A natural Frenet frame for null curves on the lightlike cone in Minkowski space R24$\mathbb{R} ^{4}_{2}$

In this paper, we investigate the representation of curves on the lightlike coneQ2 in Minkowski space R2 by structure functions. In addition, with this representation, we classify all of the null curves on the lightlike coneQ2 in four types, and we obtain a natural Frenet frame for these null curves. Furthermore, for this natural Frenet frame, we calculate curvature functions of a null curve, especially the curvature function κ2 = 0, and we show that any null curve on the lightlike cone is a helix. Finally, we find all curves with constant curvature functions.


Introduction
The study of semi-Riemannian manifolds plays an important role in differential geometry and physics, especially in the theory of relativity. In a semi-Riemannian manifold, the induced metric on a lightlike submanifold is degenerate. In general relativity, lightlike submanifolds usually appear to be some smooth parts of the achronal boundaries, for example, event horizon of Kruskal and Kerr black holes and the compact Cauchy horizons in Taub-NUT spacetime [5,14]. One of the simplest examples of lightlike submanifolds is the lightlike cone Q n q in Minkowski space R n q . In differential geometry, one of the most important and applicable tools to analyse a curve is orthonormal frame. For a regular curve in Euclidean space R n , we can use 1st, 2nd, . . . , nth derivative vectors to construct Frenet frame [8]. Abazari, Bohner, Sağer, and Yayli [2] studied the relationship between Frenet elements of the stationary acceleration curve in four-dimensional Euclidean space. Also, by Frenet elements, they have provided a necessary and sufficient condition for a curve on a timelike surface that is an acceleration curve.
Bonnor [4] introduced Cartan frame as the most useful frame, and he used this frame to study null curves. Bejancu [3] gave a method for consideration of a null curve in semi-Riemannian manifold. Ferrández, Giménez and Lucas [7] generalized the Cartan frame to Lorentzian space form. Abazari, Bohner, Sağer, and Sedaghatdoost [1] studied some properties for spacelike curves in lightlike cones of index 1. Liu [12] studied curves in the lightlike cone and gave an asymptotic frame field along the curve and defined cone curvature functions for this frame field. To study the behavior of a curve in two-and threedimensional lightlike cone, Liu and Meng [13] defined structure functions for a spacelike curve, and by these structure functions, they obtained representation formulas of spacelike curves in the lightlike cones Q 2 1 and Q 3 1 of Lorentzian space R 3 1 and R 4 1 , respectively. Also, Külahci, Bektaş and Ergüt [11] considered AW(k)-type curves in the 3-dimensional lightlike cone, and recently Külahci [10] considered spacelike normal curves on the lightlike cones Q 2 1 and Q 3 1 . Since for a null curve that lies on the lightlike cone, any order derivative vectors are null vectors [15], for a null curve x on the lightlike cone Q 3 2 , there exists a natural Frenet frame {x, x , N, W } [6,15]. Sun and Pei [15] considered null curves on the lightlike cone and unit semi-Euclidean 3-sphere of Minkowski space R 4 2 , and they obtained some results on AW(k)-type curves and null Bertrand curves on the lightlike cone Q 3 2 . In this paper, we obtain representation formulas for any curve on the lightlike cone Q 3 2 by structure functions, and by this representation, we classify all null curves on the lightlike cone. Furthermore, for a null curve on the lightlike cone, we construct a natural Frenet frame and calculate its curvature functions. Also, we show that the structure functions and the curvature functions of a null curve on the lightlike cone Q 3 2 satisfy a special secondorder differential equation, and by this natural Frenet frame, we conclude that any null curve on the lightlike cone is a helix.

Preliminaries
Let R n be n-dimensional Euclidean space. For two vectors v = (v 1 , . . . , v n ), w = (w 1 , . . . , w n ) and q ∈ N ∩ [1, n], we define the bilinear form v, w q := - which is a semi-Riemannian manifold. The resulting semi-Riemannian space is called Minkowski n-space of index q. If n = 4, then this is the simplest example of a relativistic spacetime [14]. In Minkowski space R n q , we say that a nonzero vector v ∈ R n q is spacelike, null, or timelike if v, v q is positive, zero, or negative, respectively. Also, the vector 0 is spacelike. The norm of v ∈ R n q is defined by v := √ | v, v |. In Minkowski space R n q , there exist three types of hypersurfaces, called (n -1)-pseudo-sphere or de Sitter (n -1)-space for and lightlike cone for The curve x : I → R n q is called a spacelike curve (null curve or timelike curve) if for any t 0 ∈ I, the velocity vector dx dt | t=t 0 of the curve is spacelike (null or timelike). Thus, a space-like or timelike curve can be parameterized by arc length in the sense that | dx dt , dx dt q | = 1. Therefore, the arc length parameter of x : I → R n q is denoted by s, and we have dx ds , dx ds q = ε, with ε = +1 for a spacelike curve and ε = -1 for a timelike curve. For a null curve with any parameter t, we have ε = 0 [8]. For a null curve x : where ξ (t) is the tangent vector, N(t) is the unique transversal vector to ξ (t), W (t) is the unique transversal vector to x(t) [15, (2.1)]. The vectors in this natural Frenet frame satisfy the equations which we call the natural Frenet equations of the null curve x on the lightlike cone Q 3 2 , where and the functions h, κ 1 , κ 2 are called the curvature functions of x [6]. Duggal and Jin [6, showed that x has a parameter such that the curvature function h = 0 for the null curve in R n 2 , and, in addition, Sun and Pei [15] proved the following. Hence, we can get the natural Frenet equations of a null curve on the lightlike cone Q 3 2 as x (t) = ξ (t), In this paper, the semi-Riemannian manifold is Minkowski space R 4 2 , and especially, we consider its lightlike cone Q 3 2 .
3 Representation formulas of cone curves in R 4 2 H. Liu [13] has obtained representation formulas of the spacelike curves in the lightlike cones Q 2 1 and Q 3 1 and proved the following two theorems.
Theorem 3.1 (See [13]) Let x : I → Q 2 1 ⊂ R 3 1 be a spacelike curve in Q 2 1 parameterized by arc length. Then x can be written as for some nonconstant function f , which is called the structure function of x. The structure function f and the cone curvature function κ of x satisfy Theorem 3.2 (See [13]) Let x : I → Q 3 1 ⊂ R 4 1 be a spacelike curve in Q 3 1 parameterized by arc length. Then x can be written as for some functions f and g, which are called structure functions of x. Here, ρ satisfies The structure functions f , g, the cone curvature function κ, and the cone torsion function τ of x satisfy First, in the following lemma, we give a result for spacelike and timelike curves in the lightlike cone Q 3 2 . By this virtue, we will be able to give a representation formula for such curves.
and from x 2 If one of the functions x 1 - and hence, Since μ, λ ∈ {-1, 1}, we obtain x , which is a contradiction.
be a spacelike or timelike curve parameterized by arclength in the lightlike cone Q 3 2 . Then we can write x as for some functions f and g. Here, ρ satisfies where ε = +1 for spacelike curves and ε = -1 for timelike curves.
Proof Since x, x 2 = 0, we get (5). Because the curve x is spacelike or timelike, from Lemma 3.3, we can define the nonzero smooth functions where Let x , x 2 = ε, where ε = +1 for spacelike curves and ε = -1 for timelike curves. Then so that and by (5), we have and thus From x , x 2 = 0, we have and hence so that From (8) and (9), we conclude By replacing (7) in (10), we have i.e., i.e., i.e., Since f and g are not zero, the proof is complete.
Unlike in Theorem 3.4, for a null curve in the lightlike cone, some of the functions x 1 -x 3 , x 1 + x 3 , x 4x 2 , x 4 + x 2 may be zero. Thus, in the following, we state these cases separately.
and the others are not zero.
Proof First, we prove i. Without loss of generality, we suppose which is a null straight line. Next, we prove ii. Let x be a null curve in the lightlike cone Q 3 2 . Without loss of generality, we suppose Thus x 2 1x 2 3 = 0, and x 2 4x 2 2 = 0, and this contradicts x, x 2 = 0.
In the following theorem, we classify all null curves in the lightlike cone Q 3 2 , which are not a straight line.  = (f , g, f , -g).
Here, the functions f and g are linearly independent. Type 2. If the functions x 1 + x 3 and x 2 + x 4 are not zero, then where the smooth functions ρ, f , and g are and satisfy (fgf g)ρ = 0. In this case, we have two types of curves: Type 2.1. If the functions f and g are linearly independent, then where λ is a real constant.  Type 2. In the proof of Theorem 3.4, from x 1 + x 3 = 0, x 4 + x 2 = 0, and x, x 2 = ε, we conclude In a similar way, if we set ε = 0, then we can prove Thus ρ = 0 or fgf g = 0. Type 2.1. If ρ = 0 and fgf g = 0, then ρ ≡ λ is a real constant and the functions f and g in the first equation of (11) are linearly independent, and hence Type 2.2. If ρ = 0 and fgf g = 0, then the functions f and g in the first equation of (11) are linearly dependent. Thus f = λg, where λ is a real constant, and hence x = (λ + ρ, 1λρ, λρ, 1 + λρ)g.
If ρ = 0 and fgf g = 0, then x is a straight line, contradicting the assumption.
In Table 1, we summarize all cases on the basis of (6). For a Frenet null curve in the lightlike cone Q 3 2 , there exists a nonunique natural Frenet frame {x, ξ , N, W } that satisfies (1) [15]. In the following, we construct the vector fields N and W for one of the natural Frenet frames. Now we define the natural orthogonal vector field to a null vector field in R 4 2 and prove some of its properties. Cases The type of the curve The following lemma is a direct consequence of Definition 3.8.

.
Proof Since the vector fields N and W are defined by natural orthogonal vectors to x and x , we have Moreover, since x, x 2 = 0, by Lemma 3.9 i., for any type of null curves, we have x, ξ 2 = N, W 2 = 0.
In order to obtain the vector fields N and W , it is thus sufficient to calculate x ⊥ , x 2 in all types of null curves. For a null curve of Type 1.1, we have From Theorem 3.6, f and g are not proportional, so = 0, and by Lemma 3.9 ii., we have Similarly, the calculations for null curves of Type 1.2 and Type 1.3 are valid. For a curve of Type 2.1, by direct calculations, we have This equality for a null curve of Type 2.1, by Lemma 3.9 ii., yields For a null curve of Type 2.2, we have x = Ag, where A = (λ + ρ, 1λρ, λρ, 1 + λρ).
and for a null curve of Type 2.2 are Proof By using (3) and direct calculations, we can obtain the curvature functions. For example, the curvature functions for a null curve of Type 2.1 are For a null curve of Type 2.2, let x = Ag. Then where B := 1 ρ A , A 13 ⊥ = B ⊥ 13 ρ , and B, A ⊥ 13 2 = -2(1 + λ 2 ). Therefore, Since x, x 2 = x , x 2 = 0, by Lemma 3.9 i., x ⊥ , x ⊥ 2 = x ⊥ , x ⊥ 2 = 0, so for any type of null curve, we have κ 2 = N , W = 0.
If we use the parameters presented in Proposition 2.1 for the natural Frenet frame obtained in Theorem 3.10, together with Theorem 3.11 that gives κ 2 = 0 for any type of null curve, the natural Frenet equations (4) are where κ 1 is now denoted by κ. Proof For any straight line, the statement is true. Thus, we assume that the curvature function κ(s) = 0. Let β be a constant vector such that x (s), β 2 = l for anys ∈ I. Thus, x(s), β 2 = ls + l 0 and x (s), β 2 = 0. From (12), we conclude that κ(s) x(s), β 2 = 0, and from κ(s) = 0, we get l = l 0 = 0 so that β = r 1 (s)x(s) + r 2 (s)x (s).
Theorem 3.13 Any null curve on the lightlike cone Q 3 2 is a helix.
Proof For a straight line, the tangent vector has constant angle with any constant vector. Let β be a constant vector such that x (s), β 2 = l. Then, by Lemma 3.12, By differentiation of (13), we have Again by the second equation of (12), since x i (s), i = 1, . . . , 4, and r 2 (s) satisfy the same differential equation, we can choose r 2 (s) = x i (s), i = 1, . . . , 4. In this situation, for r 2 (s) := x 1 (s), the constant vector β is The equations x 1 2 + x 2 2 = x 3 2 + x 4 2 and Kula and Yayli, in [9, Proposition 6.1], proved that, if x : R → R 4 satisfies a second-order linear homogeneous differential equation, then the image of x lies in a two-dimensional subspace of R 4 . Thus, we can prove the following corollary. Proof Since x satisfies the second-order linear homogeneous differential equation in (12), by [9, Proposition 6.1], the proof is complete.

Curves with special curvature functions
In this section, we classify all curves with constant curvature functions on the lightlike cone Q 3 2 . These curves are solutions of a second-order differential equation with constant coefficients. From now on, we assume that the parameters of the curve x(s) satisfies Proposition 2.1, i.e., h = 0. Also, since from Theorem 3.11, κ 2 = 0 for a null curve on the lightlike cone Q 3 2 with respect to the natural Frenet frame obtained in Theorem 3.10, there exists only one curvature function κ. Also, by the second equation of (12), a null curve x on the lightlike cone Q 3 2 satisfies the second-order differential equation Hence, with initial conditions x(s 0 ) = (a 1 , a 2 , a 3 , a 4 ) ∈ Q 3 2 and x (s 0 ) = (b 1 , b 2 , b 3 , b 4 ) ∈ Q 3 2 , the differential equation (14) has a unique solution. Thus, we have proved the following result. For constant curvature functions κ(s) = κ 0 , the solutions of the differential equation (14) are for κ 0 > 0, and x(s) = a 1 sin( √ -κ 0s ) + a 2 cos( √ -κ 0s ), b 1 sin( √ -κ 0s ) + b 2 cos( √ -κ 0s ), for κ 0 < 0, where it is possible to calculate the coefficients a i , b i , c i , and d i , i = 1, 2, from the initial conditions x(s 0 ) and x (s 0 ).
The following example appears in [16,Example 5.1], where the authors have obtained a natural Frenet frame and its curvature functions for the given null curve in the way of Duggal and Bejancu [6]. In this example, we calculate our natural Frenet frame by Theorem 3.10.