Null Darboux Curve Pairs in Minkowski 3-Space

: Based on the fundamental theories of null curves in Minkowski 3-space, the null Darboux mate curves of a null curve are deﬁned which can be regarded as a kind of extension for Bertrand curves and Mannheim curves in Minkowski 3-space. The relationships of null Darboux curve pairs are explored and their expression forms are presented


Introduction
The associated curves or the curve pairs, i.e., two curves related to each other at the corresponding points, play important roles in the curve theory of differential geometry. The most fascinating examples are Bertrand curves and Mannheim curves in three-dimensional space. Taking Euclidean 3-space as an example, a Bertrand curve shares its normal line with another curve and its curvature κ, torsion τ satisfy λκ + µτ = 1 for some constants λ and µ [1]; the principal normal line of a Mannheim curve coincides with the binormal line of another curve and its curvature κ, torsion τ satisfy κ = λ(κ 2 + τ 2 ) for some constant λ [2]. Over years, many mathematicians extended the notions of curve pairs, such as Bertrand curve, Mannheim curve, evolute and involute and so on from Euclidean space to Lorentz-Minkowski space [3][4][5].
The Darboux vector comes to mind naturally when we consider the fact that most curve pairs are proposed from the frame of a space curve. The Darboux vector of a space curve describes the direction of rotation axis of a Cartan frame. Explicitly, for a curve r(s) framed by {T(s), N(s), B(s)} in three-dimensional space, the Darboux vector D(s) is the axis around which the Frenet frame rotates when r(s) does real-time spirals, and D(s) satisfies Darboux equations as follows Motivated by the definitions of Bertrand curve and Mannheim curve, we can consider another kind of associated curve by setting a condition that two space curves share the same Darboux vector field at the corresponding points in Minkowski 3-space. It is well known that there are three kinds of typical vectors, i.e., space-like, time-like and null (lightlike) vectors, and the curves are classified into space-like, time-like and null (light-like) curves according to the causal character of their tangent vectors, correspondingly. Among them, the null curve is quite different because the norm of its tangent vector vanishes everywhere [6][7][8].
One of the authors found a kind of representation form of null curves and some special null curves or curve pairs are discussed [5]. Based on previous works, the null Darboux curve pairs in three-dimensional Minkowski space are investigated. In Section 2, some basic facts for space-like, time-like and null (light-like) curves are recalled. Meanwhile, the null Darboux curve and its Darboux mate curve are defined explicitly. In Sections 3-5, the space-like Darboux mate curves, time-like Darboux mate curves and null Darboux mate curves of a null curve are studied, respectively.
All geometric objects are smooth and regular unless otherwise stated.

Preliminaries
Let E 3 1 be a Minkowski 3-space with natural Lorentzian metric in terms of the natural coordinate system (x 1 , Then their scalar product is given by and the exterior product by where {e 1 , e 2 , e 3 } is an orthonormal basis in E 3 1 . One can have A vector υ∈E 3 1 is said to be space-like if υ, υ > 0 or υ = 0; time-like if υ, υ < 0; null (light-like) if υ, υ = 0 respectively, which is called the causal character of the vector. An arbitrary curve r(t) is space-like, time-like or light-like if its velocity vector is space-like, time-like or light-like. At the same time, the space-like curves in E 3 1 can be classified into the first kind space-like curve, the second kind space-like curve and the pseudo null curve according to the causal character of their principal normal vectors [6,9].  where the function κ(s) is called the null curvature function of r(s).

Remark 1.
Hereafter, a null geodesic in E 3 1 is excluded.

Proposition 3 ([5]
). Let r(s) be a null curve parameterized by null arc length s in E 3 1 . Then r(s) can be written as where f (s) is the structure function which satisfies Similar to the definitions of Bertrand curve and Mannheim curve, we can define a new kind of curve pair with the Darboux vector of a null curve as follows: Definition 1. Let r(s) be a null curve with Darboux vector field D(s), andr(s) another space curve with Darboux vector fieldD(s) in E 3 1 . Ifr(s) shares the same Darboux vector field as r(s), then r(s) is called a null Darboux curve andr(s) its Darboux mate curve.

Remark 2.
A null Darboux curve r(s) and its Darboux mate curver(s) can be related bỹ for some non-zero function λ(s), which is called the distance function between r(s) andr(s).

Remark 3.
For convenience, we recall the Darboux vector fields of space curves in E 3 1 as follows:

Null Darboux Curve and Its Space-Like Darboux Mate Curves
Let r(s) be a null Darboux curve framed by {T, N, B} andr(s) its space-like Darboux mate curve framed by {T,Ñ,B}. From Remark 2,r(s) can be expressed bỹ where λ(s) is the distance function. Taking derivative on both sides of (1) with respect to the null arc length s, we get Taking the scalar product on both sides of (2), we have This means that Therefore, (2) can be simplified with the help of (4) as

The First Kind Space-Like Darboux Mate Curves
Letr(s) be the first kind space-like Darboux mate curve of r(s). From Definition 1 and Remark 3, we knowD = 0 D, Taking the scalar product on both sides of (6), we obtaiñ whereκ,τ are the curvature and torsion ofr(s), κ is the null curvature of r(s).
In the following, we explore the explicit representations of a null Darboux curve and its first kind space-like Darboux mate curve.
Theorem 5. Let r(s) be a null Darboux curve with a first kind space-like Darboux mate curve. Then it can be represented as 1 , u(s) and v(s) are given by (17) and (18).
if c > 0, then Z 1 (s) is the cylinder function, J 1 (s) is the Bessel function of the first kind and Y 1 (s) is the Bessel function of the second kind [10].

Corollary 1. Letr(s) be the first kind space-like Darboux mate curve of a null Darboux curve r(s).
Then it can be represented as where 0 = a = λ (s), c ∈ R, C 1 , C 2 , C 3 ∈ E 3 1 , u(s) and v(s) are stated as Theorem 5.
Proof. From the expression form of the null Darboux curve r(s) in Theorem 5, through calculations, the Darboux vector D(s) of r(s) is obtained as From Theorem 1 and Remark 2, the conclusion can be achieved easily.

The Second Kind Space-Like Darboux Mate Curves
Letr(s) be the second kind space-like Darboux mate curve of r(s). From Definition 1 and Remark 3, we knowD = 0 D, ( 0 = ±1), i.e., whereκ,τ are the curvature and torsion ofr(s), κ is the null curvature of r(s).
Taking the scalar product on both sides of (19), we obtaiñ Furthermore, by taking the scalar product on both sides of (5) and (19), we obtaiñ From Equation (3), 2λ [(λκ) − 1] must be positive, we getκ = 0 identically. In this case,r(s) is a straight line which has no meaning.

Theorem 6.
A second kind space-like curve can not be the Darboux mate curve of a null Darboux curve.

Theorem 7.
A pseudo null curve can not be the Darboux mate curve of a null Darboux curve.

Null Darboux Curve and Its Time-Like Darboux Mate Curves
Let r(s) be a null Darboux curve framed by {T, N, B} andr(s) its time-like Darboux mate curve framed by {T,Ñ,B}. From Remark 2,r(s) can be expressed bỹ where λ(s) is the distance function. Taking derivative on both sides of (25) with respect to the null arc length s, we get Taking the scalar product on both sides of (26), we have It means that Therefore, (26) can be simplified with the help of (28) as From Definition 1 and Remark 3, we knowD = 0 D, ( 0 = ±1), i.e., Taking the scalar product on both sides of (30), we obtaiñ whereκ,τ are the curvature and torsion ofr(s), κ is the null curvature of r(s).
Taking the scalar product on both sides of (29) and (30), we have From (31) and (32), we also have Substituting (29), (32) and (33) into (30), thenB can be written as Taking the exterior product on both sides of (29) and (34), we get Differentiating (35) with respect to the null arc length s, we get Taking the scalar product on both sides of (36), we obtain ( ds ds ) 2 = 1. Then from (27), Consequently, (29) can be simplified as Taking derivative on both sides of (38) with respect to the null arc length s, we get Considering (33), (35) and (39), we have 0 = − 1 , λ = 0 and (λκ) = 0. Thus λ(s) = as + b, (a = 0, b ∈ R). Through appropriate transformation, we can let b = 0. Then from (37), the null curvature κ(s) can be written as From (32) and (33), the curvature and the torsion ofr(s) can be expressed as Meanwhile, the frame {T,Ñ,B} can be expressed by {T, N, B} as follows where 0 , 3 = ±1. Based on the above discussions, we have the following results.

Null Darboux Curve and Its Null Darboux Mate Curves
Taking the scalar product on both sides of (40), we obtainκ = κ, where κ andκ are the null curvatures of r(s) andr(s), respectively.
Meanwhile, from Remark 2,r(s) can be expressed bỹ where λ(s) is the distance function.
Considering (40), (49) andκ = κ, we havẽ Taking the exterior product on both sides of (49) and (50), we obtaiñ On the other hand, differentiating (49) with respect to the null arc length s, we have From (51) and (52), we have 1 − λκ = 1. It implies that the null curvature κ is a constant. Summarizing the above process, we have the following conclusions.
Theorem 12. The distance function λ(s) between a null Darboux curve r(s) and its null Darboux mate curver(s) is a non-zero constant. In the following, we investigate the explicit representations of a null Darboux curve and its null Darboux mate curve.
Theorem 15. Let r(s) be a null Darboux curve andr(s) its null Darboux mate curve. Then 1.