PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 99(113) (2016), 77–98 DOI: 10.2298/PIM1613077G

ON GENERALIZED NULL MANNHEIM CURVES IN MINKOWSKI SPACE-TIME

Milica Grbović, Kazim Ilarslan, and Emilija Nešović

Abstract. We define generalized null Mannheim curves in Minkowski spacetime and characterize them and their generalized Mannheim mate curves in terms of curvature functions, and obtain relations between their frames. We provide examples of such curves.

1. Introduction

In the Euclidean space 𝔼^{3} there are many associated curves (Bertrand mates, Mannheim mates, spherical images, evolutes, the principal-direction curves, etc.) the frame’s vector fields of which satisfy some extra conditions. In particular, Mannheim curves in 𝔼^{3} are defined by the property that their principal normal lines coincide with the binormal lines of their mate curves at the corresponding points [4, 7, 11]. The parameter equation of a Mannheim curve α in 𝔼^{3} is given in [4] by α(t) = (∫ h(t) sin(t)dt, ∫ h(t) cos(t)dt, ∫ h(t)g(t)dt), where g: I → ℝ is any smooth function and the function h: I → R is given by

h = \frac{(1 + g^{2} + g^{′2})^{3} + (1 + g^{2})^{3}(g + g′′)^{2}}{(1 + g^{2})^{3/2} (1 + g^{2} + g^{′2})^{5/2}} .

Mannheim curves and their partner curves in 3-dimensional space forms are studied in [2]. In the Euclidean 4-space, the notion of Mannheim curves is generalized in [12] as follows. A special Frenet curve α: I → 𝔼^{4} is called a generalized Mannheim curve, if there exists a special Frenet curve α^{∗}: I^{⋆} → 𝔼^{4} and a bijection ϕ : α → α^{⋆} such that the principal normal line of α at each point of α lies in the plane spanned by the first binormal and the second binormal line of α^{∗}. The generalized spacelike Mannheim curves in Minkowski space-time, the Frenet frame of which contains only non-null vectors, are characterized in [5].

In this paper, we define the generalized null Mannheim curves in Minkowski space-time. We obtain the relations between the curvature functions and the frames of the generalized null Mannheim curves and the generalized Mannheim mate curves of theirs. We give the necessary conditions for the null Cartan curve α with the second curvature κ_{2} ≠ 0 and its mate curve α^{⋆} = α + (1/2κ_{2})N to be the generalized null Mannheim curve and the generalized Mannheim mate curve, respectively. In particular, we prove that there are no generalized null Mannheim curves the generalized Mannheim mate curve of which is a partially null or a pseudo null curve. Finally, we characterize the generalized null Mannheim curves in terms of the normal curves and give some examples.

2010 Mathematics Subject Classification: Primary 53C50; Secondary 53C40.

Key words and phrases: generalized Mannheim curve, null Cartan curve, Minkowski spacetime.

Communicated by Stevan Pilipović.

2. Preliminaries

The Minkowski space-time 𝔼_{1}^{4} is the Euclidean 4-space 𝔼^{4} equipped with indefinite flat metric given by g = −dx_{1}^{2} + dx_{2}^{2} + dx_{3}^{2} + dx_{4}^{2}, where (x_{1}, x_{2}, x_{3}, x_{4}) is a rectangular coordinate system of 𝔼_{1}^{4}. Recall that a vector v ∈ 𝔼_{1}^{4}\{0} can be spacelike if g(v, v) > 0, timelike if g(v, v) < 0 and null (lightlike) if g(v, v) = 0. In particular, the vector v = 0 is said to be a spacelike. The norm of a vector v is given by ∥v∥ = \sqrt{|g(v, v)|}. Two vectors v and w are said to be orthogonal, if g(v, w) = 0. An arbitrary curve α in 𝔼_{1}^{4}, can locally be spacelike, timelike or null (lightlike), if all its velocity vectors α′(s) are respectively spacelike, timelike or null [13]. A non-null curve α is parametrized by the arc-length function s (or has the unit speed), if g(α′(s), α′(s)) = ±1. A pseudo-arc length function (or the canonical parameter) is defined in [1] by s(t) = ∫_{0}^{t} g(α′′(u), α′′(u))^{\frac{1}{4}} du. A null curve α is said to be parameterized by the pseudo-arc length function s, if g(α′′(s), α′′(s)) = 1 [1, 3].

Definition 2.1. A non-geodesic null curve α : I → 𝔼_{1}^{4} parameterized by the pseudo-arc length function s is called a Cartan curve, if there exists a unique positively oriented Cartan frame {T, N, B_{1}, B_{2}} along α and three smooth functions κ_{1}, κ_{2} and κ_{3} satisfying the Cartan equations [3].

(2.1)  [ \begin{matrix}T′ \\ N′ \\ B′_{1} \\ B′_{2} \end{matrix}] = [ \begin{matrix}0 & κ_{1} & 0 & 0 \\ −κ_{2} & 0 & −κ_{1} & 0 \\ 0 & κ_{2} & 0 & κ_{3} \\ −κ_{3} & 0 & 0 & 0 \end{matrix} ] [ \begin{matrix}T \\ N \\ B_{1} \\ B_{2} \end{matrix} ] .

The curvature functions κ_{1}(s) = 1, κ_{2}(s) and κ_{3}(s) are respectively called the first, the second and the third Cartan curvature of α. Cartan’s frame vector fields T, N, B_{1} and B_{2} are respectively called the tangent, the principal normal, the first binormal and the second binormal vector field, and they satisfy the conditions

g(T, T) = g(B_{1}, B_{1}) = 0, g(N, N) = g(B_{2}, B_{2}) = 1,

g(T, N) = g(T, B_{2}) = g(N, B_{1}) = g(N, B_{2}) = g(B_{1}, B_{2}) = 0, g(T, B_{1}) = 1.

Definition 2.2. A spacelike or a timelike unit speed smooth curve α: I → 𝔼_{1}^{4} is called a Frenet curve, if there exists a unique positively oriented orthonormal or pseudo-orthonormal Frenet frame {T, N, B_{1}, B_{2}} along α and three smooth functions κ_{1} ≠ 0, κ_{2} and κ_{3} satisfying the corresponding Frenet equations.

The smooth functions κ_{1} ≠ 0, κ_{2} and κ_{3} are respectively called the first, the second and the third Frenet curvatures of α. Let {T, N, B_{1}, B_{2}} be the moving Frenet frame along the unit speed Frenet curve α: I → 𝔼_{1}^{4}, consisting of the tangent, the principal normal, the first binormal and the second binormal vector field, respectively. Depending on the causal character of Frenet’s vector fields, we have three types of the Frenet equations.

Type 1. If α is a timelike or a spacelike Frenet curve whose orthonormal Frenet frame {T, N, B_{1}, B_{2}} contains only non-null vector fields, the Frenet equations are given by [10]

(2.2)  [ \begin{matrix}T′ \\ N′ \\ B′_{1} \\ B′_{2} \end{matrix}] = [ \begin{matrix}0 & ϵ_{2}κ_{1} & 0 & 0 \\ −ϵ_{1}κ_{1} & 0 & ϵ_{3}κ_{2} & 0 \\ 0 & −ϵ_{2}κ_{2} & 0 & −ϵ_{1}ϵ_{2}ϵ_{3}κ_{3} \\ 0 & 0 & −ϵ_{3}κ_{3} & 0 \end{matrix} ] [ \begin{matrix}T \\ N \\ B_{1} \\ B_{2} \end{matrix} ] ,

where g(T, T) = ϵ_{1}, g(N, N) = ϵ_{2}, g(B_{1}, B_{1}) = ϵ_{3}, g(B_{2}, B_{2}) = ϵ_{4}, ϵ_{1}ϵ_{2}ϵ_{3}ϵ_{4} = −1, ϵ_{i} ∈ {−1, 1}, i ∈ {1, 2, 3, 4}.

Type 2. If α is a pseudo null Frenet curve, i.e. a spacelike Frenet curve with a pseudo-orthonormal frame {T, N, B_{1}, B_{2}} and null vector fields N and B_{2}, the Frenet formulae read [14]

(2.3)  [ \begin{matrix}T′ \\ N′ \\ B′_{1} \\ B′_{2} \end{matrix}] = [ \begin{matrix}0 & k_{1} & 0 & 0 \\ 0 & 0 & k_{2} & 0 \\ 0 & k_{3} & 0 & −k_{2} \\ −k_{1} & 0 & −k_{3} & 0 \end{matrix} ] [ \begin{matrix}T \\ N \\ B_{1} \\ B_{2} \end{matrix} ] ,

where the first curvature k_{1}(s) = 1. Moreover, the following conditions are satisfied:

g(T, T) = g(B_{1}, B_{1}) = 1, g(N, N) = g(B_{2}, B_{2}) = 0,

g(T, N) = g(T, B_{1}) = g(T, B_{2}) = g(N, B_{1}) = g(B_{1}, B_{2}) = 0, g(N, B_{2}) = 1.

Type 3. If α is a partially null Frenet curve, i.e. a spacelike Frenet curve with a pseudo-orthonormal frame {T, N, B_{1}, B_{2}} and null vector fields B_{1} and B_{2}, the Frenet formulae read [14]

(2.4)  [ \begin{matrix}T′ \\ N′ \\ B′_{1} \\ B′_{2} \end{matrix}] = [ \begin{matrix}0 & k_{1} & 0 & 0 \\ −k_{1} & 0 & k_{2} & 0 \\ 0 & 0 & k_{3} & 0 \\ 0 & −k_{2} & 0 & −k_{3} \end{matrix} ] [ \begin{matrix}T \\ N \\ B_{1} \\ B_{2} \end{matrix} ] ,

where the third curvature κ_{3}(s) = 0. In particular, there holds

g(T, T) = g(N, N) = 1, g(B_{1}, B_{1}) = g(B_{2}, B_{2}) = 0,

g(T, N) = g(T, B_{1}) = g(T, B_{2}) = g(N, B_{1}) = g(N, B_{2}) = 0, g(B_{1}, B_{2}) = 1.

Recall that a normal curve in 𝔼_{1}^{4} is defined in [8] as a curve the position vector of which always lies in its normal space T^{⊥} , which represents the orthogonal complement of the tangent vector field T of the curve.

3. The generalized null Mannheim curves in 𝔼_{1}^{4}

It is proved in [6] that there are no null Mannheim curves in 𝔼_{1}^{3}. In relation to that, throughout this section we will assume that the null Cartan curve α in 𝔼_{1}^{4} has the third curvature κ_{3}(s) ≠ 0. Note that when κ_{3}(s) ≠ 0, the second curvature κ_{2}(s) of α can be equal to zero (when α is a null cubic), or different from zero. We firstly define the generalized null Mannheim curves as follows.

Definition 3.1. A null Cartan curve α: I → 𝔼_{1}^{4} is called a generalized null Mannheim curve if there exists a null Cartan curve or a Frenet curve α^{⋆}: I^{⋆} → 𝔼_{1}^{4} and a bijection ϕ: α → α^{⋆} such that at the corresponding points of the curves, the principal normal line of α is included in the plane spanned by the first binormal line and the second binormal line of α^{⋆}, under bijection ϕ.

The curve α^{∗} is called the generalized Mannheim mate curve of α.

Remark 3.1. The generalized Mannheim mate curve of the generalized spacelike Mannheim curve in 𝔼_{1}^{4}, the Frenet frame of which contains only the non-null vector fields, is defined in [5] as the spacelike Frenet curve. Unfortunately, such definition is not correct, since it can have arbitrarily causal character.

By the principal normal (binormal) line, we mean the straight line in direction of the principal normal (binormal) vector field. Throughout this section, let {T, N, B_{1}, B_{2}} and {T^{⋆}, N^{⋆}, B_{1}^{⋆}, B_{2}^{⋆}} denote the Cartan frame of the generalized null Mannheim curve α and the corresponding (Cartan or Frenet) frame of the generalized Mannheim mate curve α^{⋆} of α, respectively. Since the principal normal vector N(s) of α lies in the plane spanned by {B_{1}^{⋆}, B_{2}^{⋆}}, it satisfies the equation N(s) = a(s)B_{1}^{⋆}(s) + b(s)B_{2}^{⋆}(s), for some differentiable functions a(s) and b(s). Depending on the causal character of the plane span{B_{1}^{⋆}, B_{2}^{⋆}}, we distinguish three cases: (A) span{B_{1}^{⋆}, B_{2}^{⋆}} is a spacelike plane; (B) span{B_{1}^{⋆}, B_{2}^{⋆}} is a timelike plane; (C) span{B_{1}^{⋆}, B_{2}^{⋆}} is a lightlike plane. In what follows, we consider these three cases separately.

(A) span{B_{1}^{⋆}, B_{2}^{⋆}} is a spacelike plane.

Theorem 3.1. Let α: I → 𝔼_{1}^{4} be the generalized null Mannheim curve and α^{⋆}: I^{⋆} → 𝔼_{1}^{4} the generalized Mannheim mate curve of α such that the principal normal line of α lies in the spacelike plane spanned by {B_{1}^{⋆}, B_{2}^{⋆}}. Then α^{⋆} is the timelike Frenet curve such that the curvatures of α and α^{⋆} satisfy the relations

(3.1)  κ_{1} = 1, κ_{2} = \frac{1}{2λ} , |κ_{1}^{⋆}| = |κ_{2}^{⋆}| = |κ_{3}| ≠ 0, |κ_{3}^{⋆}| = \frac{1}{λ}, λ ∈ ℝ_{0}^{+} ,

and the corresponding frames of α and α^{⋆} are related by

(3.2)  T^{⋆} = \frac{1}{2\sqrt{λ}} (T − 2λB_{1}),
N^{⋆} = − sgn(κ_{1}^{⋆}) sgn(κ_{3})B_{2},
B_{1}^{⋆} = sgn(κ_{1}^{⋆}) sgn(κ_{2}^{⋆})\frac{1}{2\sqrt{λ}} (T + 2λB_{1}),
B_{2}^{⋆} = sgn(κ_{2}^{⋆}) sgn(κ_{3})N.

Proof. Since the principal normal line of α lies in the spacelike plane spanned by {B_{1}^{⋆}, B_{2}^{⋆}}, α^{⋆} is a spacelike or a timelike curve the Frenet frame of which satisfies the equations (2.2), where ϵ_{1}^{⋆} = −ϵ_{2}^{⋆}, ϵ_{3}^{⋆} = ϵ_{4}^{⋆} = 1. In particular, the curve α^{⋆} can be parameterized by

(3.3)  α^{⋆}(f(s)) = α(s) + λ(s)N(s),

where s is the pseudo-arc length parameter of α, s^{⋆} = f(s) = ∫_{0}^{s} ∥α^{⋆}′(t)∥dt is the arc-length parameter of α^{⋆} and f: I ⊂ ℝ → I^{⋆} ⊂ ℝ and λ are some smooth functions. We distinguish two cases: (A.1) κ_{2} = 0 and (A.2) κ_{2} ≠ 0.

(A.1) κ_{2} = 0. Differentiating relation (3.3) with respect to s and applying (2.1), we find

(3.4)  T^{⋆}f′ = T + λ′N − λB_{1}.

By taking the scalar product of (3.4) with N = aB_{1}^{⋆} + bB_{2}^{⋆}, we find λ′ = 0. Substituting this in (3.4), we get

(3.5)  T^{⋆}f′ = T − λB_{1}, λ ∈ R_{0},

where ℝ_{0} denotes ℝ\{0}. From (3.5) we have g(T^{⋆}f′, T^{⋆}f′) = ϵ_{1}^{⋆}f^{′2} = −2λ and therefore

(3.6)  f^{′2} = −2ϵ_{1}^{⋆}λ = constant ≠ 0.

Differentiating relation (3.5) with respect to s and using (2.1), (2.2) and (3.6) we find ϵ_{2}^{⋆}κ_{1}^{⋆}N^{⋆}f′^{2} = N − λκ_{3}B_{2}. By taking the scalar product of the last equation with N = aB_{1}^{⋆} + bB_{2}^{⋆}, we obtain a contradiction.

(A.2) κ_{2} ≠ 0. Differentiating relation (3.4) with respect to s and using (2.1), we find

(3.7)  T^{⋆}f′ = (1 − λκ_{2})T + λ′N − λB_{1}.

By taking the scalar product of (3.7) with N = aB_{1}^{⋆} + bB_{2}^{⋆} yields

(3.8)  λ′ = 0.

Substituting (3.8) in (3.7), we get

(3.9)  T^{⋆}f′ = (1 − λκ_{2})T − λB_{1}.

Differentiating relation (3.9) with respect to s and using (2.1) and (2.2), we obtain

(3.10)  ϵ_{2}^{⋆}κ_{1}^{⋆}N^{⋆}f^{′2} + T^{⋆}f′′= (1 − λκ_{2})′T + (1 − 2λκ_{2})N − λκ_{3}B_{2}.

By taking the scalar product of (3.10) with N = aB_{1}^{⋆} + bB_{2}^{⋆} and using (3.8), it follows that

(3.11)  κ_{2} = \frac{1}{2λ} = constant, λ ∈ ℝ_{0}.

Moreover, by using (3.9) we obtain

(3.12)  g(T^{⋆}f′, T^{⋆}f′) = ϵ_{1}^{⋆}f^{′2} = −2λ(1 − λκ_{2}).

Substituting (3.11) in (3.12) yields

(3.13)  f^{′2} = −ϵ_{1}^{⋆}λ = constant ≠ 0.

Relations (3.10), (3.11) and (3.13) imply

(3.14)  ϵ_{2}^{⋆}κ_{1}^{⋆}N^{⋆}f′^{2} = −λκ_{3}B_{2}.

Since a spacelike vector B_{2} is collinear with N^{⋆}, it follows that T^{⋆} is a timelike vector. Consequently, α^{⋆} is a timelike curve. Substituting ϵ_{1}^{⋆} = −1 = −ϵ_{2}^{⋆} in (3.13) we get

(3.15)  f^{′2} = λ, λ ∈ R_{0}^{+} ,

where ℝ_{0}^{+} denotes ℝ^{+} \{0}. Therefore,

(3.16)  f′(s) = ∥α^{⋆}′(s)∥ = \sqrt{λ}.

Substituting (3.11) and (3.16) in (3.9), we obtain

(3.17)  T^{⋆} = \frac{1}{2\sqrt{λ}} (T − 2λB_{1}).

Next, substituting ϵ_{2}^{⋆} = 1 and (3.15) in (3.14), we get κ_{1}^{⋆}N^{⋆} = −κ_{3}B_{2}. It follows that

(3.18)  κ_{1}^{⋆} = −κ_{3}, N^{⋆} = B_{2},

or

(3.19)  κ_{1}^{⋆} = κ_{3}, N^{⋆} = −B_{2}.

Assume that (3.18) holds. Differentiating the equation N^{⋆} = B_{2} with respect to s and applying (2.1) and (2.2), it follows that

(3.20)  (κ_{1}^{⋆}T^{⋆} + κ_{2}^{⋆}B_{1}^{⋆})f′ = −κ_{3}T.

By taking the scalar product of (3.20) with N = aB_{1}^{⋆} + bB_{2}^{⋆}, we find aκ_{2}^{⋆}f′ = 0. If κ_{2}^{⋆} = 0, relation (3.20) implies that a timelike vector T^{⋆} is collinear with a null vector T, which is a contradiction. Consequently, a = 0 and hence N = bB_{2}^{⋆}. The condition g(N, N) = 1 gives b^{2} = 1 and thus

(3.21)  N = B_{2}^{⋆},

or

(3.22)  N = −B_{2}^{⋆}.

Assume that (3.21) holds. Differentiating relation (3.21) with respect to s and using (2.1) and (2.2), we get −κ_{2}T − B_{1} = −κ_{3}^{⋆}f′B_{1}^{⋆}. The last relation together with (3.11) and (3.15) yield

(3.23)  B_{1}^{⋆} = \frac{1}{λ\sqrt{λ}κ_{3}^{⋆}} ( \frac{1}{2}T + λB_{1}) .

On the other hand, by using relations (3.16), (3.17), (3.18) and (3.20), we find

(3.24)  B_{1}^{⋆} = \frac{κ_{1}^{⋆}}{\sqrt{λ}κ_{2}^{⋆}} ( \frac{1}{2}T + λB_{1}) .

Since λ > 0, from (3.23) and (3.24), we get sgn(κ_{1}^{⋆}) sgn(κ_{2}^{⋆}) = sgn(κ_{3}^{⋆}). By using (3.17), (3.18), (3.21), (3.24) and the condition det(T^{⋆}, N^{⋆}, B_{1}^{⋆}, B_{2}^{⋆}) = 1, we find

(3.25)  κ_{1}^{⋆} = −κ_{2}^{⋆}.

Therefore, sgn(κ_{3}^{⋆}) = −1. From (3.23) and the condition g(B_{1}^{⋆}, B_{1}^{⋆}) = 1, we get κ_{3}^{⋆ 2} = \frac{1}{λ^{2}} . Consequently,

(3.26)  κ_{3}^{⋆} = −\frac{1}{λ}, λ ∈ ℝ_{0}^{+} .

Substituting (3.25) in (3.24), we find

(3.27)  B_{1}^{⋆} = −\frac{1}{2\sqrt{λ}} (T + 2λB_{1}).

Now, relations (3.11), (3.18), (3.25) and (3.26) imply that (3.1) holds, where κ_{1}^{⋆} = −κ_{2}^{⋆} = −κ_{3}. By using the last relation and the relations (3.17), (3.18), (3.21) and (3.27), it follows that (3.2) is satisfied. Similarly, when (3.19) and (3.21), or (3.18) and (3.22), or (3.19) and (3.22) hold, we also obtain that (3.1) and (3.2) are satisfied. This completes the proof of the theorem. □

In the following theorem we give the necessary conditions for the null Cartan curve α and its associated curve α^{⋆} = α + (1/2κ_{2})N to be the generalized null Mannheim curve and the generalized Mannheim mate curve, respectively.

Theorem 3.2. Let α: I → 𝔼_{1}^{4} be a null Cartan curve with a non-zero constant second curvature κ_{2} ∈ ℝ_{0}^{+} and the Cartan frame {T, N, B_{1}, B_{2}}. If the curve α^{⋆}: I^{⋆} → 𝔼_{1}^{4} defined by α^{⋆} = α + (1/2κ_{2})N is the Frenet curve, then α is the generalized null Mannheim curve and α^{⋆} the generalized timelike Mannheim mate curve of α.

Proof. Assume that the curve α^{⋆} defined by

(3.28)  α^{⋆}(s) = α(s) + \frac{1}{2κ_{2}} N(s), κ_{2} ∈ ℝ_{0}^{+} ,

is the Frenet curve, where s is the pseudo-arc length parameter of α and s^{⋆} = f(s) = ∫_{0}^{s} ∥α^{⋆}′(t)∥dt is the arc-length parameter of α^{⋆}. Putting λ = 1/κ_{2}, κ_{2} ∈ ℝ_{0}^{+} , it can be easily checked that g(α^{⋆}′(s), α^{⋆}′(s)) = −λ. This means that α^{⋆} is a timelike curve. Consequently, f(s) = \sqrt{λ}s. Differentiating relation (3.28) with respect to s, and using (2.1) and f′ = \sqrt{λ}, we obtain

(3.29)  T^{⋆} = \frac{1}{2\sqrt{λ}} (T − 2λB_{1}).

Differentiating the last relation with respect to s and using (2.1) and (2.2), we find κ_{1}^{⋆}N^{⋆} = −κ_{3}B_{2}. It follows that

(3.30)  κ_{1}^{⋆} = −κ_{3}, N^{⋆} = B_{2},

or

(3.31)  κ_{1}^{⋆} = κ_{3}, N^{⋆} = −B_{2}.

Assume that relation (3.30) holds. Differentiating the relation N^{⋆} = B_{2} with respect to s and using (2.1) and (2.2), we get

(3.32)  (κ_{1}^{⋆}T^{⋆} + κ_{2}^{⋆}B_{1}^{⋆})f′ = −κ_{3}T.

The last relation implies (−κ_{1}^{⋆2} + κ_{2}^{⋆2})f^{′2} = 0. It follows that

(3.33)  |κ_{1}^{⋆}| = |κ_{2}^{⋆}|.

Substituting (3.29) in (3.32) and using (3.30), (3.33) and f′ = \sqrt{λ}, we get

(3.34)  B_{1}^{⋆} = \frac{sgn(κ_{1}^{⋆}) sgn(κ_{2}^{⋆})}{2\sqrt{λ}} (T + 2λB_{1}).

Differentiating the last relation with respect to s and using (2.1), (2.2), (3.30) and λ = 1/2κ_{2}, we get

(3.35)  B_{2}^{⋆} = \frac{sgn(κ_{1}^{⋆}) sgn(κ_{2}^{⋆})}{λκ_{3}^{⋆}} N.

Relations (3.29), (3.30), (3.34), (3.35) and the condition det(T^{⋆}, N^{⋆}, B_{1}^{⋆}, B_{2}^{⋆}) = 1 imply κ_{3}^{⋆} = −1/λ, λ ∈ ℝ_{0}^{+} . Substituting the last relation in (3.35), it follows that

(3.36)  B_{2}^{⋆} = − sgn(κ_{1}^{⋆}) sgn(κ_{2}^{⋆})N.

By using (3.29), (3.30), (3.34) and (3.36), we obtain that the frames of α and α^{⋆} are related by

T^{⋆} = \frac{1}{2\sqrt{λ}} (T − 2λB_{1}),
N^{⋆} = B_{2},
B_{1}^{⋆} = \frac{sgn(κ_{1}^{⋆}) sgn(κ_{2}^{⋆})}{2\sqrt{λ}} (T + 2λB_{1}),
B_{2}^{⋆} = − sgn(κ_{1}^{⋆}) sgn(κ_{2}^{⋆})N.

Since the principal normal N of α lies in the spacelike plane spanned by {B_{1}^{⋆}, B_{2}^{⋆}}, it follows that α is the generalized null Mannheim curve and α^{⋆} the generalized timelike Mannheim mate curve of α. Assuming that relation (3.31) holds, we also obtain that N lies in the spacelike plane spanned by {B_{1}^{⋆}, B_{2}^{⋆}}. This completes the proof of the theorem. □

(B) span{B_{1}^{⋆}, B_{2}^{⋆}} is a timelike plane.

In this case, we obtain two theorems depending on the causal character of the basis vectors B_{1}^{⋆} and B_{2}^{⋆} of a timelike plane span{B_{1}^{⋆}, B_{2}^{⋆}}. It is known that a timelike plane can be spanned by the spacelike and the timelike mutually orthogonal unit vectors or else by two linearly independent null vectors. Theorems 3.3 and 3.4 can be proved in a similar way as Theorems 3.1 and 3.2 respectively, so we omit their proofs.

Theorem 3.3. Let α: I → 𝔼_{1}^{4} be the generalized null Mannheim curve and α^{⋆}: I^{⋆} → 𝔼_{1}^{4} the generalized Mannheim mate curve of α such that the principal normal line of α lies in the timelike plane spanned by non-null vectors {B_{1}^{⋆}, B_{2}^{⋆}}. Then α^{⋆} is the spacelike Frenet curve such that the curvatures of α and α^{⋆} satisfy the relations κ_{1} = 1, κ_{2} = 1/2λ, |κ_{1}^{⋆}| = |κ_{2}^{⋆}| = |κ_{3}|, |κ_{3}^{⋆}| = −\frac{1}{λ}, λ ∈ ℝ_{0}^{−} , and the corresponding frames of α and α^{⋆} are related by

T^{⋆} = \frac{1}{2\sqrt{|λ|}} (T − 2λB_{1}),
N^{⋆} = sgn(κ_{1}^{⋆}) sgn(κ_{3})B_{2},
B_{1}^{⋆} = sgn(κ_{1}^{⋆}) sgn(κ_{2}^{⋆})\frac{1}{2\sqrt{|λ|}} (T + 2λB_{1}),
B_{2}^{⋆} = sgn(κ_{2}^{⋆}) sgn(κ_{3})N.

Theorem 3.4. Let α: I → 𝔼_{1}^{4} be a null Cartan curve with a non-zero constant second curvature κ_{2} ∈ ℝ_{0}^{−} and the Cartan frame {T, N, B_{1}, B_{2}}. If the curve α^{⋆}: I^{⋆} → 𝔼_{1}^{4} defined by α^{⋆} = α + (1/2κ_{2})N is the Frenet curve, then α is the generalized null Mannheim curve and α^{⋆} the generalized spacelike Mannheim mate curve of α.

In the case when a timelike plane span{B_{1}^{⋆}, B_{2}^{⋆}} is spanned by two linearly independent null vectors B_{1}^{⋆} and B_{2}^{⋆}, we obtain the following theorem.

Theorem 3.5. There are no generalized null Mannheim curves in 𝔼_{1}^{4} the generalized Mannheim mate curve of which is a partially null Frenet curve.

Proof. Assume that there exists the generalized null Mannheim curve α: I → 𝔼_{1}^{4} the generalized Mannheim mate curve α^{⋆}: I^{⋆} → 𝔼_{1}^{4} of which is a partially null Frenet curve. Then the principal normal line of α lies in the timelike plane spanned by null vector fields B_{1}^{⋆} and B_{2}^{⋆}. In particular, the curve α^{⋆} can be parameterized by

(3.37)  α^{⋆}(f(s)) = α(s) + λ(s)N(s),

where s is the pseudo-arc length parameter of α, s^{⋆} = f(s) = ∫_{0}^{s} ∥α^{⋆}′(t)∥dt the arclength parameter of α^{⋆} and f: I ⊂ ℝ → I^{⋆} ⊂ ℝ and λ are some smooth functions. We distinguish two cases: (B.1) κ_{2} = 0 and (B.2) κ_{2} ≠ 0.

(B.1) κ_{2} = 0.

Differentiating relation (3.37) with respect to s and applying (2.1), we obtain

(3.38)  T^{⋆}f′ = T + λ′N − λB_{1}.

By taking the scalar product of (3.38) with N = aB_{1}^{⋆} + bB_{2}^{⋆}, we find λ′ = 0. Substituting this in (3.38) it follows that

(3.39)  T^{⋆}f′ = T − λB_{1}, λ ∈ R_{0}.

From (3.39) we have

(3.40)  g(T^{⋆}f′, T^{⋆}f′) = f^{′2} = −2λ = constant ≠ 0.

Differentiating relation (3.39) with respect to s and using (2.1), (2.4) and (3.40), it follows that

(3.41)  κ_{1}^{⋆}N^{⋆}f^{′2} = N − λκ_{3}B_{2}.

By taking the scalar product of (3.41) with N = aB_{1}^{⋆} + bB_{2}^{⋆}, we obtain a contradiction.

(B.2) κ_{2} ≠ 0.

Differentiating relation (3.37) with respect to s and using (2.1), we find

(3.42)  T^{⋆}f′ = (1 − λκ_{2})T + λ′N − λB_{1}.

By taking the scalar product of (3.42) with N yields

(3.43)  λ′ = 0.

Substituting (3.43) in (3.42) we get

(3.44)  T^{⋆}f′ = (1 − λκ_{2})T − λB_{1}, λ ∈ ℝ_{0}.

Differentiating relation (3.44) with respect to s and using (2.1) and (2.4), we obtain

(3.45)  κ_{1}^{⋆}N^{⋆}f^{′2} + T^{⋆}f′′= (1 − λκ_{2})′T + (1 − 2λκ_{2})N − λκ_{3}B_{2}.

By taking the scalar product of (3.45) with N = aB_{1}^{⋆} + bB_{2}^{⋆}, it follows that

(3.46)  κ_{2} = \frac{1}{2λ} , λ ∈ ℝ_{0}.

By using (3.44) we obtain

(3.47)  g(T^{⋆}f′, T^{⋆}f′) = f^{′2} = −2λ(1 − λκ_{2}).

Substituting (3.46) in (3.47) it follows that

(3.48)  f′^{2} = −λ = constant, λ ∈ ℝ_{0}^{−} .

From (3.45), (3.46) and (3.48) we obtain

(3.49)  κ_{1}^{⋆}N^{⋆} = κ_{3}B_{2}.

Relation (3.49) implies κ_{1}^{⋆} = κ_{3}, N^{⋆} = B_{2} or κ_{1}^{⋆} = −κ_{3}, N^{⋆} = −B_{2}. Differentiating the relation N^{⋆} = ±B_{2} with respect to s and applying (2.1) and (2.4), it follows that

(3.50)  (−κ_{1}^{⋆}T^{⋆} + κ_{2}^{⋆}B_{1}^{⋆})f′ = ∓κ_{3}T.

By taking the scalar product of (3.50) with N = aB_{1}^{⋆} + bB_{2}^{⋆} we find bκ_{2}^{⋆}f′ = 0. If κ_{2}^{⋆} = 0, relation (3.50) implies that a spacelike vector T^{⋆} is collinear with a null vector T, which is a contradiction. It follows that b = 0, so N = aB_{1}^{⋆}. This means that a spacelike vector N is collinear with a null vector B_{1}^{⋆}, which is impossible. □

(C) span{B_{1}^{⋆}, B_{2}^{⋆}} is a lightlike plane.

In this case, we obtain two theorems depending on the causal character of the basis vectors of a lightlike plane span{B_{1}^{⋆}, B_{2}^{⋆}}, which can be spanned by a null vector B_{1}^{⋆} and a spacelike vector B_{2}^{⋆}, or else by a spacelike vector B_{1}^{⋆} and a null vector B_{2}^{⋆}.

Theorem 3.6. Let α: I → 𝔼_{1}^{4} be the generalized null Mannheim curve and α^{⋆}: I^{⋆} → 𝔼_{1}^{4} the generalized Mannheim mate curve of α, such that the principal normal line of α lies in the lightlike plane spanned by a null vector B_{1}^{⋆} and a spacelike vector B_{2}^{⋆}. Then α^{⋆} is a null Cartan curve such that one of the two statements hold:

ON GENERALIZED NULL MANNHEIM CURVES 87

(i) the curvatures of α and α^{⋆} satisfy the relations

(3.51) κ_{2} = \frac{1 − sinh^{2} ( \frac{\sqrt{2}}{2}s)}{cosh^{2} ( \frac{\sqrt{2}}{2}s)} , κ_{2}^{⋆} = 0, |κ_{3}| = cosh^{6} ( \frac{\sqrt{2}}{2}s) , |κ_{3}^{⋆}| = \frac{1}{cosh^{2} ( \frac{\sqrt{2}}{2}s)} ,

and the corresponding Cartan frames of α and α^{⋆} are related by

(3.52)

T^{⋆} = \frac{sinh^{2}(\frac{\sqrt{2}}{2}s)}{cosh^{4}(\frac{\sqrt{2}}{2}s)}T + \frac{\sqrt{2}sinh(\frac{\sqrt{2}}{2}s)}{cosh^{3}(\frac{\sqrt{2}}{2}s)}N − \frac{1}{cosh^{2}(\frac{\sqrt{2}}{2}s)}B_{1},
N^{⋆} = − sgn(κ_{3}^{⋆})B_{2},
B_{1}^{⋆} = − cosh^{2} ( \frac{\sqrt{2}}{2}s) T,
B_{2}^{⋆} = − sgn(κ_{3}^{⋆})( \frac{\sqrt{2}sinh( \frac{\sqrt{2}}{2}s)}{cosh( \frac{\sqrt{2}}{2}s)} T + N) ;

(ii) the curvatures of α and α^{⋆} satisfy the relations

(3.53)  κ_{2} = \frac{2λ − λ^{′2}}{2λ^{2}} ≠ 0, κ_{2}^{⋆} = \frac{X}{λλ^{′2}f^{′2}} ≠ 0, |κ_{3}| = \frac{\sqrt{λ^{2}f^{′4} − X^{2}}}{λ^{2}} ,
|κ_{3}^{⋆}| = \frac{( \frac{λ′X}{2λ^{2} f^{′2}} )′+ \frac{2(λ^{2} f^{′4}−X^{2} )−λX}{2λ^{3} f^{′2}} + (( \frac{X}{λ′f^{′2}} )′+ \frac{X}{λ^{′2}f^{′2}} )( \frac{2λ−λ^{′2}}{2λ^{2}} )}{f^{′2}} .

where X(s) = λλ′′− λ^{′2} − λ, f′ = e^{∫ \frac{λλ′′+λ^{′2} −λ}{λλ′} ds} and λ(s) ≠ constant satisfy the differential equation

\frac{X^{2}[λ^{2}f^{′4} + λλ′X′− (3λλ′′+ 2λ^{′2} − 3λ)X]^{2}}{λ^{4}λ^{′2}f^{′′4}(λ^{2}f^{′4} − X^{2})}

+ 2[( \frac{λ′X}{2λ^{2}f^{′2}} )′+ \frac{2(λ^{2}f^{′4} − X^{2}) − λX}{2λ^{3}f^{′2}} ][( \frac{X}{λ′f^{′2}} )′+ \frac{X}{λ^{′2}f^{′2}} ] = 0,

while the corresponding Cartan frames of α and α^{⋆} are related by

T^{⋆} = \frac{λ^{′2}}{2λf′} T + \frac{λ′}{f′}N − \frac{λ}{f′} B_{1},
N^{⋆} = − sgn(κ_{3}^{⋆})( \frac{λ′(λλ′′− λ^{′2} − λ)}{2λ^{2}f^{′2}} T + ( \frac{λλ′′− λ^{′2} − λ}{λ′f^{′2}} ) B_{1}

− \frac{\sqrt{λ^{2}f^{′4} − (λλ′′− λ^{′2} − λ)^{2}}}{λf^{′2}} ) B_{2}, (3.54)

B_{1}^{⋆} = xT + yB_{1} + zB_{2},
B_{2}^{⋆} = −\frac{sgn(κ_{3}^{⋆})}{f′κ_{3}^{⋆}}[(x′− zκ_{3} − κ_{2}^{⋆}mf′)T + (x + yκ_{2})N
+ (y′− κ_{2}^{⋆}nf′)B_{1} + (yκ_{3} + z′− κ_{2}^{⋆}pf′)B_{2}],

where x, y, z are given by

x = −\frac{1}{f′} ( ( \frac{λ′X}{2λ^{2}f^{′2}} )′+ \frac{2(λ^{2}f^{′4} − X^{2}) − λX}{2λ^{3}f^{′2}} ) ,
y = −\frac{1}{f′} ( ( \frac{X}{λ′f^{′2}} )′+ \frac{X}{λ^{′2}f^{′2}} ) ,
z = −\frac{1}{f′} ( \frac{X[λ^{2}f^{′4} + λλ′X′− (3λλ′′+ 2λ^{′2} − 3λ)X]}{λ^{2}λ′f^{′2}\sqrt{λ^{2}f^{′4} − X^{2}}}  ) .

Proof. By assumption the principal normal line of α lies in the lightlike plane spanned by a null vector B_{1}^{⋆} and a spacelike vector B_{2}^{⋆}. Therefore, α^{⋆} is a null Cartan curve the Cartan frame of which satisfies the relation (2.1). The curve α^{⋆} has parametrization of the form

(3.55)  α^{⋆}(f(s)) = α(s) + λ(s)N(s),

where s and s^{⋆} = f(s) are the pseudo-arc length parameters of α and α^{⋆} respectively and f: I ⊂ ℝ → I^{⋆} ⊂ ℝ and λ are some smooth functions. We distinguish two subcases: (C.1) κ_{2} = 0 and (C.2) κ_{2} ≠ 0.

(C.1) κ_{2} = 0.

Differentiating relation (3.55) with respect to s and using (2.1) we obtain

(3.56)  T^{⋆}f′ = T + λ′N − λB_{1}.

Since g(T^{⋆}f′, T^{⋆}f′) = λ^{′2} − 2λ = 0, it follows that

(3.57)  λ(s) = \frac{(s + c)^{2}}{2} , c ∈ ℝ.

Differentiating relation (3.56) with respect to s and using (2.1) and (3.57), we obtain

(3.58)  N^{⋆}f^{′2} + T^{⋆}f′′= 2N − 2(s + c)B_{1} − \frac{1}{2}(s + c)^{2}κ_{3}B_{2}.

By taking the scalar products of (3.56) and (3.58) with N = aB_{1}^{⋆} ± B_{2}^{⋆}, we respectively find

(3.59)  af′ = s + c, c ∈ ℝ,
(3.60)  af′′ = 2.

Relations (3.59) and (3.60) imply

(3.61)  a = \frac{1}{c_{1}(s + c)} , f′ = c_{1}(s + c)^{2}, c_{1} ∈ ℝ_{0}^{+} , c ∈ ℝ.

Substituting (3.57) and (3.61) in (3.56), we obtain

(3.62)  T^{⋆} = \frac{1}{c_{1}(s + c)^{2}} T + \frac{1}{c_{1}(s + c)} N − \frac{1}{2c_{1}} B_{1}.

Substituting (3.61) in (3.58) and using (3.62), we get

(3.63)  N^{⋆} = −\frac{2}{c_{1}^{2}(s + c)^{5}} T − \frac{1}{c_{1}^{2}(s + c)^{3}} B_{1} − \frac{κ_{3}}{2c_{1}^{2}(s + c)^{2}} B_{2}.

Now we distinguish two subcases: (C.1.1) κ_{2}^{⋆} = 0 and (C.1.2) κ_{2}^{⋆} ≠ 0.

(C.1.1) κ_{2}^{⋆} = 0;

Differentiating relation (3.58) with respect to s and using (2.1), we obtain

f′′′T^{⋆} + 3f′f′′N^{⋆} − f^{′3}B_{1}^{⋆} = \frac{κ_{3}^{2}}{2} (s + c)^{2}T − 4B_{1} − (s + c)( 3κ_{3} + \frac{(s + c)}{2} κ′_{3}) B_{2}.

By taking the scalar product of the last relation with N = aB_{1}^{⋆} ± B_{2}^{⋆}, we get af′′′= 0. This implies a = 0 or f′′′= 0, which is a contradiction with (3.61).

(C.1.2) κ_{2}^{⋆} ≠ 0; Differentiating relation (3.63) with respect to s and using (2.1), we find

(3.64)  (−κ_{2}^{⋆}T^{⋆} − B_{1}^{⋆})f′ = \frac{20 + κ_{3}^{2}(s + c)^{4}}{2c_{1}^{2}(s + c)^{6}} T − \frac{2}{c_{1}^{2}(s + c)^{5}} N

+ \frac{3}{c_{1}^{2}(s + c)^{4}} B_{1} − \frac{κ′_{3}}{2c_{1}^{2}(s + c)^{2}} B_{2}.

By taking the scalar product of the last relation with N = aB_{1}^{⋆} ± B_{2}^{⋆} and using (3.61), it follows that

(3.65)  κ_{2}^{⋆} = \frac{2}{c_{1}^{2}(s + c)^{6}} , c_{1} ∈ ℝ_{0}^{+} , c ∈ ℝ.

Substituting (3.62) and (3.65) in (3.64), we get

−B_{1}^{⋆}f′ = \frac{24 + κ_{3}^{2}(s + c)^{4}}{2c_{1}^{2}(s + c)^{6}} T + \frac{2}{c_{1}^{2}(s + c)^{4}} B_{1} − \frac{κ′_{3}}{2c_{1}^{2}(s + c)^{2}} B_{2}.

The last relation and the condition g(B_{1}^{⋆}, B_{1}^{⋆}) = 0 imply [2(24 + κ_{3}^{2}(s + c)^{4})/c_{1}^{4}(s + c)^{10}] + [κ_{3}^{′2}/4c_{1}^{4}(s + c)^{4}] = 0, which is a contradiction.

(C.2) κ_{2} ≠ 0;

Differentiating relation (3.55) with respect to s and using (2.1), we obtain

(3.66)  T^{⋆} = ( \frac{1 − λκ_{2}}{f′} ) T + \frac{λ′}{f′}N − \frac{λ}{f′} B_{1}.

Since g(T^{⋆}, T^{⋆}) = (λ^{′2} − 2λ(1 − λκ_{2}))/f^{′2} = 0, it follows that

(3.67)  κ_{2} = \frac{2λ − λ^{′2}}{2λ^{2}} .

Next we prove that λ ≠ constant. If λ = constant ≠ 0, relation (3.67) gives κ_{2} = \frac{1}{λ}. Substituting this in (3.66) we obtain T^{⋆}f′ = −λB_{1}. By taking the scalar product of the last equation with N = aB_{1}^{⋆}±B_{2}^{⋆}, we find af′ = 0. Therefore, a = 0 and thus N = ±B_{2}^{⋆}. Differentiating the last relation with respect to s and using (2.1), we get −κ_{2}T − B_{1} = ±κ_{3}^{⋆}f′T^{⋆}. The last relation implies g(−κ_{2}T − B_{1}, −κ_{2}T − B_{1}) = 2κ_{2} = 0, which is a contradiction. Consequently, λ ≠ constant.

Putting

(3.68)  u = \frac{1 − λκ_{2}}{f′} v = \frac{λ′}{f′} w = −\frac{λ}{f′} ,

relation (3.66) becomes

(3.69)  T^{⋆} = uT + vN + wB_{1}.

Differentiating the last relation with respect to s and using (2.1), we find

(3.70)  N^{⋆} = ( \frac{u′− vκ_{2}}{f′} ) T + ( \frac{u + v′+ wκ_{2}}{f′} ) N + ( \frac{w′− v}{f′} ) B_{1} + \frac{wκ_{3}}{f′} B_{2}.

By taking the scalar product of (3.70) with N = aB_{1}^{⋆} ± B_{2}^{⋆}, we get

(3.71)  u + v′+ wκ_{2} = 0.

Substituting (3.67) and (3.68) in (3.71), we obtain

(3.72)  \frac{f′′}{f′} = \frac{λλ′′+ λ^{′2} − λ}{λλ′} .

Consequently, f′ = ce^{∫ \frac{λλ′′+λ^{′2} −λ}{λλ′} ds}, c ∈ ℝ_{0}^{+} . Taking c = 1, we find

(3.73)  f′ = e^{∫ \frac{λλ′′+λ^{′2} −λ}{λλ′}  ds}.

Substituting (3.71) in (3.70) yields

(3.74)  N^{⋆} = ( \frac{u′− vκ_{2}}{f′} ) T + ( \frac{w′− v}{f′} ) B_{1} + \frac{wκ_{3}}{f′} B_{2}.

Next, the condition g(N^{⋆}, N^{⋆}) = 1 and the relation (3.74) imply κ_{3}^{2} = [f^{′2}− 2(u′− vκ_{2})(w′− v)]/w^{2}. Substituting (3.67) and (3.68) in the last relation, we get

(3.75)  |κ_{3}| = \frac{\sqrt{λ^{2}f^{′4} − (λλ′′− λ^{′2} − λ)^{2}}}{λ^{2}} .

Putting

(3.76)  m = \frac{u′− vκ_{2}}{f′} , n = \frac{w′− v}{f′} , p = \frac{wκ_{3}}{f′} ,

relation (3.74) becomes

(3.77)  N^{⋆} = mT + nB_{1} + pB_{2}.

Differentiating the last relation with respect to s and using (2.1), we get

(3.78) (−κ_{2}^{⋆}T^{⋆} − B_{1}^{⋆})f′ = (m′− pκ_{3})T + (m + nκ_{2})N + n′B_{1} + (nκ_{3} + p′)B_{2}.

Now we distinguish two subcases: (C.2.1) κ_{2}^{⋆} = 0 and (C.2.2) κ_{2}^{⋆} ≠ 0.

(C.2.1) κ_{2}^{⋆} = 0.

Then relation (3.78) reads

(3.79)  B_{1}^{⋆} = −\frac{1}{f′} [(m′− pκ_{3})T + (m + nκ_{2})N + n′B_{1} + (nκ_{3} + p′)B_{2}].

By taking the scalar product of (3.79) with N = aB_{1}^{⋆} ± B_{2}^{⋆}, we get m + nκ_{2} = 0. Substituting (3.76) in the last relation, we find u′ − 2vκ_{2} + w′κ_{2} = 0. Next, substituting (3.67), (3.68) and (3.72) in the last equation, we obtain the second order differential equation in terms of λ, which reads λλ′′−λ^{′2}−λ = 0. The general solution of the last differential equation is given by λ = (2/c_{1}) cosh^{2} ( \frac{\sqrt{c_{1}}}{2} (s + c_{2})), c_{1}, c_{2} ∈ ℝ, c_{1} ≠ 0. Taking c_{1} = 2 and c_{2} = 0, we get

(3.80)  λ = cosh^{2}(\frac{\sqrt{2}}{2}s).

Substituting (3.80) in (3.67) and (3.75), we respectively get

(3.81)  κ_{2} = \frac{1 − sinh^{2}(\frac{\sqrt{2}}{2}s)}{cosh^{2}(\frac{\sqrt{2}}{2}s)} , |κ_{3}| = cosh^{6}(\frac{\sqrt{2}}{2}s).

Next, substituting (3.80) in (3.73), we obtain

(3.82)  f′ = cosh^{4}(\frac{\sqrt{2}}{2}s).

From (3.80), (3.81), (3.82) and (3.66) we have

(3.83)  T^{⋆} = \frac{sinh^{2} ( \frac{\sqrt{2}}{2}s)}{cosh^{4} ( \frac{\sqrt{2}}{2}s)} T + \frac{\sqrt{2}sinh( \frac{\sqrt{2}}{2}s)}{cosh^{3} ( \frac{\sqrt{2}}{2}s)} N − \frac{1}{cosh^{2} ( \frac{\sqrt{2}}{2}s)} B_{1}.

Differentiating the previous equation with respect to s, using (2.1) and (3.82), we obtain N^{⋆} = −κ_{3}/ cosh^{6} ( \frac{\sqrt{2}}{2}s)B_{2}. It follows that

(3.84)  N^{⋆} = B_{2}, κ_{3} = − cosh^{6} ( \frac{\sqrt{2}}{2}s) ,

or

(3.85)  N^{⋆} = −B_{2}, κ_{3} = cosh^{6} ( \frac{\sqrt{2}}{2}s) .

Assume that (3.84) holds. Differentiating the relation N^{⋆} = B_{2} with respect to s, using (2.1), (3.82) and (3.84), we find

(3.86)  B_{1}^{⋆} = − cosh^{2} ( \frac{\sqrt{2}}{2}s) T.

Differentiating relation (3.86) with respect to s and using (2.1) and (3.82), we get

(3.87)  κ_{3}^{⋆}B_{2}^{⋆} = −\frac{\sqrt{2}sinh( \frac{\sqrt{2}}{2}s)}{cosh^{3} ( \frac{\sqrt{2}}{2}s)} T − \frac{1}{cosh^{2} ( \frac{\sqrt{2}}{2}s)} N.

Relation (3.87) and the condition g(B_{2}^{⋆}, B_{2}^{⋆}) = 1 imply

(3.88)  |κ_{3}^{⋆}| = \frac{1}{cosh^{2} ( \frac{\sqrt{2}}{2}s)} .

Next, by using (3.83), (3.84), (3.86), (3.87) and the condition det(T^{⋆}, N^{⋆}, B_{1}^{⋆}, B_{2}^{⋆}) = 1, we obtain

(3.89)  κ_{3}^{⋆} = −\frac{1}{cosh^{2} ( \frac{\sqrt{2}}{2}s)} .

Substituting (3.89) in (3.87) yields

(3.90)  B_{2}^{⋆} = \frac{\sqrt{2}sinh( \frac{\sqrt{2}}{2}s)}{cosh( \frac{\sqrt{2}}{2}s)} T + N.

Finally, relations (3.81) and (3.88) imply that (3.51) holds. By using relations (3.83), (3.84), (3.86) and (3.90), we obtain that (3.52) holds. Assuming that (3.85) holds, in a similar way it can be proved that (3.51) and (3.52) are satisfied. This proves statement (i).

(C.2.2) κ_{2}^{⋆} ≠ 0;

Since relation (3.73) holds, it follows that the curvatures κ_{2} and κ_{3} in relations (3.67) and (3.75) are expressed only in terms of λ. Next, we show that another two curvatures κ_{2}^{⋆} and κ_{3}^{⋆} can also be expressed only in terms of λ, such that λ ≠ constant satisfies the corresponding differential equation. Substituting (3.69) in (3.78), we obtain

(3.91)  B_{1}^{⋆} = − \frac{1}{f′} [(m′− pκ_{3} + uf′κ_{2}^{⋆})T + (m + nκ_{2} + vf′κ_{2}^{⋆})N
+ (n′+ wf′κ_{2}^{⋆})B_{1} + (nκ_{3} + p′)B_{2}].

By taking the scalar product of (3.91) with N = aB_{1}^{⋆} ± B_{2}^{⋆}, we find

(3.92)  κ_{2}^{⋆} = −\frac{m + nκ_{2}}{vf′} .

Substituting (3.67), (3.68), (3.73) and (3.76) in (3.92), it follows that

(3.93)  κ_{2}^{⋆} = \frac{λ + λ^{′2} − λλ′′}{λλ^{′2}e^{2 ∫ \frac{λλ′′+λ^{′2} −λ}{λλ′}  ds}} .

Substituting (3.92) in (3.91), we find

(3.94) B_{1}^{⋆} = −\frac{1}{f′} [(m′− pκ_{3} + uf′κ_{2}^{⋆})T + (n′+ wf′κ_{2}^{⋆})B_{1} + (nκ_{3} + p′)B_{2}].

By using relation (3.94) and the condition g(B_{1}^{⋆}, B_{1}^{⋆}) = 0, we get

(3.95)  (nκ_{3} + p′)^{2} + 2(m′− pκ_{3} + uf′κ_{2}^{⋆})(n′+ wf′κ_{2}^{⋆}) = 0.

Assuming that sgn(κ_{3}) = 1 and putting

(3.96)  X = λλ′′− λ^{′2} − λ,

relation (3.75) becomes κ_{3} = \sqrt{λ^{2}f^{′4} − X^{2}} /λ^{2}. By using the last relation and relations (3.68), (3.75), (3.76), (3.93), a straightforward calculation yields

nκ_{3} + p′ = \frac{X[λ^{2}f^{′4} + λλ′X′− (3λλ′′+ 2λ^{′2} − 3λ)X]}{λ^{2}λ′f^{′2}\sqrt{λ^{2}f^{′4} − X^{2}}} , (3.97)

m′− pκ_{3} + uf′κ_{2}^{⋆} = ( \frac{λ′X}{2λ^{2}f^{′2}} )′+ \frac{2(λ^{2}f^{′4} − X^{2}) − λX}{2λ^{3}f^{′2}} , (3.98)

n′+ wf′κ_{2}^{⋆} = ( \frac{X}{λ′f^{′2}} )′+ \frac{X}{λ^{′2}f^{′2}} , (3.99)

where f′′ and X are given by (3.73) and (3.96), respectively. Substituting the relations (3.97), (3.98) and (3.99) in (3.95), we obtain the third order differential equation only in terms of λ, which reads

\frac{X^{2}[λ^{2}f^{′4} + λλ′X′− (3λλ′′+ 2λ^{′2} − 3λ)X]^{2}}{λ^{4}λ^{′2}f^{′4}(λ^{2}f^{′4} − X^{2})}

+ 2[( \frac{λ′X}{2λ^{2}f^{′2}} )′+ \frac{2(λ^{2}f^{′4} − X^{2}) − λX}{2λ^{3}f^{′2}} ][( \frac{X}{λ′f^{′2}} )′+ \frac{X}{λ^{′2}f^{′2}} ] = 0.

Putting

(3.100)  \begin{aligned}x & = −\frac{m′− pκ_{3} + uf′κ_{2}^{⋆}}{f′} , \\ y & = −\frac{n′+ wf′κ_{2}^{⋆}}{f′} , \\ z & = −\frac{nκ_{3} + p′}{f′} , \end{aligned}

relation (3.94) becomes

(3.101)  B_{1}^{⋆} = xT + yB_{1} + zB_{2}.

Differentiating relation (3.101) with respect to s and using (2.1), we obtain

(3.102) f′(κ_{2}^{⋆}N^{⋆} + κ_{3}^{⋆}B_{2}^{⋆}) = (x′− zκ_{3})T + (x + yκ_{2})N + y′B_{1} + (yκ_{3} + z′)B_{2}.

Next, substituting (3.77) in (3.102) we get

(3.103)  B_{2}^{⋆} = \frac{1}{f′κ_{3}^{⋆}} [(x′− zκ_{3} − κ_{2}^{⋆}mf′)T + (x + yκ_{2})N
+ (y′− κ_{2}^{⋆}nf′)B_{1} + (yκ_{3} + z′− κ_{2}^{⋆}f′p)B_{2}],

where x, y, z are given by (3.100). By taking the scalar product of (3.103) with N = aB_{1}^{⋆} ± B_{2}^{⋆}, we find

|κ_{3}^{⋆}| = \frac{x + yκ_{2}}{f′} .

By using (3.69), (3.77), (3.101), (3.103) and the condition det(T^{⋆}, N^{⋆}, B_{1}^{⋆}, B_{2}^{⋆}) = 1, we get

(3.104)  κ_{3}^{⋆} = \frac{x + yκ_{2}}{f′} .

Substituting (3.67) and (3.100) in (3.104), we get

(3.105)  κ_{3}^{⋆} = −\frac{( \frac{λ′X}{2λ^{2} f^{′2}} )′+ \frac{2(λ^{2} f^{′4}−X^{2} )−λX}{2λ^{3} f^{′2}} + (( \frac{X}{λ′f^{′2}} )′+ \frac{X}{λ^{′2}f^{′2}} )( \frac{2λ−λ^{′2}}{2λ^{2}} )}{f^{′2}} .

Finally, relations (3.67), (3.75), (3.93) and (3.105) imply that (3.53) holds. By using (3.67), (3.68), (3.69), (3.76), (3.77), (3.101) and (3.103), we obtain that the Cartan frames of α and α^{⋆} are related by (3.54). Moreover, a straightforward calculation shows that N = (λ′/f′)B_{1}^{⋆} + B_{2}^{⋆}. Assuming that sgn(κ_{3}) = −1, we also obtain that (3.53) and (3.54) hold. This proves statement (ii) of the theorem. □

Remark 3.2. Note that the statement (i) of Theorem 3.6 is the special case of statement (ii), when X = 0.

When a lightlike plane span{B_{1}^{⋆}, B_{2}^{⋆}} is spanned by a spacelike vector B_{1}^{⋆} and a null vector B_{2}^{⋆}, we get the following theorem.

Theorem 3.7. There are no generalized null Mannheim curves in 𝔼_{1}^{4} the generalized Mannheim mate curve of which is a pseudo null Frenet curve.

Proof. Assume that there exists the generalized null Mannheim curve α in E_{1}^{4} the generalized Mannheim mate curve α^{⋆}: I^{⋆} → 𝔼_{1}^{4} of which is a pseudo null Frenet curve. The curve α^{⋆} can be parameterized by

(3.106)  α^{⋆}(f(s)) = α(s) + λ(s)N(s),

where s is the pseudo-arc length parameter of α, s^{⋆} = f(s) = ∫_{0}^{s} ∥α^{⋆}′(t)∥dt is the arc-length parameter of α^{⋆} and f: I ⊂ ℝ → I^{⋆} ⊂ ℝ and λ are some smooth functions on I. We distinguish two cases: (C^{∗}.1) κ_{2} = 0 and (C^{∗}.2) κ_{2} ≠ 0.

(C^{∗}.1) κ_{2} = 0.

Differentiating relation (3.106) with respect to s and applying (2.1), we obtain

(3.107)  T^{⋆}f′ = T + λ′N − λB_{1}.

By taking the scalar product of (3.107) with N = ±B_{1}^{⋆} + bB_{2}^{⋆}, we find λ′ = 0. Substituting this in (3.107), we get

(3.108)  T^{⋆}f′ = T − λB_{1}, λ ∈ ℝ_{0},

where ℝ_{0} denotes ℝ\{0}. Relation (3.108) implies

(3.109)  g(T^{⋆}f′, T^{⋆}f′) = f^{′2} = −2λ = constant.

Differentiating relation (3.108) with respect to s and using (2.1), (2.3) and (3.109), it follows that

N^{⋆}f^{′2} = N − λκ_{3}B_{2}.

The last relation gives g(N^{⋆}f^{′2}, N^{⋆}f^{′2}) = 1 + λ^{2}κ_{3}^{2} = 0, which is a contradiction. (C^{∗}.2) κ_{2} ≠ 0. Differentiating relation (3.106) with respect to s and using (2.1), we obtain

T^{⋆}f′ = (1 − λκ_{2})T + λ′N − λB_{1}. By taking the scalar product of the last relation with N = ±B_{1}^{⋆} + bB_{2}^{⋆}, we get λ′ = 0. Consequently,

(3.110)  T^{⋆} = ( \frac{1 − λκ_{2}}{f′} ) T − \frac{λ}{f′} B_{1}, λ ∈ ℝ_{0}.

By using the condition g(T^{⋆}, T^{⋆}) = 1 and relation (3.110), we find

(3.111)  f^{′2} = −2λ(1 − λκ_{2}).

Differentiating relation (3.110) with respect to s, using (2.1), (2.3) and (3.111), we obtain

(3.112)  N^{⋆} = −\frac{λκ′_{2}}{2f^{′2}} T + ( \frac{1 − 2λκ_{2}}{f^{′2}} ) N + \frac{λ^{3}κ′_{2}}{f^{′4}} B_{1} − \frac{λκ_{3}}{f^{′2}} B_{2}.

Assuming that κ_{2} = constant ≠ 0, relation (3.112) implies that a null vector N^{⋆} is a linear combination of two mutually orthogonal spacelike vectors N i B_{2}, which is impossible. Therefore,

(3.113)  κ_{2} ≠ constant.

Relation (3.112) and the condition g(N^{⋆}, N^{⋆}) = 0 imply

(3.114)  κ_{3}^{2} = \frac{λ^{4}κ_{2}^{′2} − f^{′2}(1 − 2λκ_{2})^{2}}{λ^{2}f^{′2}}

Substituting (3.111) in (3.114), we get

(3.115)  κ_{3}^{2} = \frac{λ^{3}κ_{2}^{′2} + 2(1 − λκ_{2})(1 − 2λκ_{2})^{2}}{−2λ^{2}(1 − λκ_{2})}

Putting

(3.116)  m = −\frac{λκ′_{2}}{2f^{′2}} , n = \frac{1 − 2λκ_{2}}{f^{′2}} , p = \frac{λ^{3}κ′_{2}}{f^{′4}} , q = −\frac{λκ_{3}}{f^{′2}} ,

relation (3.112) becomes N^{⋆} = mT + nN + pB_{1} + qB_{2}. Differentiating the last relation with respect to s and using (2.1) and (2.3), we find

κ_{2}^{⋆}B_{1}^{⋆}f′ = (m′− nκ_{2} − qκ_{3})T + (m + n′+ pκ_{2})N + (p′− n)B_{1} + (pκ_{3} + q′)B_{2}.

By taking the scalar product of the last relation with N = ±B_{1}^{⋆} + bB_{2}^{⋆} it follows that ±κ_{2}^{⋆}f′ = m + n′+ pκ_{2}. Assuming that κ_{2}^{⋆}f′ = m + n′+ pκ_{2}, we obtain

(3.117)  B_{1}^{⋆} = ( \frac{m′− nκ_{2} − qκ_{3}}{κ_{2}^{⋆}f′} ) T + N + ( \frac{p′− n}{κ_{2}^{⋆}f′} ) B_{1} + ( \frac{pκ_{3} + q′}{κ_{2}^{⋆}f′} ) B_{2}.

The condition g(B_{1}^{⋆}, B_{1}^{⋆}) = 1 and relation (3.117) yield

(3.118)  2(m′− nκ_{2} − qκ_{3})(p′− n) + (pκ_{3} + q′)^{2} = 0.

By using (3.115) and (3.116), we find

m′− nκ_{2} − qκ_{3} = \frac{κ′′_{2}λ^{2}(1 − λκ_{2}) + 2λ^{3}κ_{2}^{′2} + 2(1 − λκ_{2})^{2}(1 − 2λκ_{2})}{4λ^{2}(1 − λκ_{2})^{2}}  ,
(3.119)  p′− n = \frac{κ′′_{2}λ^{2}(1 − λκ_{2}) + 2λ^{3}κ_{2}^{′2} + 2(1 − λκ_{2})^{2}(1 − 2λκ_{2})}{4λ(1 − λκ_{2})^{3}}  .

From the last two relations we get

(3.120)  m′− nκ_{2} − qκ_{3} = \frac{(1 − λκ_{2})}{λ} (p′− n).

Substituting (3.120) in (3.118), we obtain

(3.121)  2( \frac{1 − λκ_{2}}{λ} ) (p′− n)^{2} + (pκ_{3} + q′)^{2} = 0.

Since (3.113) holds, relation (3.121) implies

(3.122)  p′− n = 0,
(3.123)  pκ_{3} + q′ = 0.

By using (3.114) and (3.116), a straightforward calculation yields

(3.124) pκ_{3} + q′ = \frac{κ′_{2}[κ′′_{2}λ^{2}(1 − λκ_{2}) + 2λ^{3}κ_{2}^{′2} − (1 − λκ_{2})(1 − 2λκ_{2})(1 + 2λκ_{2})]}{−4λκ_{3}(1 − λκ_{2})^{3}}

By using (3.119) and (3.122), we have

(3.125)  κ′′_{2}λ^{2}(1 − λκ_{2}) + 2λ^{3}κ_{2}^{′2} + 2(1 − λκ_{2})^{2}(1 − 2λκ_{2}) = 0.

Similarly, by using (3.123) and (3.124), we get

(3.126)  κ′′_{2}λ^{2}(1 − λκ_{2}) + 2λ^{3}κ_{2}^{′2} + (1 − λκ_{2})(1 − 2λκ_{2})(−1 − 2λκ_{2}) = 0.

Finally, relations (3.125) and (3.126) imply κ_{2} = 1/2λ = constant or κ_{2} = 1/λ = constant, which is a contradiction with (3.113). □

In order to characterize the generalized null Mannheim curves in 𝔼_{1}^{4} in terms of the normal curves, we recall the next theorem given in [9].

Theorem 3.8. Let α(s) be a null curve in 𝔼_{1}^{4} parameterized by pseudo-arc s and with curvatures κ_{1}(s) = 1, κ_{2}(s) ≠ 0, κ_{3}(s) ≠ 0. Then α is a normal curve if and only if one of the following statements holds:

(i) α lies in pseudosphere 𝕊_{1}^{3}(r), r ∈ ℝ_{0}^{+} ; (ii) the third curvature κ_{3}(s) is a non-zero constant; (iii) the second binormal component of the position vector α is a non-zero

constant, i.e. g(α, B_{2}) = c_{0}, c_{0} ∈ ℝ_{0}.

By using Theorems 3.1–3.4, 3.6 and 3.8, we get the following corollaries.

Corollary 3.1. Every null Cartan helix in 𝔼_{1}^{4} is a normal generalized null Mannheim curve the generalized Mannheim mate curve of which is a timelike or a spacelike helix.

Corollary 3.2. There are no normal generalized null Mannheim curves the generalized Mannheim mate curve of which is a null Cartan curve.

4. Some examples

Example 4.1. Consider the null Cartan curve α in 𝔼_{1}^{4} with parameter equation

α(s) = ( ∫ e^{s} cosh( \frac{\sqrt{2}}{2e^{s}}) ds, − ∫ e^{s} sinh( \frac{\sqrt{2}}{2e^{s}}) ds,

− ∫ e^{s} sin( \frac{\sqrt{2}}{2e^{s}}) ds, ∫ e^{s} cos( \frac{\sqrt{2}}{2e^{s}}) ds)

where s is the pseudo-arc length parameter. The curvatures of α are given by κ_{1}(s) = 1, κ_{2}(s) = −1/2, κ_{3}(s) = 1/2e^{2s}. The principal normal vector N(s) of α is given by

N(s) = α′′(s) = e^{s}( cosh( \frac{\sqrt{2}}{2e^{s}}) , − sinh( \frac{\sqrt{2}}{2e^{s}}) , − sin( \frac{\sqrt{2}}{2e^{s}}) , cos( \frac{\sqrt{2}}{2e^{s}}) )

+ \frac{\sqrt{2}}{2}( − sinh( \frac{\sqrt{2}}{2e^{s}}) , cosh( \frac{\sqrt{2}}{2e^{s}}) , cos( \frac{\sqrt{2}}{2e^{s}}) , sin( \frac{\sqrt{2}}{2e^{s}}) ) .

Define the curve α^{⋆}: I → 𝔼_{1}^{4} by α^{⋆}(s) = α(s) − N(s). Since g(α^{⋆}′(s), α^{⋆}′(s)) = 1, α^{⋆} is a unit speed spacelike curve. The curvatures of α^{⋆} are given by κ_{1}^{⋆}(s) = κ_{2}^{⋆}(s) = 1/2e^{2s}, κ_{3}^{⋆}(s) = 1. Moreover, by using a straightforward calculation, it follows that the Frenet frame of α^{⋆} is given by

T^{⋆}(s) = \frac{\sqrt{2}}{2}( sinh( \frac{\sqrt{2}}{2e^{s}}) , − cosh( \frac{\sqrt{2}}{2e^{s}}) , − cos( \frac{\sqrt{2}}{2e^{s}}) , − sin( \frac{\sqrt{2}}{2e^{s}}) )

− \frac{e^{−s}}{2} ( cosh( \frac{\sqrt{2}}{2e^{s}}) , − sinh( \frac{\sqrt{2}}{2e^{s}}) , sin( \frac{\sqrt{2}}{2e^{s}}) , − cos( \frac{\sqrt{2}}{2e^{s}}) ) ,

N^{⋆}(s) = \frac{\sqrt{2}}{2}( sinh( \frac{\sqrt{2}}{2e^{s}}) , − cosh( \frac{\sqrt{2}}{2e^{s}}) , cos( \frac{\sqrt{2}}{2e^{s}}) , sin( \frac{\sqrt{2}}{2e^{s}}) ) ,
B_{1}^{⋆}(s) = −\frac{\sqrt{2}}{2}( sinh( \frac{\sqrt{2}}{2e^{s}}) , − cosh( \frac{\sqrt{2}}{2e^{s}}) , − cos( \frac{\sqrt{2}}{2e^{s}}) , − sin( \frac{\sqrt{2}}{2e^{s}}) )

+ \frac{e^{−s}}{2} ( cosh( \frac{\sqrt{2}}{2e^{s}}) , − sinh( \frac{\sqrt{2}}{2e^{s}}) , sin( \frac{\sqrt{2}}{2e^{s}}) , − cos( \frac{\sqrt{2}}{2e^{s}}) )

+ e^{s}( cosh( \frac{\sqrt{2}}{2e^{s}}) , − sinh( \frac{\sqrt{2}}{2e^{s}}) , − sin( \frac{\sqrt{2}}{2e^{s}}) , cos( \frac{\sqrt{2}}{2e^{s}}) ) ,

B_{2}^{⋆}(s) = e^{s}( cosh( \frac{\sqrt{2}}{2e^{s}}) , − sinh( \frac{\sqrt{2}}{2e^{s}}) , − sin( \frac{\sqrt{2}}{2e^{s}}) , cos( \frac{\sqrt{2}}{2e^{s}}) )

+ \frac{\sqrt{2}}{2}( − sinh( \frac{\sqrt{2}}{2e^{s}}) , cosh( \frac{\sqrt{2}}{2e^{s}}) , cos( \frac{\sqrt{2}}{2e^{s}}) , sin( \frac{\sqrt{2}}{2e^{s}}) ) .

According to Theorem 3.4, α is the generalized null Mannheim curve and α^{⋆} the generalized spacelike Mannheim mate curve of α. Since N(s) = B_{2}^{⋆}(s), the vector N(s) lies in the timelike plane span{B_{1}^{⋆}, B_{2}^{⋆}}.

Example 4.2. Consider the null Cartan helix α in 𝔼_{1}^{4}, parameterized by the pseudo-arc length function s, with parameter equation α(s) = (sinh(s/\sqrt{2}), (1/\sqrt{3}) sin(s\sqrt{3}/\sqrt{2}), cosh(s/\sqrt{2}), (−1/\sqrt{3}) cos(s\sqrt{3}/\sqrt{2})) and the curvature functions κ_{1}(s) = 1, κ_{2}(s) = 1/2, κ_{3}(s) = \sqrt{3}/2. The principal normal vector N(s) of α is given by

N(s) = α′′(s) = ( \frac{1}{2} sinh( \frac{s}{\sqrt{2}}) , −\frac{\sqrt{3}}{2} sin( \frac{s\sqrt{3}}{\sqrt{2}}) , \frac{1}{2} cosh( \frac{s}{\sqrt{2}}) , \frac{\sqrt{3}}{2} cos( \frac{s\sqrt{3}}{\sqrt{2}}) ) .

Define the curve α^{⋆}: I → 𝔼_{1}^{4} by α^{⋆}(s) = α(s) + N(s). Since g(α^{⋆}′(s), α^{⋆}′(s)) = −1, α^{⋆} is a unit speed timelike curve. A straightforward calculation shows that the curvatures of α^{⋆} are given by κ_{1}^{⋆}(s) = κ_{2}^{⋆}(s) = \sqrt{3}/2, κ_{3}^{⋆}(s) = 1. Consequently, α^{⋆} is a timelike helix. Moreover, the Frenet frame of α^{⋆} has the form

T^{⋆}(s) = ( \frac{3}{2\sqrt{2}} cosh( \frac{s}{\sqrt{2}}) , −\frac{1}{2\sqrt{2}} cos( \frac{s\sqrt{3}}{\sqrt{2}}) , \frac{3}{2\sqrt{2}} sinh( \frac{s}{\sqrt{2}}) , −\frac{1}{2\sqrt{2}} sin( \frac{s\sqrt{3}}{\sqrt{2}}) ) ,
N^{⋆}(s) = ( \frac{\sqrt{3}}{2} sinh( \frac{s}{\sqrt{2}}) , \frac{1}{2} sin( \frac{s\sqrt{3}}{\sqrt{2}}) , \frac{\sqrt{3}}{2} cosh( \frac{s}{\sqrt{2}}) , −\frac{1}{2} cos( \frac{s\sqrt{3}}{\sqrt{2}}) ) ,
B_{1}^{⋆}(s) = ( − \frac{1}{2\sqrt{2}} cosh( \frac{s}{\sqrt{2}}) , \frac{3}{2\sqrt{2}} cos( \frac{s\sqrt{3}}{\sqrt{2}}) , −\frac{1}{2\sqrt{2}} sinh( \frac{s}{\sqrt{2}}) , \frac{3}{2\sqrt{2}} sin( \frac{s\sqrt{3}}{\sqrt{2}}) ) ,
B_{2}^{⋆}(s) = ( \frac{1}{2} sinh( \frac{s}{\sqrt{2}}) , −\frac{\sqrt{3}}{2} sin( \frac{s\sqrt{3}}{\sqrt{2}}) , \frac{1}{2} cosh( \frac{s}{\sqrt{2}}) , \frac{\sqrt{3}}{2} cos( \frac{s\sqrt{3}}{\sqrt{2}}) ) .

According to Corollary 3.1, α is the normal generalized null Mannheim curve and α^{⋆} the generalized timelike Mannheim mate curve of α, such that N(s) = B_{2}^{⋆}(s).

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Faculty of Science (Received 17 08 2014) University of Kragujevac Department of Mathematics and Informatics Kragujevac Serbia

milica_grbovic@yahoo.com
nesovickg@sbb.rs

Faculty of Sciences and Arts Department of Mathematics Kirikkale University Kirikkale Turkey

kilarslan@yahoo.com