No Null-Helix Mannheim Curves in the Minkowski Space E 31

In the study of the fundamental theory and the characterizations of space curves, the related curves for which there exist corresponding relations between the curves are very interesting and important problems. The most fascinating examples of such curve are associated curves, the curves for which at the corresponding points of them one of the Frenet vectors of a curve coincides with the one of Frenet vectors of the other curve. The well-known associated curve is Bertrand curve which is characterized as a kind of corresponding relation between the two curves. The relation is that the principal normal of a curve is the principal normal of the other curve, that is, the Bertrand curve is a curve which shares the normal line with the other curve 1 . Furthermore, Bertrand curves are not only the example of associated curves. Recently, a new definition of the associated curves was given by Liu and Wang 2 . They called these new curves as Mannheim partner curves. They showed that the curve γ1 is the Mannheim partner of the other curve γ if and only if the curvature κ1 and τ1 of γ1 satisfy the following equation:


Introduction
In the study of the fundamental theory and the characterizations of space curves, the related curves for which there exist corresponding relations between the curves are very interesting and important problems.The most fascinating examples of such curve are associated curves, the curves for which at the corresponding points of them one of the Frenet vectors of a curve coincides with the one of Frenet vectors of the other curve.The well-known associated curve is Bertrand curve which is characterized as a kind of corresponding relation between the two curves.The relation is that the principal normal of a curve is the principal normal of the other curve, that is, the Bertrand curve is a curve which shares the normal line with the other curve 1 .
Furthermore, Bertrand curves are not only the example of associated curves.Recently, a new definition of the associated curves was given by Liu and Wang 2 .They called these new curves as Mannheim partner curves.They showed that the curve γ 1 is the Mannheim partner of the other curve γ if and only if the curvature κ 1 and τ 1 of γ 1 satisfy the following equation: International Journal of Mathematics and Mathematical Sciences for some nonzero constant λ.They also study the Mannheim curves in Minkowski 3-space.Some different characterizations of Mannheim partner curves are given by Orbay and Kasap 3 .Another example is null Mannheim curves from Öztekin and Erg üt 4 .Since a null vector and a nonnull vector are linearly independent in the Minkowski space E 3 1 , they have noticed that the Mannheim partner curve of a null curve cannot be a null curve.They defined the null Mannheim curves whose Mannheim partner curves are either time-like or space-like.
In this paper, we get the necessary and sufficient conditions for the null Mannheim curves.Then, we investigate there exists no null-helix Mannheim curve in the Minkowski 3-space E 3 1 .

Preliminaries
Let E 3 1 be a 3-dimensional Lorentzian space and C a smooth null curve in E 3 1 , given by Then, the tangent vector field l γ in E 3 1 satisfies l, l 0.

2.2
Denote by TC the tangent bundle of C and TC ⊥ the TC perpendicular.Clearly, TC ⊥ is a vector bundle over C of rank 2. Since ξ is null, the tangent bundle TC of C is a subbundle of TC ⊥ of rank 1.This implies that TC ⊥ is not complementary of TC in E 3 1 | C .Thus, we must find a complementary vector bundle to TC of C in E 3 1 which will play the role of the normal bundle TC ⊥ consistent with the classical non-degenerate theory.
Suppose S TC ⊥ denotes the complementary vector subbundle to TC in TC ⊥ ; that is, we have where ⊥ means the orthogonal direct sum.It follows that S TC ⊥ is a nondegenerate vector subbundle of E 3 1 , of rank of 1.We call S TC ⊥ a screen vector bundle of C, which being nondegenerate, and we have where S TC ⊥ ⊥ is a complementary orthogonal vector subbundle to S TC ⊥ in E 3 1 | C of rank 2. We denote by F C the algebra of smooth functions on C and by Γ E the F C module of smooth sections of a vector bundle F over C. We use the same notation for any other vector bundle.
Theorem 2.1 see 5, 6 .Let C be a null curve of a Lorentzian space E 3 1 and S TC ⊥ a screen vector bundle of C.Then, there exists a unique vector bundle ntr C over C of rank 1 such that there is a unique section n ∈ Γ ntr C satisfying We where n is the null transversal vector field with respect to S TC ⊥ and γ .Let C C p be a smooth null curve, parametrized by the distinguished parameter p instead of t such that γ κ / 0 6 .Using 2.5 and 2.7 and taking into account that the screen vector bundle S TC ⊥ is Euclidean of rank 1, one obtains the following Frenet equations 1 : 2.9 Definition 2.2.Let γ be a curve in the Minkowski 3-space E 3 1 and γ a velocity of vector of γ.The curve γ is called time-like (or space-like) if γ , γ < 0 or if γ , γ > 0 .
Let T, N, B be the tangent, the principal normal, and the binormal of γ, respectively.Then, there are two cases for the Frenet formulae.

International Journal of Mathematics and Mathematical Sciences
Case 1. T and B are space-like vectors, and N is a time-like vector

2.10
Case 2. T is a time-like vector, and N and B are space-like vectors where κ and τ are called the dual curvature and dual torsion of γ, respectively 5 .Since l is null and from 3.4 , we obtain The proof is complete.

Null Mannheim Curves in
Remarks. a Theorems hold for null dual Mannheim curve with space-like dual Mannheim partner curve.b Some results in 4 unfortunately are not correct.For example, Theorem 3.3 gave necessary and sufficient conditions for null Mannheim curve, which implies that the null Mannheim curve should be a null-helix from Proposition 3.4.Moreover, Propositions in 4 are related with a null-helix partner curve.

E 3 1 Definition 3 . 1 .Theorem 3 . 2 . 2 Since
Let C : γ p be a Cartan framed null curve and C * : γ * p * a time-like or spacelike curve in the Minkowski space E 3 1 .If there exists a corresponding relationship between the space curves C and C * such that the principal normal lines of C coincides with the binormal lines of C * at the corresponding points of the curves, then C called a null Mannheim curve and C * is called a time-like or space-like Mannheim partner curve of C. The pair of {C, C * } is said to be a null Mannheim pair 2, 4 .Let C : γ p be a null Mannheim curve with time-like Mannheim partner curve C * : γ * p * , and let {l p , n p , u p } be the Cartan frame field along C and {T p * , N p * , B p * } the Frenet frame field along C * .Then, C * is the time-like Mannheim partner curve of γ if and only if its torsion τ * is constant such that τ * ∓ 1/μ , where μ is nonzero constant.Proof.Assume that γ is a null Mannheim curve with time-like Mannheim partner curve γ * .Then, by Definition 3.1, we can write γ p p * γ * p * μ p * B p * , 3.1 for some function μ p * .By taking the derivative of 3.1 with respect to p * and applying the Frenet formulae, we have l dp dp * T μ B μ −τ * N .3.u coincides with B, we get μ 0, 3.3 which means that μ is a nonzero constant.Thus, we have l dp dp * T − μτ * N. 3.4 call the vector bundle ntr C the null transversal bundle of C with respect to S TC ⊥ .Next consider the vector bundle C TC ⊕ tr C TC ⊕ ntr C ⊥ S TC ⊥ .2.7One calls tr C the transversal vector bundle of C with respect to S TC ⊥ .The vector field n in Theorem 2.1 is called the null transversal vector field of C with respect to ξ.As {ξ, n} is a null basis of Γ TC ⊕ ntr C satisfying 2.5 , any screen vector bundle S TC ⊥ of C is Euclidean.Note that for any arbitrary parameter t on C and a screen vector bundle S TC ⊥ one finds a distinguished parameter given by 3which means that γ * is a time-like curve with constant torsion.Conversely, let the torsion τ * of the time-like curve C * be a constant with τ * ∓ 1/μ for some nonzero constant μ.By considering a null curve C : γ p defined by Mannheim partner curve γ * if and only if the torsion τ of γ is nonzero constant.isatimelikecurvewhose binormal direction coincides with the principal normal of γ.Then, B p * ∓u p .Therefore, we can write Since the binormal direction of γ * coincides with the principal normal of γ, we get T, u 0. Therefore, we have μ 0 and μ is constant.By taking the derivative of 3.10 , we get If a Cartan framed null curve γ in E31 is a null Mannheim curve with time-like Mannheim partner curve γ * , the signs of κ and τ are the same.2If a Cartan framed null curve γ in E 3 1 is a null Mannheim curve with space-like Mannheim partner curve γ * , the signs of κ and τ are opposite.Proof.From 3.10 and 3.13 , * p * μ p * B p * , 3.6 we prove that γ is a null Mannheim and γ * is the time-like Mannheim partner curve of γ.By differentiating 3.6 with respect to p * , we get l dp dp * T − μτ * N. * T ∓ N, 3.8 which means that l lies in the plane which is spanned by T and N, hence l ⊥ B. The proof is complete.Theorem 3.3.A Cartan framed null curve γ in E 3 1 is a null Mannheim curve with time-like * γ * p * Proposition 3.4.If γ γ p be a generalized null-helix in E 3 1 , then, the curve can not be a Mannheim curve.