Darboux Associated Curves of a Null Curve on Pseudo-Riemannian Space Forms

: In this work, the Darboux associated curves of a null curve on pseudo-Riemannian space forms, i.e., de-Sitter space, hyperbolic space and a light-like cone in Minkowski 3-space are deﬁned. The relationships of such partner curves are revealed including the relationship of their Frenet frames and the curvatures. Furthermore, the Darboux associated curves of k-type null helices are characterized and the conclusion that a null curve and its self-associated curve share the same Darboux associated curve is obtained.


Introduction
The geometry in Minkowski space is very important and interesting in both mathematics and physics. It is well-known that there exist three kinds of curves, i.e., space-like curve, time-like curve and light-like (null) curve in Minkowski space. Many topics in classical differential geometry of Riemannian manifolds can be extended into those of Lorentz-Minkowski manifolds. However, the geometry of null curves has no Riemannian analogs because one can not define the arc length parameter of null curves in a natural way due to the norm of the light-like vector vanishing everywhere.
Bejancu [1] studied the properties of general null curves in Minkowski space. In 1998, Nersessian and Romos [2,3] have shown the importance of null curves in physics and mathematics by showing that there exists a geometric particle model associated with null curves in Minkowski space. Inoguchi and Lee [4] proved the existence of a canonical representation of null curves in Minkowski 3-space. All these works proved that it is possible to study the null curves if the appropriate method can be applied. Wang and Pei [5] defined the Darboux (rotation) vector of a null curve which describes the direction of rotation axis of a Cartan frame in Minkowski 3-space. Nesovic et al. [6] defined a k-type null Cartan slant helices lying on a time-like surface according to their Darboux frame. One of the authors and Kim [7] defined the structure function of null curves and studied the directional associated curves of a null curve in Minkowski 3-space. Making use of the structure function of null curves, most of works about null curves have been pushed forward greatly, such as the generalized null scrolls that are characterized via the structure function of null curves [8].
In this paper, the Darboux associated curves of a null curve on three pseudo-Riemannian space forms are defined and studied. In Section 2, some fundamental facts of null curves, space forms and the Darboux vector of a null curve are reviewed. In Section 3, the relationships between a null curve and its Darboux associated curve on three pseudo-Riemannian space forms, i.e., de-Sitter space, hyperbolic space and light-like cone are discussed respectively. Particularly, the Darboux associated curves of k-type null helices are characterized and some typical examples are given. Last but not least, the relationship of two null curves which share the same Darboux associated curves is found.
Throughout this paper, all geometric objects under consideration are smooth and regular unless otherwise stated.

Preliminaries
Let E 3 1 be the Minkowski 3-space with natural Lorentzian metric in terms of the natural coordinate system (x 1 , An arbitrary curve r in E 3 1 is space-like, time-like or light-like (null) if its tangent vector r is space-like, time-like or light-like (null), correspondingly. For null curves, we have  In the sequel, T(s), N(s) and B(s) is called the tangent, principal normal and binormal vector field of r(s), respectively. From Equation (1), it is easy to know that κ(s) = − 1 2 r (s), r (s) . The function κ(s) is called the null curvature of r(s) which is an invariant under Lorentzian transformations [4]. Hence, a null curve is only determined by its null curvature.
In [7], the authors introduced the structure function and the representation formula of a null curve. Namely, Proposition 2. [7] Let r = r(s) : I → E 3 1 be a null curve. Then r can be written as where f is called the structure function of r. And the null curvature of r can be expressed by For a null curve r(s) with Frenet frame {T(s), N(s), B(s)} and null curvature κ(s), the Darboux (rotation) vector field D(s) along r(s) is defined as D(s) = κ(s)T(s) + B(s) (see details in [5]). Definition 1. [9] Let p be a fixed point in E 3 1 and r > 0 be a constant. Then the pseudo-Riemannian space forms, i.e., the de-Sitter space S 2 1 (p, r), the hyperbolic space H 2 0 (p, r) and the light-like cone Q 2 1 (p) are defined as The point p is called the center of S 2 1 (p, r), H 2 0 (p, r) and Q 2 1 (p). When p is the origin and r = 1, we simply denote them by S 2 1 , H 2 and Q 2 1 .

Definition 2.
Let r = r(s) : I → E 3 1 be a null curve with Darboux (rotation) vector field D(s). Then the curvē r(s) : is called the Darboux associated curve of r(s) on M 2 (ε 0 ).
. If D is light-like, then the Darboux associated curve of r(s) degenerates asr(s) = B(s) ∈ Q 2 1 .

Main Results
In this section, we study the Darboux associated curve of a null curve on de-Sitter space S 2 1 , hyperbolic space H 2 and light-like cone Q 2 1 , respectively.
the curvatureκ(s) ofr(s) can be given bȳ Proof of Theorem 1. When κ > 0, i.e.,r(s) ∈ S 2 1 for all s, we have Differentiating Equation (8) by using Equations (4) and (1), it gives wheres is the arc length ofr. Taking scalar product on both sides of Equation (9), we obtain Equation (10) Hence, Equation (9) can be rewritten by Differentiating Equation (11), we get From Equation (12), we can find easilyκ Taking scalar product on both sides of Equation (13), we obtain α = ±N and Thus, the curvatureκ(s) ofr(s) is given by Equation (7).

Darboux Associated Curve of a Null Curve on Hyperbolic Space
Theorem 2. Let r = r(s) : I → E 3 1 be a null curve with Frenet frame {T, N, B} and null curvature κ(s) < 0, r(s) its Darboux associated curve on H 2 with Frenet frame {α, β, γ} and curvatureκ(s). Then the Frenet frames of r(s) andr(s) satisfy the curvatureκ(s) ofr(s) can be given bȳ Proof of Theorem 2. When κ < 0, i.e.,r(s) ∈ H 2 for all s, by similar calculations in Theorem 1, we can find the conclusions easily. Proof of Theorem 3. When κ = 0, i.e.,r(s) = B(s) ∈ Q 2 1 for all s. From Remark 2, the conclusion is straightforward.

Darboux Associated Curves of K-Type Null Helices on Pseudo-Riemannian Space Forms
First of all, let us review the definition of k-type null helix in E 3 1 and the relevant results.    when a = c 2 − 1 and c = 0, ±1, we have
Proof of Theorem 6. From Proposition 1 and Theorem 5, through direct calculations, the Frenet frame of r(s) can be expressed as follows: 1. when a = c 2 − 1 and c = 0, ±1, we have

Therefore, the Darboux vector D(s) = κ(s)T(s) + B(s) of r(s) can be written by
when a = −1 and c = 0, we have
Proof of Theorem 7. By Definition 2 and Theorem 6, the Darboux associated curver(s) of r(s) can be obtained easily.
From Theorem 1, Theorem 2 and Theorem 4, we have the following result.

Corollary 1.
Let r = r(s) : I → E 3 1 be a null curve. Then r(s) is a 2-type null helix if and only if its Darboux associated curver(s) has nonzero constant curvature.
From Theorem 7, the Darboux associated curver(s) can be obtained as which satisfies r,r = −1, i.e.,r(s) ∈ H 2 .(see Figure 3 and 4)   Last but not least, we study the null curves which share the same Darboux associated curves on pseudo-Riemannian space forms. Theorem 8. Let r i : I → E 3 1 be null curves with non-constant null curvatures κ i (i = 1, 2). If the Darboux associated curvesr i of r i satisfyr 1 =r 2 , then the null curvatures κ i of r i satisfy κ 1 = κ 2 or κ 1 = 1 κ 2 .
In 2015, one of the authors and Kim defined the self-associated curves of a null curve as follows: the null curvatureκ ofr satisfyκ = 1 κ .

Corollary 2.
Let r = r(s) : I → E 3 1 be a null curve andr its self-associated curve. Then they share the same Darboux associated curve on pseudo-Riemannian space forms.
Proof of Corollary 2. From Definition 2 and Theorem 9, the Darboux associated curver ofr can be written byr wherer is the Darboux associated curve of r. Then the result is achieved easily.

Remark 5.
Even the result in Corollary 2 shows that two null curves whose null curvature are reciprocal have the same Darboux associated curves, but it does not mean the opposite of conclusion in Theorem 8 holds.