Local geometric properties of the lightlike Killing magnetic curves in de Sitter 3-space

Abstract: In this article, we mainly discuss the local differential geometrical properties of the lightlike Killing magnetic curve γ(s) in S1 with a magnetic field V. Here, a new Frenet frame {γ,T, N, B} is established, and we obtain the local structure of γ(s). Moreover, the singular properties of the binormal lightlike surface of the γ(s) are given. Finally, an example is used to understand the main results of the paper.


Introduction
Since Einstein discovered the general theory of relativity in 1905, many scientists have studied Minkowski space systematically. The existence of the lightlike vector is the principal difference between Minkowski space and Euclidean space. Meanwhile, in Minkowski space, a lightlike curve has some special properties. In physic, the geometric particle model was constructed by using the lightlike curve [1]. A. Ferrandez et al. [2] considered the equations of the particles in 3-dimensional lightlike curves. The second author and D. Pei [3] studied the differential geometric properties of the lightlike curves on Λ 3 1 . Also, D. Pei etc. [3] pointed out that de Sitter 3-space was a crucial model of the physical universe in Minkowski space. Some properties of spacelike curves in S 3 1 were studied by T. Fusho and S. Izumiya in [4]. Y. Li and Z. Wang [5] studied the geometric properties of lightlike tangent developables.
Following the action of the Lorentz force produced through the magnetic field F, the trajectory of the charged particle is called the magnetic curve. Under certain conditions, the magnetic curve is regarded as the extension of the geodesic [6]. Magnetic curves describe the movement of charged particles in several physical scenarios and form magnetic flux in the background magnetic field [7].
In recent years, many researchers have studied magnetic curves in different spaces [8][9][10][11][12]. In a Riemannian 3-space (M 3 , g), Z. Bozkurt [8] used a new variational method to research the magnetic flow with the Killing magnetic field. The results of classification for the Killing magnetic trajectories on the Minkowski 3-space was obtained in [9]. In 3-dimensional massive gravity, G. Clémen [10] considered a black hole with a lightlike Killing vector. M. I. Munteanu [6] introduced the magnetic curves in Euclidean space and used different methods to study the corresponding Killing magnetic curves.
With the deepening of theoretical research, the application of the singularity theory is more and more extensive [13][14][15][16][17][18][19][20][21][22]. Z. Wang [13,20,21] considered the singularity classifications of ruled lightlike surfaces in S 3 1 . However, very little has been researched about the differential geometric properties of the lightlike Killing magnetic curves. The second author [14,15] studied the singularity types of the Killing magnetic curves and the lightlike Killing magnetic curves in R 3 1 . Here, the classifications of the singularity of the lightlike Killing magnetic curves are considered in S 3 1 . The content of the article is summarized as follows. Firstly, the second part defines γ(s) (in the following text, we use γ(s) to represent the lightlike Killing magnetic curve), the related concepts of the magnetic curve, and Frenet formulas of γ(s). Section 3 shows the major results of the paper (Theorem 3.1), which gives the singularity classification of γ(s) ∈ S 3 1 with V. In the fourth section, the height function of γ(s) is used to obtain the singularity classification (Proposition 4.1). Section 5 introduces the unfolding of the height function and proves the Theorem 3.1. To enrich the local theory, the local structure of γ(s) ∈ S 3 1 with V is given in the sixth section. In the last section, to better understand the main results of this article, an example of γ(s) with V is given.

Preliminaries
The relevant definitions of the R 4 1 and S 3 1 are described in [13]. In this section, some definitions related to magnetic curves are introduced. We establish the Frenet frame {γ, T, N, B} and obtain the Frenet-Sernet formula.
We define V as a Killing vector field and F V = ι V dv g , where ι is an inner product. By we can get the Lorentz force of the F V . Thus, we obtain the Lorentz force equation defined as γ(s) is called a Killing magnetic curve [6][7][8][9][10][11][12]. In the following, we suppose γ(s) as a lightlike Killing curve with Killing field V. Since γ(s) ∈ S 3 1 , γ(s), γ(s) = 1, so γ(s), γ (s) = 0. We now define Therefore, we obtain the Frenet-Sernet formula of γ(s) as follows: Remark 2.2. If k 1 (s) = 0, then γ(s) is a straight line, and we omit it here.
For any v 0 ∈ S 3 1 , we call the set : By definition in [13], we can get the major results.

The height function
Here, we define a function on γ : I → S 3 1 , and get a geometric invariant σ(s) of the tangent indicatrix of γ(s) in S 3 1 with V. For γ : I → S 3 1 , we call the function . Then we can draw the following conclusions: and σ(s 0 ) = σ (s 0 ) = 0.

Unfolding of functions
Based on the unfolding theory of the height function germ, we prove Theorem 3.1. It is described in detail in the book [17,20].
Here, a significant set concerning the unfolding is given. The discriminant set of F is the set We get the vital result [17]. This result is the singular classification theorem, which is the same as Theorem 8. By Proposition 4.1, the discriminant set of Then, the following results are proved. Proof. Assuming that γ(s) = {γ 1 (s), γ 2 (s), γ 3 (s), γ 4 (s)}, therefore, we have the 2-jet of ∂H(s,v) ∂v i at s 0 as follows:

We now define
where v ∈ D H is a singular point. Thus then, the rank of A is equal to 3. So the theorem holds.  1, 2, 3) if and only if h v 0 has A k -singularity at s 0 . Apply the singularity theory [17], we complete the proof by the conclusion of Proposition 4.1 and Theorem 5.1.
(2) the local structure of γ(s) at γ(s 0 ) onto the principal normal vector space (ξ = 0) is as shown in the figure (see Figure 4).
then the projection of γ(s) (see Figure 7), the tangent indicatrix of γ(s) (see Figure 8), and the binormal lightlike surface (see Figure 9) onto the 3-dimension tangent space as follows.
The singular locus of the binormal lightlike surface onto the 3-dimension tangent space (see Figure  10

Conclusions
In the previous paper [14,15], we had obtained the singularities of the Killing and null Killing magnetic curves in Minkowski space. In this paper, we observed the singularity properties of the lightlike Killing magnetic curves in S 3 1 , which were degenerate curves. By considering the lightlike tangent vector, we constructed a new Frenet equations of a lightlike Killing magnetic curve γ(s) using the transversality theorem. Under the view of the contact theory, we obtain some local geometric properties of the lightlike Killing magnetic curves in S 3 1 . We made a profound research on the contact between rectifying surface and basic sphere space by the height functions. And we rightfully obtained the geometric invariants of a lightlike Killing magnetic curve, which is used to describe the properties of lightlike Killing magnetic curve. In the following research, some other local geometrical properties of the lightlike Killing magnetic curve in nullcone will be considered.