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

1.   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 ${\Lambda}^{3}_{1}$. 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 $\mathbb{S}^{3}_{1}$ 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 $\boldsymbol{ F}$, 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 $(\boldsymbol{ M}_3, g)$, 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. Cl$\acute{e}$men ^[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 $\mathbb{S}^{3}_{1}$. 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 $\mathbb{R}^{3}_{1}$. Here, the classifications of the singularity of the lightlike Killing magnetic curves are considered in $\mathbb{S}^{3}_{1}$.

The content of the article is summarized as follows. Firstly, the second part defines $\mathit{\boldsymbol{\gamma }}(s)$ (in the following text, we use $\mathit{\boldsymbol{\gamma }}(s)$ to represent the lightlike Killing magnetic curve), the related concepts of the magnetic curve, and Frenet formulas of $\mathit{\boldsymbol{\gamma }}(s)$. Section 3 shows the major results of the paper (Theorem 3.1), which gives the singularity classification of $\mathit{\boldsymbol{\gamma }}(s) \in \mathbb{S}^{3}_{1}$ with $\boldsymbol{ V}$. In the fourth section, the height function of $\mathit{\boldsymbol{\gamma }}(s)$ 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 $\mathit{\boldsymbol{\gamma }}(s) \in \mathbb{S}^{3}_{1}$ with $\boldsymbol{ V}$ is given in the sixth section. In the last section, to better understand the main results of this article, an example of $\mathit{\boldsymbol{\gamma }}(s)$ with $\boldsymbol{ V}$ is given.

2.   Preliminaries

The relevant definitions of the $\mathbb{R}^{4}_{1}$ and $\mathbb{S}^{3}_{1}$ are described in ^[13]. In this section, some definitions related to magnetic curves are introduced. We establish the Frenet frame $\{\mathit{\boldsymbol{\gamma }}, \boldsymbol{ T}, \boldsymbol{ N}, \boldsymbol{ B}\}$ and obtain the Frenet-Sernet formula.

We define $\boldsymbol{ V}$ as a Killing vector field and $F_{V} = \iota_{V}dv_{g}$, where $\iota$ is an inner product. By

we can get the Lorentz force of the $F_{V}$. Thus, we obtain the Lorentz force equation defined as

$\mathit{\boldsymbol{\gamma }}(s)$ is called a Killing magnetic curve ^{[6,7,8,9,10,11,12]}.

Definition 2.1. For a Killing magnetic curve $\mathit{\boldsymbol{\gamma }}(s)\in\mathbb{S}^{3}_{1}$, if $\langle\mathit{\boldsymbol{\gamma }}'(s), \mathit{\boldsymbol{\gamma }}'(s)\rangle = 0$, we call $\mathit{\boldsymbol{\gamma }}(s)$ the lightlike Killing magnetic curve.

In the following, we suppose $\mathit{\boldsymbol{\gamma }}(s)$ as a lightlike Killing curve with Killing field $\boldsymbol{ V}$.

Since $\mathit{\boldsymbol{\gamma }}(s)\in\mathbb{S}^{3}_{1}$, $\langle\mathit{\boldsymbol{\gamma }}(s), \mathit{\boldsymbol{\gamma }}(s)\rangle = 1$, so $\langle\mathit{\boldsymbol{\gamma }}(s), \mathit{\boldsymbol{\gamma }}'(s)\rangle = 0$. We now define

then

There exists a lightlike transversal vector $\boldsymbol{ B}(s),$ satisfying

Therefore, we obtain the Frenet-Sernet formula of $\mathit{\boldsymbol{\gamma }}(s)$ as follows:

where $k_1(s) = \langle\boldsymbol{ T}'(s), \boldsymbol{ N}(s)\rangle = - \langle\boldsymbol{ N}'(s), \boldsymbol{ T}(s)\rangle = \Vert V\wedge\mathit{\boldsymbol{\gamma }}'(s)\Vert$, $k_2(s) = \langle\boldsymbol{ N}'(s), \boldsymbol{ B}(s)\rangle = - \langle\boldsymbol{ B}'(s), \boldsymbol{ N}(s)\rangle.$

Remark 2.2. If $k_1(s) = 0$, then $\mathit{\boldsymbol{\gamma }}(s)$ is a straight line, and we omit it here.

3.   The classification of singularity

Here, we define the tangent indicatrix of $\mathit{\boldsymbol{\gamma }}(s)$ as $\Phi:I\rightarrow \mathbb{S}^{3}_{1}$ given by

where $\varepsilon = \pm1$,

We define a surface

by

we call $\mathcal{BNS}(s, \mu)$ the binormal lightlike surface of the tangent indicatrix of $\mathit{\boldsymbol{\gamma }}(s)$.

For any $\boldsymbol{ v}_{0}\in \mathbb{S}^{3}_{1}$, we call the set :

the lightlike magnetic bundle through $\boldsymbol{ v}_{0}$.

Also, a geometric invariant $\sigma(s)$ of $\mathit{\boldsymbol{\gamma }}(s)$ is given in $\mathbb{S}^{3}_{1}$ with $\boldsymbol{ V}$ by

By definition in ^[13], we can get the major results.

Theorem 3.1. For a curve $\mathit{\boldsymbol{\gamma }}(s)$ in $\mathbb{S}^{3}_{1}$ with $\boldsymbol{ V}$, when $\varrho(s_0)\neq 0$, giving a vector $\boldsymbol{ v}_{0} = \mathcal{BNS}(s_{0}, \mu_{0})$ and the magnetic bundle $\mathcal{NMB}(\boldsymbol{ v}_{0})$, we have the following conclusions:

$\rm(1)$ As for $\mathit{\boldsymbol{\gamma }}(s)$ and $\mathcal{NMB}(\boldsymbol{ v}_{0})$, there are at lowest 2-point contact at $s_0$.

$\rm(2)$ As for $\mathit{\boldsymbol{\gamma }}(s)$ and $\mathcal{NMB}(\boldsymbol{ v}_{0})$, there are at lowest 3-point contact at $s_0$ if and only if

where $\varepsilon = \pm1$ and $\sigma(s_{0})\neq 0$. With the above conclusions, the germ of image $\mathcal{BNS}(s, \mu)$ at $(s_{0}, \mu_{0})$ is locally diffeomorphic to to cuspidal edge $(C\times \mathbb{R})$ (see Figure 1).

$\rm(3)$ As for $\mathit{\boldsymbol{\gamma }}(s)$ and $\mathcal{NMB}(\boldsymbol{ v}_{0})$, there are at lowest 4-point contact at $s_0$ if and only if

and $\sigma(s_{0}) = 0, \sigma'(s_{0})\neq 0$. With the above conclusions, the germ of image $\mathcal{BNS}(s, \mu)$ at $(s_{0}, \mu_{0})$ is locally diffeomorphic to swallowtail $(SW)$(see Figure 2).

The cuspidal edge is defined as $C\times \mathbb{R} = \{(x_{1}, x_{2}, x_{3})\mid x_{1} = u, x_{2} = \pm v^{1/2}, x_{3} = v^{1/3} \}$, and the swallowtail is defined as $SW = \{(x_{1}, x_{2}, x_{3})\mid x_{1} = 3u^{4}+u^{2}v, x_{2} = 4u^{3}+2uv, x_{3} = v \}$.

4.   The height function

Here, we define a function on $\mathit{\boldsymbol{\gamma }}:I\rightarrow \mathbb{S}^{3}_{1}$, and get a geometric invariant $\sigma(s)$ of the tangent indicatrix of $\mathit{\boldsymbol{\gamma }}(s)$ in $\mathbb{S}^{3}_{1}$ with $\boldsymbol{ V}$.

For $\mathit{\boldsymbol{\gamma }}:I\rightarrow \mathbb{S}^{3}_{1}$, we call the function

by

as a height function of $\mathit{\boldsymbol{\gamma }}(s)$. Giving a vector $\boldsymbol{v_{0}} \in \mathbb{S}^{3}_{1},$ we define $h(s) = H_{v_{0}}(s) = H(s, \boldsymbol{v}_{0})$. Then we can draw the following conclusions:

Proposition 4.1. For a lightlike Killing magnetic curve $\mathit{\boldsymbol{\gamma }}(s)$ in $\mathbb{S}^{3}_{1}$ with $\boldsymbol{ V}$, when

then

$\rm(1)$ $h(s_{0}) = 0$ if and only if there exist $a, b, d\in \mathbb{R}$, such that $\boldsymbol{v} = a \mathit{\boldsymbol{\gamma }}(s_{0})+b \boldsymbol{ T}(s_{0})+d\boldsymbol{ B}(s_{0})$ and $a^{2}+2bd = 1$.

$\rm(2)$ $h(s_{0}) = \nabla_{\mathit{\boldsymbol{\gamma }}'(s)} h(s_{0}) = 0$ if and only if

$\rm(3)$ $h(s_{0}) = \nabla_{\mathit{\boldsymbol{\gamma }}'(s)}h(s_{0}) = \nabla_{\mathit{\boldsymbol{\gamma }}'(s)} \nabla_{\mathit{\boldsymbol{\gamma }}'(s)}h(s_{0}) = 0$ if and only if $d = \varepsilon\frac{k_1^{2}(s_{0})}{\varrho(s_{0})}$, then

$\rm(4)$ $h(s_{0}) = \nabla_{\mathit{\boldsymbol{\gamma }}'(s)}h(s_{0}) = \nabla_{\mathit{\boldsymbol{\gamma }}'(s)} \nabla_{\mathit{\boldsymbol{\gamma }}'(s)}h(s_{0}) = \nabla^{(3)}_{\mathit{\boldsymbol{\gamma }}'(s)}h(s_{0}) = 0$ if and only if

and $\sigma(s_{0}) = 0$.

$\rm(5)$ $h(s_{0}) = \nabla_{\mathit{\boldsymbol{\gamma }}'(s)}h(s_{0}) = \nabla_{\mathit{\boldsymbol{\gamma }}'(s)} \nabla_{\mathit{\boldsymbol{\gamma }}'(s)}h(s_{0}) = \nabla^{(3)}_{\mathit{\boldsymbol{\gamma }}'(s)}h(s_{0}) = \nabla^{(4)}_{\mathit{\boldsymbol{\gamma }}'(s)}h(s_{0}) = 0$ if and only if

and $\sigma(s_{0}) = \sigma'(s_{0}) = 0$.

Proof. We assume that $\boldsymbol{v} = a \mathit{\boldsymbol{\gamma }}(s)+b \boldsymbol{ T}(s)+c\boldsymbol{ N}(s)+d\boldsymbol{ B}(s) = 1$, where $a, b, c, d$ in $\mathbb{R}$. Since $\boldsymbol{v} \in \mathbb{S}^{3}_{1}$, $a^{2}+2bd+c^{2} = 1$. By using the Frenet formula (2.1), we have the following results:

$\rm(1)$ When $h(s_{0}) = 0$, we obtion

where $k_1(s_0)\neq 0$, then $c = 0$, $\boldsymbol{v} = a \mathit{\boldsymbol{\gamma }}(s_{0})+b \boldsymbol{ T}(s_{0})+d\boldsymbol{ B}(s_{0})$.

$\rm(2)$ When $h(s_{0}) = \nabla_{\mathit{\boldsymbol{\gamma }}'} h(s_{0}) = 0$, we have

then $b = \frac{k_2(s_{0})}{k_1(s_{0})}d$, $a = \varepsilon\sqrt{1- \frac{2k_2(s_{0})}{k_1(s_{0})} d^{2}}$, and

$\rm(3)$ When $h(s_{0}) = \nabla_{\mathit{\boldsymbol{\gamma }}'}h(s_{0}) = \nabla_{\mathit{\boldsymbol{\gamma }}'} \nabla_{\mathit{\boldsymbol{\gamma }}'}h(s_{0}) = 0$, we obtion

we can obtain $d = \varepsilon \frac{k_1^{2}(s_{0})}{\varrho(s_{0})}$, where

Hence,

$\rm(4)$ When $h(s_{0}) = \nabla_{\mathit{\boldsymbol{\gamma }}'}h(s_{0}) = \nabla_{\mathit{\boldsymbol{\gamma }}'} \nabla_{\mathit{\boldsymbol{\gamma }}'}h(s_{0}) = \nabla^{(3)}_{\mathit{\boldsymbol{\gamma }}'} h(s_{0}) = 0$, we have

we can obtain

$\sigma(s_0) = 0$ and $\sigma'(s_{0})\neq 0.$

$\rm(5)$ When $h(s_{0}) = \nabla_{\mathit{\boldsymbol{\gamma }}'}h(s_{0}) = \nabla_{\mathit{\boldsymbol{\gamma }}'} \nabla_{\mathit{\boldsymbol{\gamma }}'}h(s_{0}) = \nabla^{(3)}_{\mathit{\boldsymbol{\gamma }}'} h(s_{0}) = \nabla^{(4)}_{\mathit{\boldsymbol{\gamma }}'} h(s_{0}) = 0$,

we can obtain

and $\sigma(s_{0}) = \sigma'(s_{0}) = 0.$

5.   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 setof $F$ 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 $H(s, \boldsymbol{v}) = \langle \mathit{\boldsymbol{\gamma }}(s)-\boldsymbol{v}, V\wedge\mathit{\boldsymbol{\gamma }}'(s)\rangle$ is the set

Then, the following results are proved.

Theorem 5.1. For a curve $\mathit{\boldsymbol{\gamma }}(s)$, when $\varrho(s_{0})\neq 0$, $H(s, \boldsymbol{v}): I\times \mathbb{S}^{3}_{1}\rightarrow \mathbb{R}$ be the height function of $\mathit{\boldsymbol{\gamma }}(s)$ and $\boldsymbol{v}\in \mathfrak{D}_{H}$. $H(s, \boldsymbol{v})$ is a versal unfolding of $h_{v_0}$, when $h_{v_0}$ has $A_{k}$-singularity at $s\ (k = 1, 2, 3)$.

Proof. Assuming that $\mathit{\boldsymbol{\gamma }}(s) = \{\gamma_{1}(s), \gamma_{2}(s), \gamma_{3}(s), \gamma_{4}(s)\},$

$\boldsymbol{N}(s) = \{N_{1}, N_{2}, N_{3}, N_{4}\}, \boldsymbol{v}(s) = \{ v_{1}, v_{2}, v_{3}, v_{4}\}$ in $\mathbb{S}^{3}_{1},$ where

then

We have

where $i = 2, 3, 4$, thus

therefore, we have the 2-jet of $\frac{\partial H(s, \boldsymbol{v})}{\partial v_{i}}$ at $s_{0}$ as follows:

We now define

where $a_i = k_1(\varepsilon\frac{v_i}{v_1}N_1-N_i)$, $b_i = \varepsilon\frac{v_i}{v_1}(k_1'N_1+k_1N'_1)-k_1'N_i-k_1N_i'$, $c_i = \varepsilon\frac{v_i}{v_1}(k_1''N_1+2k_1'N_1'+k_1N_1'')-(k_1''N_i+2k_1'N_i'+k_1N_i'')$, $i = 2, 3, 4.$

Then,

Since

and

where $\boldsymbol{v}\in \mathfrak{D}_{H}$ is a singular point. Thus

then, the rank of $A$ is equal to 3. So the theorem holds.

Proof of Theorem 3.1. For the curve $\mathit{\boldsymbol{\gamma }}(s)$, when $\varrho(s_{0})\neq 0$. Suppose $\boldsymbol{ v}_{0} = \mathcal{BNS}(s_{0}, \boldsymbol{ u}_{0})$, A function $\mathfrak{H}:\mathcal{NMB}(\boldsymbol{ v}_{0})\rightarrow \mathbb{R}$ is defined by $\mathfrak{H}(u) = \langle \boldsymbol{ u}-\boldsymbol{ v}_{0}, V\wedge\boldsymbol{ u}\rangle$. Thus, we get $h_{\boldsymbol{ v}_{0}} = \mathfrak{H}(\boldsymbol{ \gamma}(s))$. Because $0$ and $\mathfrak{H}^{-1}(0) = \mathcal{NMB}(\boldsymbol{ v}_{0})$ is a regular value of $\mathfrak{H}$, as for $\boldsymbol{ \gamma}(s)$ and $\mathcal{NMB}(\boldsymbol{ v}_{0})$, there are $(k+1)$-point contact at $s_{0}$ $(k = 1, 2, 3)$ if and only if $h_{\boldsymbol{ v}_{0}}$ 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.

6.   The local structure of the lightlike magnetic curve

Here, the local structure of curve $\mathit{\boldsymbol{\gamma }}(s)$ in $\mathbb{S}^{3}_{1}$ with $\boldsymbol{ V}$ is considered. The Taylor's formula of $\mathit{\boldsymbol{\gamma }}(s_0)$ at $s = s_0$ is given by

where $\lim\limits_{\triangle s\to 0}{\boldsymbol{ \varepsilon}} = \boldsymbol{ 0}.$

Since

Then

where $\boldsymbol{\varepsilon} = \varepsilon_1\mathit{\boldsymbol{\gamma }}(s_0)+\varepsilon_2\boldsymbol{T}(s_0)+\varepsilon_3\boldsymbol{N}(s_0)+\varepsilon_4\boldsymbol{B}(s_0).$

Therefore, we have

When $\boldsymbol{\varepsilon}\to 0$, taking the first term in each of the coefficients of $\boldsymbol{T}(s_0), \boldsymbol{N}(s_0)$ and $\boldsymbol{B}(s_0)$, we obtain

We suppose $\{\alpha, \xi, \zeta, \eta\}$ be the coordinates adjacent to $\mathit{\boldsymbol{\gamma }}(s_0)$, then

which can be regarded as an approximate equation for the structure of the curve $\mathit{\boldsymbol{\gamma }} = \mathit{\boldsymbol{\gamma }}(s)$ near the point $\mathit{\boldsymbol{\gamma }}(s_0)$. We can obtain the shape of the curve near the point $s_0$ is completely determined by $k_1(s_0)$.

We consider the local structure of curve $\mathit{\boldsymbol{\gamma }}(s)$ at $\mathit{\boldsymbol{\gamma }}(s_0)$:

(1) the local structure of $\mathit{\boldsymbol{\gamma }}(s)$ at $\mathit{\boldsymbol{\gamma }}(s_0)$ onto the tangent space $(\alpha = 0)$ is

as shown in the figure (see Figure 3).

(2) the local structure of $\mathit{\boldsymbol{\gamma }}(s)$ at $\mathit{\boldsymbol{\gamma }}(s_0)$ onto the principal normal vector space $(\xi = 0)$ is

as shown in the figure (see Figure 4).

(3) the local structure of $\mathit{\boldsymbol{\gamma }}(s)$ at $\mathit{\boldsymbol{\gamma }}(s_0)$ onto the 1st binormal normal vector space $(\zeta = 0)$ is

as shown in the figure (see Figure 5).

(4) the local structure of $\mathit{\boldsymbol{\gamma }}(s)$ at $\mathit{\boldsymbol{\gamma }}(s_0)$ onto the 2nd binormal vector space $(\eta = 0)$ is

as shown in the figure (see Figure 6).

7.   Examples

Here, we consider an example of $\mathit{\boldsymbol{\gamma }}(s)$ in $\mathbb{S}^{3}_{1}$ with $\boldsymbol{ V}$ to show the major results of the paper. The graphics of $\mathit{\boldsymbol{\gamma }}(s)$, the tangent indicatrix of $\mathit{\boldsymbol{\gamma }}(s)$, the binormal lightlike surface, and the singularities are given in the following text.

Since it's impossible to draw a graphic in four dimensions, we only show projections to the 3-dimension tangent space generated by $\{\boldsymbol{T}, \boldsymbol{N}, \boldsymbol{B}\}$.

Example 7.1. Let $\mathit{\boldsymbol{\gamma }}:I\rightarrow \mathbb{S}^{3}_{1}$ be a lightlike Killing magnetic curve with $\boldsymbol{ V}$ as follows:

and

By calculation, we can get $k_1 = 1, k_2 = 1$, then

Hence, the tangent indicatrix of $\mathit{\boldsymbol{\gamma }}(s)$:

and the binormal lightlike surface :

then the projection of $\mathit{\boldsymbol{\gamma }}(s)$ (see Figure 7), the tangent indicatrix of $\mathit{\boldsymbol{\gamma }}(s)$ (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) is

By calculation, we can obtain the geometric invariant $\sigma(s) = \pm \frac{\sqrt{2}}{2}$, and $\sigma'(s) = 0$ for all $s$.

8.   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 $\mathbb{S}_{1}^{3}$, which were degenerate curves. By considering the lightlike tangent vector, we constructed a new Frenet equations of a lightlike Killing magnetic curve $\gamma(s)$ using the transversality theorem. Under the view of the contact theory, we obtain some local geometric properties of the lightlike Killing magnetic curves in $\mathbb{S}_{1}^{3}$. 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.

Acknowledgements

J. G. Sun is supported by the National Natural Science Foundation of China (No.11601520).

Conflict of interest

The authors declare no conflict of interest.