}_{\dot{\beta}}\dot{\beta}$ is a timelike vector, the geodesic curvature $\kappa^{L}_{\beta, S}$ of $\beta$ at $\beta(t)$ is defined as

Definition 4.2. Let $S\subset S^L_{\alpha}$ be a Lorentzian regular surface, $\beta:I\rightarrow S$ be a $C^{2}$-smooth regular curve. We define the intrinsic geodesic curvature $\kappa^{\infty}_{\beta, S}$ of $\beta$ at $\beta(t)$ to be

if the limit exists.

Proposition 4.3. Let $S \subset S^L_{\alpha}$ be a Lorentzian regular surface, $\beta:I\rightarrow S$ be a $C^{2}$-smooth regular curve.

(1) If $\nabla^{S, L}_{\dot{\beta}}\dot{\beta}$ is a spacelike vector, then $\kappa^{\infty}_{\beta, S}$ does not exist, if $\theta(\dot{\beta}(t))\ne0$.

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

$\lim\limits_{L\to+\infty} \frac{\kappa^{L}_{\beta, S}}{\sqrt{L}}$ does not exist, if $\theta(\dot{\beta}(t)) = 0\ \mathit{\mbox{and}}\ \frac{d}{dt}(\theta(\dot{\beta}(t)))\ne0$.

Proof. (1) If $\nabla^{S, L}_{\dot{\beta}}\dot{\beta}$ is a spacelike vector, by (3.7) and $\dot{\beta}\in TS$, we have

Thus,

and we have

Thus,

By (4.4), we have

Similarly, we have that when $\theta(\dot {\beta}(t))\neq0$,

By (4.4) and (4.9), we have

where $M_0$ does not depend on $L.$ By Definition 4.1, (4.11)–(4.13), $\kappa^{\infty}_{\beta, S}$ does not exist, if $\theta(\dot{\beta}(t)) \ne0$. When $\theta(\dot{\beta}(t)) = 0$ and $\frac{d}{dt}(\theta(\dot{\beta}(t))) = 0$, then

and

where $\bar{A} = -\bar{q}[\frac{1}{\alpha}\ddot{\beta}_{3}+\alpha e^{2{\beta_{3}}}\dot{\beta}_{1}^2]-\bar{p}[2\dot{\beta}_{3} \dot{\beta}_{1}e^{{\beta_{3}}}+e^{{\beta_{3}}}\ddot{\beta}_{1} ]$ and $\bar{B} = \frac{1}{\alpha}\dot{\beta_{3}}\bar{q}+e^{\beta_{3}}\dot{\beta_{1}}\bar{p}$. By (4.14)–(4.16) and (4.6), we get

When $\theta(\dot{\beta}(t)) = 0$, and $\frac{d}{dt}(\theta(\dot{\beta}(t)))\neq0$, we have

Therefore, (4.8) holds.

(2) If $\nabla^{S, L}_{\dot{\beta}}\dot{\beta}$ is a timelike vector, by similar calculation, we get (2).

Definition 4.4. Let $S \subset S^L_{\alpha}$ be a Lorentzian regular surface, $\beta:I\rightarrow S$ be a $C^{2}$-smooth regular curve. The signed geodesic curvature $\kappa^{L, c}_{\beta, S}$ of $\beta$ at $\beta(t)$ is defined as

Definition 4.5. Let $S \subset S^L_{\alpha}$ be a Lorentzian regular surface, $\beta:I\rightarrow S$ be a $C^{2}$-smooth regular curve. We define the intrinsic geodesic curvature $\kappa^{\infty}_{\beta, S}$ of $\beta$ at the non-characteristic point $\beta(t)$ to be

if the limit exists.

Proposition 4.6. Let $S \subset S^L_{\alpha}$ be a Lorentzian regular surface.

(1) If $\beta:I\rightarrow S$ is a $C^{2}$-smooth regular spacelike curve, then

(2) If $\beta:I\rightarrow S$ is a $C^{2}$-smooth regular timelike curve, then

Proof. For (4.1), by (4.10), we have

By (4.4) and (4.21), we have

Thus, if $\theta(\dot{\beta}(t))\ne 0$, (4.17) holds. When $\theta(\dot{\beta}(t)) = 0$ and $\frac{d}{dt}(\theta(\dot{\beta}(t))) = 0$, we get

So, $\kappa^{\infty, c}_{\beta, S} = 0.$ When $\theta(\dot{\beta}(t)) = 0$ and $\frac{d}{dt}(\theta(\dot{\beta}(t)))\neq0$, we have

We get

(2) If $\beta:I\rightarrow S$ is a $C^{2}$-smooth regular timelike curve, by similar calculation, we get (2).

5.   Lorentzian surface and a Gauss-Bonnet theorem in $S^L_{\alpha}$

In this section, we will compute the expression of the sub-Lorentzian limit of the Gaussian curvature of Lorentzian surfaces in $S^L_{\alpha}$. Then, we will prove a Gauss-Bonnet theorem for Lorentzian surfaces in $S^L_{\alpha}$. To do this, we define the second fundamental form $II^{L}$ of the embedding of $S$ into $S^L_{\alpha}$ by

We have the following theorem.

Theorem 5.1. For the embedding of $S$ into $S^L_{\alpha}$, the second fundamental form $II^L$ of the embedding of $S$ is given by

where

Proof. Since $\langle F_{1}, N_{L}\rangle_{L} = 0, \langle F_{2}, N_{L}\rangle_{L} = 0,$ we have

Using the definition of the connection, the identities in (2.4) and grouping terms, we have

Since $-\bar{p}^{2}+\bar{q}^{2} = 1$, we have $-\bar{p}E_{i}\bar{p}+\bar{q}E_{i}\bar{q} = 0, i = 1, 2, 3.$ Thus, $\bar{q}E_{1}\bar{q} = \bar{p}E_{1}\bar{p},$ $\bar{q}E_{2}\bar{q} = \bar{p}E_{2}\bar{p}$. Next, we compute the inner product of this with $N_{L}$, and we have

We get

To compute $h_{12}$, using the definition of the connection, we obtain

We get

and therefore

To compute $h_{21}$, using the definition of the connection, we obtain

We get

Therefore,

Since $\langle\nabla_{F_{2}}N_{L}, F_{2}\rangle_{L} = -\langle\nabla_{F_{2}}F_{2}, N_{L}\rangle_{L},$ we use the definition of connection, the identities in (2.4) and grouping terms. Taking the inner product with $N_{L}$ and under some simplifications similar to Theorem 4.3 in ^[23], we have

and then we get

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

Let

By the Gauss equation, we have

Proposition 5.2. The horizontal mean curvature $\mathcal{H}_{\infty}$ of $S \subset S_\alpha$ away from characteristic point is given in the following form:

Proof. By

we get (5.2).

Proposition 5.3. Away from characteristic points, we have

where

Proof. We compute

and then

By Proposition 2.2, we find

By the second fundamental form and ${\nabla _H}\left({{{\bar r}_L}} \right) = {L^{-\frac{1}{2}}}{\nabla _H}\left({\frac{{{E_3}h}}{{\left| {{\nabla _H}h} \right|}}} \right) + O\left({{L^{ - 1}}} \right)$ as $L\rightarrow +\infty$, we get

By (5.1), (5.6) and (5.7), we get the desired equation.

We get a Gauss-Bonnet theorem for Lorentzian surface in $S^L_\alpha$ as follows.

Theorem 5.4. Let $S\subset S^L_\alpha$ be a regular Lorentzian surface with finitely many boundary components $(\partial S)_i, i\in \{1, \cdots, n\}$, given by Euclidean $C^2$-smooth regular and closed spacelike curves $\beta_i:[0, 2\pi]\rightarrow (\partial S)_i.$ Let $A$ be Gaussian curvature of $\Sigma$ in Proposition 5.3 and $\kappa^{\infty, c}_{\beta_i, S}$ the sub-Lorentzian signed geodesic curvature of $\beta_i$ relative to $\Sigma$ in Proposition 4.6. Supposing that the characteristic set $C(S)$ be the empty set, $d\sigma_S$ is defined by (8.5), and $ds$ is defined by (8.1) in Appendix A. Then,

Proof. By the discussions in ^[15], suppose that all points satisfy $\theta(\dot{\beta(t)})\ne 0$ on $\beta_i$. Therefore, using Proposition 4.6, we obtain

According to the Gauss-Bonnet theorem (see [4], page 90 Theorem 1.4), we get

Therefore, by (5.8), (5.9), (8.6), (5.3) and (8.4), we get

Let $L$ go to infinity and use the dominated convergence theorem, and we get the desired result.

6.   Spacelike surface and a Gauss-Bonet theorem in $S^L_\alpha$

In this section, we will prove a Gauss-Bonet theorem for spacelike surface in $S^L_\alpha.$ Let

Let $p^2-q^2 > 0, \mbox{when} \ L\rightarrow +\infty,$ and we have $p^2-q^2-r^2 > 0.$ We define

In particular, $\bar{p}^{2}-\bar{q}^{2} = 1$. These functions are well defined at every non-characteristic point. Let

and then $N_{L}$ is the unit timelike normal vector to $S$, $F_{1}$ and $F_{2}$ are the unit spacelike vector of $S$. $\{F_{1}, F_{2}\}$ is the orthonormal basis of $S.$ We call $S$ a spacelike surface in Lorentzian $\alpha$-Sasakian space. We define a linear transformation on $TS$ by $J_L:TS\rightarrow TS$, and the transformation is well defined:

For every $U, V \in TS$, we define $\nabla^{S, L}_{U}V = \pi\nabla^{L}_{U}V$ where $\pi:T S^L_\alpha \rightarrow TS$ is the projection. Then, $\nabla^{S, L}$ is the Levi-Civita connection on $S$ with respect to the metric $g_{L}$. By (3.9), (6.2) and

we have

Therefore, when $\theta(\dot{\beta}(t)) = 0$, we have

Definition 6.1. Let $S \subset S^L_{\alpha}$ be a regular spacelike surface, $\beta:I\rightarrow S$ be a $C^{2}$-smooth regular curve. We define the geodesic curvature $\kappa^{L}_{\beta, S}$ of $\beta$ at $\beta(t)$ by

Definition 6.2. Let $S\subset S^L_{\alpha}$ be a regular spacelike surface, $\beta:I\rightarrow S$ be a $C^{2}$-smooth regular curve. The intrinsic geodesic curvature $\kappa^{\infty}_{\beta, S}$ of $\beta$ at $\beta(t)$ is defined as

if the limit exists.

Proposition 6.3. Let $S \subset S^L_{\alpha}$ be a regular spacelike surface, $\beta:I\rightarrow S$ be a $C^{2}$-smooth spacelike curve, and then we have the following assertions:

Proof. By (3.7) and $\dot{\beta}\in TS$, we have

By (6.5), we have

Similarly, if $\theta(\dot {\beta}(t))\neq0$,

By (6.5) and (6.10), we have

where $M_0$ does not depend on $L.$ By (6.7) and (6.11)-(6.13), (6.8) holds. When $\theta(\dot{\beta}(t)) = 0$ and $\frac{d}{dt}(\theta(\dot{\beta}(t))) = 0$,

and

By (6.14)-(6.16) and (6.7), we get

When $\theta(\dot{\beta}(t)) = 0$, and $\frac{d}{dt}(\theta(\dot{\beta}(t)))\neq0$, we have

Therefore, (6.9) holds.

Proposition 6.4. Let $S \subset S^L_{\alpha}$ be a regular spacelike surface. $\beta:I\rightarrow S$ is a $C^{2}$-smooth regular spacelike curve, and then

Proof. By (6.3) and (6.10), we have

By (6.5) and (6.19), we have

Therefore, if $\theta(\dot{\beta}(t))\ne 0$, (6.17) holds. When $\theta(\dot{\beta}(t)) = 0$ and $\frac{d}{dt}(\theta(\dot{\beta}(t))) = 0$, we get

So, $\kappa^{\infty, c}_{\beta, S} = 0.$ When $\theta(\dot{\beta}(t)) = 0$ and $\frac{d}{dt}(\theta(\dot{\beta}(t)))\neq0$, we have

We get

In the following, we investigate the sub-Lorentzian limit of the Gaussian curvature of spacelike surfaces in $S^L_{\alpha}$. The second fundamental form $II^{L}$ of the embedding of $S$ into $S^L_{\alpha}$ is defined as

Similar to Theorem 4.3 in ^[23], we have the following.

Theorem 6.5. For the embedding of $S$ into $S^L_{\alpha}$, the second fundamental form $II^L$ of the embedding of $S$ is given by

where

Proof. We combine

and $\langle \nabla^{L}_{F_{i}}N_{L}, F_{j}\rangle_{L} = -\langle\nabla^{L}_{F_{i}}F_{j}, N_{L}\rangle_{L}, i, j = 1, 2.$ By direct calculation, we obtain

Since $\bar{p}^{2}-\bar{q}^{2} = 1$, we have $\bar{p}E_{i}\bar{p}-\bar{q}E_{i}\bar{q} = 0, i = 1, 2.$ Then, $\bar{q}E_{1}\bar{q} = \bar{p}E_{1}\bar{p},$ $\bar{q}E_{2}\bar{q} = \bar{p}E_{2}\bar{p}$. Next, we compute the inner product of this with $N_{L}$, and we have

We obtain

Similarly, we have

Thus, Theorem 6.5 holds.

By the Gauss equation, we have

Proposition 6.6. The horizontal mean curvature $\mathcal{H}_{\infty}$ of $S \subset S_\alpha$ away from characteristic point is given in the following form:

Proof. By

we get (6.22).

Proposition 6.7. Away from characteristic points, we have

where

Proof. We compute

and then

So,

By the second fundamental form and ${\nabla _H}\left({{{\bar r}_L}} \right) = {L^{-\frac{1}{2}}}{\nabla _H}\left({\frac{{{E_3}h}}{{\left| {{\nabla _H}h} \right|}}} \right) + O\left({{L^{ - 1}}} \right)$ as $L\rightarrow +\infty$, we get

By (6.21), (6.26) and (6.27), we get the desired equation.

Theorem 6.8. Let $S\subset S^L_\alpha$ be a regular spacelike surface with finitely many boundary components $(\partial S)_i, i\in \{1, \cdots, n\}$, given by Euclidean $C^2$-smooth regular and closed spacelike curves $\beta_i:[0, 2\pi]\rightarrow (\partial S)_i.$ Suppose that $C$ is defined by (6.24), $d\sigma_{S}$ is defined by (8.8) and $\kappa^{\infty, c}_{\beta_i, S}$ is the sub-Lorentzian signed geodesic curvature of $\beta_i$ relative to $S$. If the characteristic set $C(S)$ is the empty set, then

Proof. By the discussions in ^[15], we may assume that there are no points satisfying $\theta(\dot{\beta(t)}) = 0$ and $\frac{d}{dt}(\theta(\dot{\beta(t)}))\ne 0$ on $\beta_i$. Therefore, using Proposition 6.3, we obtain

According to the Gauss-Bonnet theorem, we get

Therefore, by (6.28), (6.29), (8.9), (6.23), (8.3) and (8.4), we get

Let $L$ go to infinity and use the dominated convergence theorem, and we get the desired result.

7.   Conclusions

This paper proved two Gauss-Bonnet theorems for the Lorentzian surfaces and spacelike surfaces in a Lorentzian $\alpha$-Sasakian manifold by using the method of the Lorentzian approximation scheme. For Lorentzian surfaces, we derive the expressions of the intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on Lorentzian surfaces and the intrinsic Gaussian curvature of Lorentzian surfaces away from characteristic points in the Lorentzian $\alpha$-Sasakian manifold. Similarly, we get the corresponding results for the spacelike surface.

Appendix

To prove Gauss-Bonnet theorems, we need to define the Lorentzian length measure and the Lorentzian surface measure. Let us first consider the case of a regular spacelike curve $\beta:I\rightarrow S^L_\alpha$, and we define the length measure $d{s_L} = {\| {\dot \beta} \|_L}dt.$

Lemma 7.1. Let $\beta:I\rightarrow S^L_\alpha$ be a $C^2$-smooth spacelike curve. Let

Then,

When $\theta({\dot\beta(t)})\neq0$, we have

With the situation of $\theta({\dot\beta(t)}) = 0$, we have

Proof. We know that

and similar to the proof of Lemma 6.1 in ^[15], we can prove

so we get (8.2). When $\theta({\dot\beta(t)})\neq0$, we have

Using the Taylor expansion, we can prove

From the definition of $ds_L$ and $\theta({\dot\beta(t)}) = 0$, we get

Proposition 7.2. Let $S\subset S^L_\alpha$ be a regular Lorentzian $C^2$-smooth surface. Let $d\sigma_{S, L}$ denote the surface measure on $S$ with respect to the Lorentzian metric $g_{L}.$ Let

Then,

If $S = f(D)$ with $f = f(h_1, h_2) = (f_1, f_2, f_3):D\subset{{\mathbb{R}}}^2\rightarrow {S^L_\alpha}$, then

Proof. It is well known that

We define ${F_1}^\ast: = g_L(F_1, \cdot), \ {F_2}^\ast: = g_L(F_2, \cdot),$ and then

Therefore,

Recall

and the Taylor expansion

and we get (8.6). By (2.2), we have

and

Let

We know that $d{\sigma _{S, L}} = \sqrt {\det \left({{g_{ij}}} \right)} d{h_1}d{h_2}, \ {g_{ij}} = {g_L}\left({{f_{{h_i}}}, {f_{{h_j}}}} \right),$ and

so by the dominated convergence theorem, we get

Proposition 7.3. Let $S\subset S^L_\alpha$ be a spacelike $C^2$-smooth surface. Let $d\sigma_{S, L}$ denote the surface measure on $S$ with respect to the metric $g_{L}.$ Suppose that

Then,

Proof. It is well known that

Then,

Therefore,

Recall

and the Taylor expansion

and we get (8.9).

Acknowledgments

This research was funded by the Project of Science and Technology of Heilongjiang Provincial Education Department (Grant No. 1354ZD008), the Reform and Development Foundation for Local Colleges and Universities of the Central Government (Excellent Young Talents project of Heilongjiang Province, Grant No. ZYQN2019071) and the Natural Science Foundation of Heilongjiang Province of China (Grant No. LH2021A020).

Conflict of interest

The authors declare that there are no conflicts of interests in this work.