On the Geometrical Properties of the Lightlike Rectifying Curves and the Centrodes

: This paper mainly focuses on some notions of the lightlike rectifying curves and the centrodes in Minkowski 3-space. Some geometrical characteristics of the three types of lightlike curves are obtained. In addition, we obtain the conditions of the centrodes of the lightlike curves are the lightlike rectifying curves. Meanwhile, a detailed analysis between the N -type lightlike slant helices and the centrodes of lightlike curves is provided in this paper. We give the projections of the lightlike rectifying curves to the timelike

Motivated from [6,7,9,[17][18][19][20][21][22][23], there are many new geometrical properties of the lightlike curves compared with the spacelike curves and the timelike curves. In physics, the most important property of a lightlike curve is clear from the following fact: a classical relativistic string is a single lightlike curve [19,20]. Penrose indicated that null curves were null geodesics [22]. Hiscock revealed that the null hypersurfaces were a part of horizon [23].
However, until recently, to the best of our knowledge, there has been little information available in the literature concerning the lightlike rectifying curves. In the present paper, we pursue and describe the geometrical characteristics of lightlike rectifying curves in Minkowski 3-space. In addition, we obtain the relationship between centrodes of the lightlike curves and the N-type lightlike slant helices.
We organize the present manuscript as follows: the second section contains the basic notions of Minkowski 3-space and the Frenet frame of a lightlike curve in R 3 1 . We focus on geometrical characteristics of the three types of lightlike curves (lightlike rectifying curves, lightlike normal curves, and lightlike osculating curves) in Section 3. The main conclusions (Theorems 1 and 4) describe the geometrical properties of the lightlike rectifying curves. For the remainder of this article, we show the relationship between the N-type lightlike slant helices and the centrodes of the lightlike curves in Section 4. In Section 5, we deal with the projection equations of the lightlike rectifying curves to the timelike planes. Finally, two examples and graphs are used to certify our main conclusions.

Preliminaries
} be a three-dimensional vector space. For any vectors x = (x 1 , x 2 , x 3 ) and y = (y 1 , y 2 , y 3 ) in R 3 , the pseudo scalar product of x and y is defined to be x, y = −x 1 y 1 + x 2 y 2 + x 3 y 3 . (R 3 , , ) is called Minkowski 3-space and written by R 3 1 . A vector x in R 3 1 \ {0} is called a spacelike vector, a lightlike vector or a timelike vector if x, x is positive, zero, or negative, respectively. The norm of a vector x ∈ R 3 1 is defined by x = | x, x |. For any two vectors x and y in R 3 1 , we call x pseudo-perpendicular to y if x, y = 0. The pseudo vector product of vectors x and y is defined by where {e 1 , e 2 , e 3 } is the canonical basis of R 3 1 . One can easily show that a, x ∧ y = det(a, x, y). For a real number c, we define the hyperplane with pseudo normal vector n by HP(n, c) = {x ∈ R 3 1 | x, n = c}. We call HP(n, c) a spacelike hyperplane, a timelike hyperplane or a lightlike hyperplane if n is a timelike, spacelike or lightlike vector, respectively. The Frenet Equations of α(s) are given as follows [18]: where κ 1 (s) = t(s), n (s) and κ 2 (s) = b(s), n (s) are called the curvature function and torsion function of α(s), respectively.

Three Types of the Lightlike Curves
In this section, we consider some geometrical properties of the three types of lightlike curves (lightlike rectifying curves, lightlike normal curves, and lightlike osculating curves). Meanwhile, a necessary and sufficient condition of the lightlike rectifying curves is given in Theorem 4.
In the following, we would like to obtain the geometrical properties of the other two types of lightlike curves in R 3 1 .

α(s) = λ(s)t(s) + µ(s)n(s).
Differentiate the equation It is worth mentioning that the important applications of the rectifying curves used in mathematic and physics [4,9,11,12] in R 3 1 . Proof. Let α(t) be a lightlike rectifying curve with κ 1 (t) = 0. According to Theorem 1, the distance function ρ of the curve satisfies ρ 2 = C 1 + C 2 e t for some constants C 1 , C 2 . Making a suitable parameter t, we have ρ 2 = C + e t for constant C. Define a curve in S 2 1 by Differentiate the equation with respect to t, where v =| β (t) | is the pseudo unit speed function of the spherical curve β(t).
Since β(s) and β (s) are orthonormal vector fields, we can obtain that α(s) is lightlike curve and β(s) is a timelike unit curve in H 2 0 .

Remark 1. By the same method as [8], if β(s)
is a unit speed curve in S 2 1 or H 2 0 , for any constants C and s 0 , we have that α(s) = Ce s+s 0 β(s) is a lightlike rectifying curve in R 3 1 .

The Centrodes of Lightlike Curves
In this section, the centrodes of lightlike curves and the N-type lightlike slant helices in R 3 1 are given. Meanwhile, we obtain the relationship between the centrodes of lightlike curves and the N-type lightlike slant helices. Theorem 5. The centrode of a lightlike curve α(s) is a lightlike rectifying curve, when the curvature κ 1 (s) = Ce −s and the torsion κ 2 (s) = κ 1 (s). As the same, when the centrode of a lightlike curve α(s) satisfies the torsion κ 2 (s) = Ce −s and the curvature κ 1 (s) = κ 2 (s), the lightlike curve α(s) is also a lightlike rectifying curve, where C is a constant.

Using the Frenet Equations
Substitute v d , Hence, by Theorem 6, the curve α(s) is an N-type lightlike slant helix.

The Projections of the Lightlike Rectifying Curves
The projections of the non-lightlike curves onto the lightlike planes in R 3 1 were given in [9]. The lightlike curves belong to one of the lightlike planes in R 3 1 . In this section, we study the projections of the lightlike rectifying curves onto the timelike planes.
Theorem 8. Let α(s) be a lightlike rectifying curve in R 3 1 and an orthogonal projection β(s) on the timelike plane. Then, we can obtain the parametrical α(s) as where C is a constant.
In the following text, we only consider case 2, case 1 is a special case of case 2. When µ = C 1 s + C 2 , the arclength parameter of β is and When = 1 or = −1, the lightlike rectifying curve α(s) = e s γ(s) belongs to S 2 1 or H 2 0 , respectively. In addition, By the conditions γ(s), γ(s) = and γ (s), γ(s) = 0, we have

Some Examples
In this section, we give some examples about the lightlike rectifying curves to certify our main conclusions. The graphics of the lightlike rectifying curves and the centrodes are described in the following graphs. In addition, we give the projection graph of a lightlike rectifying curve to the timelike plane in Example 1.   the centrode d(s) of the lightlike curve α(s) is in Figure 3.

Conclusions
This paper considered the geometrical properties of three types of lightlike curves in Minkowski 3-space (lightlike rectifying curves, lightlike normal curves, and lightlike osculating curves). The lightlike rectifying curves are mainly studied. In addition, the conditions that the centrodes of the lightlike curves are the lightlike rectifying curves are obtained. Furthermore, we obtain the relationship between the N-type lightlike slant helices and the centrodes of lightlike curves. In addition, the projections of the lightlike rectifying curves to the timelike planes are researched.
In the following research, we will continue study the geometric properties of the lightlike rectifying curves in high dimensional spaces, such as in 4-space. Some unique geometric properties of curves in high dimensional space are desired to be obtained in the future.