Abstract

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

1. Introduction

In recent years, there has been done a lot of research concerning Gauss-Bonnet theorems in three-dimensional Lie groups with sub-Riemannian geometric structures. One of the reasons for this success is the discovery of the method of Riemannian approximations. Initial work by Balogh et al. proved a Heisenberg version of the Gauss-Bonnet theorem with the help of the method of Riemannian approximations [1, 2]. Hereafter the above work, many research on Gauss-Bonnet theorem have been gained; we refer the monograph [36]. In particular, Wang and Wei proved Gauss-Bonnet theorems on the affine group [3], the group of rigid motions of the Minkowski plane [3, 4], the BCV spaces [5], and the Lorentzian Heisenberg group [4, 6]. Inspired by their work, we proved Gauss-Bonnet theorems in the rototranslation group [7, 8], Lorentzian Sasakian space forms, and the group of rigid motions of Minkowski plane with the general left-invariant metric [9, 10]. Riemannian approximations can be extended to the case for any Lie group equipped with left-invariant Lorentzian metric named Lorentzian approximations. In particular, one can consider a sequence of Lorentzian manifolds denoted by , where a family of metrics , for , is essentially obtained as an anisotropic blow-up of the Lorentzian metric Then, one can define sub-Lorentzian objects as limits of horizontal objects in , since the intrinsic horizontal geometry does not change with Some typical works of Lorentzian approximations in the Lorentzian Heisenberg group are obtained in [4, 6]. For Lorentzian Sasakian space forms, see [9].

In this paper, we consider sub-Lorentzian geometry of curves and surfaces on . The group of rigid motions of Minkowski plane has two left-invariant Lorentzian metrics and [11, 12]. Onda proved that the metric is a Lorentz Ricci soliton [12]. In [13], Patrangenaru proved that any left-invariant metric on is isometric to one of the metric with and In [14], the metric was denoted by with the metric provides a natural -parametric deformation family of which is the model space of solve geometry in the eight model geometries of Thurston, which makes very interesting and important [14, 15]. However, very little is known about sub-Lorentzian geometry of In this paper, we focus on the general left-invariant Lorentzian metric on , where the coframe By using the method of Lorentzian approximations, we consider a sequence of Lorentzian manifolds , where , for , is an anisotropic blow-up of the Lorentzian metric Using the Koszul formula, we calculate the expressions of Levi-Civita connection and curvature tensor in the Lorentzian approximants of in terms of the basis We define the notions of the intrinsic curvature for regular curves, the intrinsic geodesic curvature of regular curves on spacelike surfaces and Lorentzian surfaces, and the intrinsic Gaussian curvature of spacelike surfaces and Lorentzian surfaces away from characteristic points. Furthermore, we derive the expressions of those curvatures and prove two generalized Gauss-Bonnet theorems in .

This paper is organized as follows. In Section 2, we introduce the Lorentzian approximations of and calculate the expressions of corresponding Levi-Civita connection in terms of the basis . Furthermore, we define the notions of geodesic curvature and intrinsic geodesic curvature of curves in . We get the expressions of those curvatures and give an example. In Sections 3 and 4, we compute intrinsic geodesic curvatures of regular curves on Lorentzian surfaces and the intrinsic Gaussian curvature of Lorentzian surfaces in We also give two examples. In Section 5, we get the first Gauss-Bonnet theorem in In Section 6, we compute intrinsic geodesic curvature of curves on spacelike surfaces and the intrinsic Gaussian curvature of spacelike surfaces in and we get the second Guass-Bonnet theorem.

2. Curvatures for Curves in Lorentzian Approximations

In this section, some basic notions in the Lorentzian group of rigid motions of the Minkowski plane group will be introduced. Let be the motion group of the Minkowski -space. This consists of all matrices of the form

Topologically, is a Lie group which is diffeomorphic to Its Lie algebra has a basis consisting of for which

From the above equations in Equation (2), we get that

We denote Let be the horizontal distribution on and

Then, . We consider the left-invariant Lorentzian metrics given by and its anisotropic blow-up , for We call the Lorentzian approximants of and denote by throughout the paper. It is easy to check that is the Lorentzian metric on and are pseudo-orthonormal basis on with respect to Then, the Levi-Civita connection of is given by following proposition.

Proposition 1. The Levi-Civita connection relative to the coordinate frame of is described as follows:

Proof. To derive the above expressions, it is useful to recall the following Koszul identity from the famous proof of the unique of Riemannian connection: where By Equations (3) and (14), the first formula in Equation (11) is as follows: When , we compute . It follows that and Hence, . Similarly, and Moreover, other expressions follow some similar computations.
A parametrized curve in is a map , where is an open interval in , where and each function has derivatives of two orders, for all . Such is called -smooth. The regular curve is called regular provided that there does not exist with We call a spacelike curve, timelike curve, or null curve if is a spacelike vector, timelike vector, or null vector at any , respectively.

Definition 2. For an arbitrary -smooth regular curve , , we say that is a horizontal point of provided the following function satisfies the following:

Definition 3. For an arbitrary -smooth regular curve , , we define the geodesic curvature of at in the following way. if is a spacelike vector. if is a timelike vector.

Proposition 4. There are formulae of curvature for -smooth regular curve (1)If is a spacelike vector, then where In particular, if is a horizontal point of , (2)If is a timelike vector, then In particular, if is a horizontal point of ,

Proof. By Equation (8), we have the following: By Proposition 1 and Equation (25), we get the following: Using Equations (25) and (26), we have the following: By Equations (14), (25), and (29), we obtain the following:

By the definition of , we get the desired formulae of curvature.

Definition 5. Let be a -smooth regular curve, we define the intrinsic curvature of at to be if the limit exists.
To derive the expression of intrinsic curvature, we need the following notion: for continuous functions ,

Proposition 6. Let be a -smooth regular curve in . (1)If is a spacelike vector and , then we have the following formula of If , thenIf and , then If and , then (2)If is a timelike vector and , then we have the following formula of If , thenIf and , then

Proof. (1)If is a spacelike vector, when , we have the following:Therefore, If , by Equation (7), we have the following: By Equation (20) and , we have the following: When and , we have the following: If and , by (Equation (15)), we get the following: (2)By some similar computations, we get (2).

Example 7. Let be a -smooth regular curve, where is an open interval in and . We compute the following: By Equation (29), we have the following: