Lorentzian Approximations and Gauss–Bonnet Theorem for with the Second Lorentzian Metric

Abstract

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

1. Introduction

In 2021, Wang and Wei studied Gauss–Bonnet theorems in the affine group, the group of rigid motions of the Minkowski plane, BCV spaces, and the twisted Heisenberg group by using the method of Riemannian approximations in References [1, 2], which were first took by Balogh, Tyson, and Vecchi to prove a Heisenberg version of the Gauss–Bonnet theorem in References [3, 4]. Riemannian approximations can be extended to the case for any Lie group equipped with left-invariant Lorentzian metric named Lorentzian approximations. Some typical work of Lorentzian approximations in Lorentzian Heisenberg group is obtained in Reference [5, 6]. For Lorentzian Sasakian space forms, we refer to Reference [7], and for semiaffine transformations of the Euclidean plane and Borel subgroup of the group , we refer to References [8–10]. In Reference [11], the authors classified parallel surfaces in the groups of rigid motions of Minkowski plane. Inspired by the abovementioned work, we proved Gauss–Bonnet theorems in with the general left-invariant metric in References [12, 13], where is one of the three-dimensional unimodular Lie groups classified by Milnor in Reference [14].

One motivation for this paper was our observation in Reference [15] that there are two classes of left-invariant Lorentzian metrics and on the group of rigid motions of Minkowski 2-space , where satisfies a Ricci soliton equation and is geodesically complete and flat. A natural question is how to prove Gauss–Bonnet theorem for Lorentzian surface in with the second left-invariant Lorentzian metric . In this paper, we try to solve this question by employing the method of the Lorentzian approximation scheme. To do this, we define the notions of the intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on Lorentzian surface, and the intrinsic Gaussian curvature of Lorentzian surface away from characteristic points. Furthermore, we derive the expressions of those curvatures and prove Gauss–Bonnet theorem for the Lorentzian surface in with the second left-invariant Lorentzian metric .

The paper is organized in the following way: In Section 2, we introduce some notations on the Lorentzian approximations of . Furthermore, we define the notions of curvature and intrinsic curvature of curves in the Lorentzian approximations of . We then get the expressions of those curvatures. In Section 3, we compute intrinsic geodesic curvatures of regular curves on Lorentzian surfaces and the intrinsic Gaussian curvature of Lorentzian surface in the Lorentzian approximations of . Finally, we prove the Gauss–Bonnet theorem for the Lorentzian surface in . In Section 4, we summarize the main results.

2. Lorentzian Approximations for

Let be the group of rigid motions of Minkowski 2-space in Reference [15]. This consists of all matrices of the following form:

Topologically, is diffeomorphic to under the map

It is Lie algebra that has a basis consisting offor which

Put , , and . Then, we get the following equation:

We consider left-invariant Lorentzian metric which has a pseudo-orthonormal basis , given by

Let our frame be defined by , , and , thenand . Let be the horizontal distribution on . Let

Then, . For the constant , let , and be the Lorentzian metric on in Reference [15]. We call the Lorentzian approximations of rigid motions of the Minkowski plane and write instead of . Then, are pseudo-orthonormal bases on with respect to , with brackets

A nonzero vector is called to be space-like, null, or time-like if , , and , respectively. We define the norm of the vector by . Let be a regular curve, where is an open interval in . The regular curve is called a space-like curve, time-like curve, or null curve if is a space-like vector, time-like vector, or null vector at any , respectively.

We assume that is the Levi-Civita connection on with respect to . By the Koszul formula, we have the following equation:where . By (9) and (10), we have the following:

Lemma 1. The Levi-Civita connection on relative to the coordinate frame is given byDefine the curvature of the connection byWe get the following proposition.

Proposition 1. The curvatures of the connection for are given by

Proof. It is a direct computation usingTakingfor example, we computeHence,

Definition 1. Let be a -smooth curve, we say that is regular if for every . Moreover, we say that is a horizontal point of ifwhere .
The curvature for a curve with an arbitrary parametrization is given as follows:

Definition 2. Let be a -smooth regular curve.(1)If is a space-like vector, we define the curvature of at by(2)If is a time-like vector, we define the curvature of at byOne can get the expression of curvature for curve as follows:

Lemma 2. Suppose that is a -smooth regular curve.(1)If is a space-like vector, then which and . In particular, if is a horizontal point of ,(2)If is a time-like vector, thenIn particular, if is a horizontal point of ,

Proof. By using equation (7), we have the following equation:where and . By Lemma 1 and (25), we obtain the following equation:By (25) and (26), we have the following equation:By Definition 2, (25), and (27), we get Lemma 2.

Definition 3. Let be a -smooth regular curve. We define the intrinsic curvature of at to beif the limit exists.
We introduce the following notation: for continuous functions ,

Lemma 3. Let be a -smooth regular curve. Then(1)if is a space-like vector, then if and . If , and , Therefore, this situation does not exist.(2)If is a time-like vector, thenTherefore, this situation does not exist.if and . If , and ,

Proof. (1) If is a space-like vector.
Using the notation introduced in (29), when , we have the following equations:ThusSo by Definition 2, we have (30). If and , we get (31). If and , thenThen, this situation does not exist.(2) If is a time-like vector, by similar calculation, we get (2).

3. Lorentzian Surface and Gauss–Bonnet Theorem in

In this section, we will compute intrinsic geodesic curvatures of regular curves on Lorentzian surfaces and the intrinsic Gaussian curvature of Lorentzian surface in the Lorentzian approximations of . Furthermore, we will prove the Gauss–Bonnet theorem for the Lorentzian surface in .

We say that a surface is regular if is a -smooth compact and oriented surface. In particular, we will assume that there exists a -smooth function such thatand . Let . A point is called characteristic if .

Our computations will be local and away from characteristic points of .

We define . Since , we say is a horizontal space-like surface. When , then .

Then we define

In particular, . These functions are well defined at every noncharacteristic point. Letthen is the unit space-like normal vector to , is a unit space-like vector, and is a unit time-like vector of . is the pseudo-orthonormal basis of . We call a Lorentzian surface in .

Let , we define , if is a -smooth space-like curve. We define , if is a -smooth time-like curve. Then, and have the same orientation with .

For every , we define where is the projection. Then, is the Levi-Civita connection on with respect to the metric . By (26), (42), and the followingwe have

Therefore, when , we have the following equation:

Definition 4. Let be a Lorentzian regular surface and be a -smooth regular curve.(1)If is a space-like vector, the geodesic curvature of at is defined as follows:(2)If is a time-like vector, the geodesic curvature of at is defined as follows:

Definition 5. Let be a Lorentzian regular surface and be a -smooth regular curve. We define the intrinsic geodesic curvature of at to beif the limit exists.

Lemma 4. Let be a regular Lorentzian surface. Let be a -smooth regular curve.(1)If is a space-like vector, then Therefore, this situation does not exist.(2)If is a time-like vector, thenTherefore, this situation does not exist.

Proof. (1) If is a space-like vector. By (25) and , we have the following equation:By (44), we have the following equation:Similarly, we have that when ,By (44) and (54), we have the following equation:where does not depend on . By Definition 4 and (55)-(57), we get (49).
When and , thenBy (58)-(60) and Definition 4, we get .
When and , we have the following equations:so we get (50).
(2) If is a time-like vector, by similar calculation, we get (2).

Definition 6. Let be a Lorentzian regular surface and be a -smooth regular curve. The signed geodesic curvature of at is defined as follows:

Definition 7. Let be a regular surface. Let be a -smooth regular curve. We define the intrinsic geodesic curvature of at the noncharacteristic point to beif the limit exists.

Lemma 5. Let be a Lorentzian regular surface.(1)If be a -smooth regular space-like curve, then(2)If be a -smooth regular time-like curve, then

Proof. For (1), by (54), we have the following equation:By (44) and (71), we have the following equations:When and , we get the following equation:So . When and , we have the following equation:So we get (68).
(2) If is a -smooth regular time-like curve, by similar calculation, we get (2).
Next, we compute the sub-Riemannian limit of the Gaussian curvature of surfaces in the . The second fundamental form of the embedding of into is defined bySimilar to Theorem 4.3 in Reference [16], we have the following theorem:

Theorem 1. For the embedding of into , the second fundamental form of the embedding of is given bywhereWe define the mean curvature of byLetBy the Gauss equation, we have

Proposition 2. The horizontal mean curvature of away from characteristic point is the following form:

Proof. Bywe get (85).

Proposition 3. Away from characteristic points, we havewhere

Proof. We computethen