Normal Partner Curves of a Pseudo Null Curve on Dual Space Forms

: In this work, a kind of normal partner curves of a pseudo null curve on dual space forms is deﬁned and studied. The Frenet frames and curvatures of a pseudo null curve and its associate normal curve on de-Sitter space, its associate normal curve on hyperbolic space, are related by some particular function and the angles between their tangent vector ﬁelds, respectively. Meanwhile, the relationships between the normal partner curves of a pseudo null curve are revealed. Last but not least, some examples are given and their graphs are plotted by the aid of a software programme.


Introduction
In differential geometry, the space associate curves for which there exist some relations between their Frenet frames or curvatures compose a large class of fascinating subjects in the curve theory, such as Bertrand curve, Mannheim curve, central trace of osculating sphere, involute-evolute curves etc. [1][2][3]. Most of the researchers aimed to explore the relationships between the partner curves. For example, in Euclidean 3-space, the classical Bertrand curves are characterized by constant distance between the corresponding points of the partner curves and by constant angle between tangent vector fields of the partner curves. Naturally, the idea of partner curves research can be moved to other spaces, such as Lorentz-Minkowski space, Galilean space and so on.
The Lorentz-Minkowski metric divides the vectors into spacelike, timelike and lightlike (null) vectors [4]. Due to the causal character of vectors, some simple problems become a little complicated and strange, such as the arc-length of null curves can not be defined similar to the definition in Euclidean space; the angles between different type of vectors need to be classified according to different conditions [5,6]. In Minkowski space, the curves are divided into spacelike, timelike and lightlike (null) curves according to the causal character of their tangent vectors. Some particular curves such as the helix, the Bertrand curve, the Mannheim curve and the normal curve, the osculating curve and the rectifying curve have been surveyed by some researchers [5,[7][8][9][10][11][12].
Based on pseudo null curves in Minkowski 3-space, in this work, we define a kind of normal partner curves of a pseudo null curve which lies on de-Sitter space and hyperbolic space, respectively. In Section 2, some fundamental facts about the pseudo null curves, the space forms and the angles between any two non-null vectors are recalled. In Section 3, the relationships between a pseudo null curve and its normal partner curves on dual space forms are explicitly expressed by some particular function and the angles between their tangent vector fields. Furthermore, the relationships between the normal partner curves of a pseudo null curve are presented through the pseudo null curve. Last but not least, some useful and interesting examples of pseudo null curves and their normal partner curves are shown vividly.
The curves in this paper are regular and smooth unless otherwise stated. 3 in terms of the natural coordinate system (x 1 , x 2 , x 3 ). Recall that a vector v is said to be spacelike, timelike and lightlike (null 1 , the vector product is given by where {e 1 , e 2 , e 3 } is an orthogonal basis in E 3 1 . An arbitrary curve r(t) : I → E 3 1 can locally be spacelike, timelike or lightlike (null) if all of its velocity vectors r (t) are spacelike, timelike or lightlike, respectively. Furthermore, the spacelike curves can be classified into three kinds according to their principal normal vectors are spacelike, timelike or lightlike, which are called the first and the second kind of spacelike curve and the pseudo null curve, respectively [13]. Among of them, the pseudo null curve is defined as the following.
where T, T = N, B = 1, N, N = B, B = T, N = T, B = 0 and T × N = N, N × B = T, B × T = B. In sequence, T, N, B is called the tangent, principal normal and binormal vector fields of r(s), respectively. The function κ(s) is called the pseudo null curvature of r(s).

Remark 1 ([8]
). Every pseudo null curve r(s) in E 3 1 is planar no matter the value of the pseudo null curvature.
The authors of [14] characterized pseudo null curves with the structure function as the following.

Proposition 2 ([14]
). Let r(s) be a pseudo null curve in E 3 1 . Then r(s) and its pseudo null curvature κ(s) can be written as where 0 = c ∈ R, g = g(s) is non-constant function which is called the structure function of r(s).

Definition 2 ([15]
). 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 lightlike 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 .
is called an associate normal curve of r(s) on de-Sitter space, an associate normal curve of r(s) on hyperbolic space for some non-zero differentiable function λ = λ(s), respectively. In a word, b 1 (s) and b 2 (s) are called normal partner curves of r(s) on dual space forms.
To serve the discussions, some fundamental facts of curves lying on space forms will be reviewed.
At last, let us recall the notion of angles between two arbitrary non-null vectors in E 3 1 .
• If u and v span a timelike vector subspace. Then we have | u, v | > u v and hence, there is a unique positive real number θ such that The real number θ is called the Lorentz timelike angle between u and v.
• If u and v span a spacelike vector subspace. Then we have | u, v | ≤ u v and hence, there is a unique The real number θ is called the Lorentz spacelike angle between u and v.

Definition 5 ([16]
). Let u be a spacelike vector and v a future pointing timelike vector in E 3 1 . Then there is a unique non-negative real number θ such that The real number θ is called the Lorentz timelike angle between u and v.

Definition 6.
[16] Let u and v be future pointing (past pointing) timelike vectors in E 3 1 . Then there is a unique non-negative real number θ such that The real number θ is called the Lorentz timelike angle between u and v.

Remark 2 ([6]
). Physically, this designation of the future pointing and past pointing timelike vectors corresponds to a choice of an arrow of time at the given point, therefore Equations (6) and (7) include all definitions of angles between non-null vectors and timelike vectors.

Remark 3.
The angles between a lightlike vector and an arbitrary spacelike vector, timelike vector or another lightlike vector which is independent to it have been defined in [6]. We do not recall the details here, because they are not involved in this paper.

Main Conclusions
In this section, the associate normal curves of a pseudo null curve on de-Sitter space and hyperbolic space, respectively will be discussed. At the same time, the relationships between the normal partner curves will be presented.

Associate Normal Curves of a Pseudo Null Curve on de-Sitter Space
Let r(s) be a pseudo null curve framed by {T, N, Case (1): δ 0 = 1, i.e., β 1 is spacelike. In order to distinguish different cases, we rewrite Equation (2) as the following: where s is the arc-length of b 1 (s), and γ + Taking derivative on both sides of γ + 1 (s) = b 1 (s) with respect to the arc-length s of r(s), we get where f = f (s) = √ 2(λ + λκ). Making inner product on both sides of Equation (8) with itself, we get ( ds substituting it into Equation (8), we get Due to T, β + 1 are spacelike vectors and T × β is spacelike, then T and β + 1 span a timelike subspace. According to Equation (4), we have where θ + 1 is the Lorentz timelike angle between T and β + 1 . Together with T, (10) can be rewritten as Differentiating Equation (11) with respect to s and by Equation (9), we have Taking inner product on both sides of Equation (12) with itself, we get (13) we have Then, by substituting Equation (14) and (12), we can obtain Case (2): δ 0 = −1, i.e., β 1 is timelike. Similar to the process of Case 1, we can rewrite Equation (2) as the following: where s is the arc-length of b 1 (s), and γ − Taking derivative on both sides of γ − 1 (s) = b 1 (s) with respect to the arc-length s of r(s), we get where f = f (s) = √ 2(λ + λκ). Making inner product on both sides of Equation (15) with itself, we get ( ds substituting it into Equation (15), we get Due to T is spacelike, β − 1 is timelike, according to Equation (6), we have where θ − 1 is the Lorentz timelike angle between T and β − 1 . Together with T, (17) can be rewritten as Differentiating Equation (18) with respect to s and by Equation (16), we have Taking inner product on both sides of Equation (19) with itself, we get Then, by substituting Equation (21) and γ − 1 = 1 Based on above discussions, we can get the following conclusions. (λN + 1 λ B) its associate normal curve on de-Sitter space framed by {α 1 , β 1 , γ 1 = b 1 }.
It is obvious that the case f = 0 is excluded in the first case of Theorems 1 and 2. In fact, when f = 0, i.e., λ + λκ = 0, by solving the differential equation, we get (λN + 1 λ B) its associate normal curve on de-Sitter space framed by the Frenet frame of r(s) and the pseudo spherical Frenet frame of b 1 (s) can be related by Proof of Corollary 1. When λ(s) = ce − κ(s)ds = c g (s) , (0 = c ∈ R), by taking derivative on both sides of γ + 1 (s) = b 1 (s) with respect to the arc-length s of r(s), we get Making inner product on both sides of Equation (22) with itself, we get ( ds ds ) 2 = 1 2λ 2 . Then, we get Obviously, the arc-lengths = c 0 g(s), (0 = c 0 ∈ R) from Equation (23). Substituting Equation (23) into Equation (22), we get Taking derivative on both sides of Equation (24) with respect to s and by Equation (23), we have Making inner product on both sides of Equation (25) with itself, we have κ + 1 = ±1. Then from γ + 1 = 1 √ 2 (λN + 1 λ B) and Equation (25), we can obtain The proof is completed.
Similar to the procedure of the associate normal curve of a pseudo null curve on de-Sitter space b 1 (s), for the associate normal curve of a pseudo null curve on hyperbolic space b 2 (s), when f = 0, i.e., λ(s) = ce − κ(s)ds = c g (s) , (0 = c ∈ R), we have the following conclusions.

Corollary 2.
Let r(s) be a pseudo null curve framed by {T, N, B} with pseudo null curvature κ(s) and structure function g(s), b 2 (s) = 1 √ 2 (λN − 1 λ B) its associate normal curve on hyperbolic space framed by the Frenet frame of r(s) and the hyperbolic Frenet frame of b 2 (s) can be related by Remark 5. The proof of Corollary 2 is omitted here since it is very similar to Corollary 1. Obviously, the results in Theorems 3 and 4 still hold for f = 0, i.e., λ(s) = ce − κ(s)ds = c g (s) , (0 = c ∈ R).

The Relationships of the Normal Partner Curves
In this section, we state the relations of the normal partner curves on dual space forms using the knowledge of linear algebra and the results obtained in Sections 3.1 and 3.2. (λN − 1 λ B) framed by {α 2 , β 2 , γ 2 = b 2 } be normal partner curves of r(s) on dual space forms.
Proof of Theorem 5. From Theorems 1 and 3, by some matrix calculations, it is easy to get the conclusions.