IJMMSInternational Journal of Mathematics and Mathematical Sciences1687-04250161-1712Hindawi Publishing Corporation58053710.1155/2011/580537580537Research ArticleNo Null-Helix Mannheim Curves in the Minkowski Space 𝔼13LeeJae Won1MollerManfred1Institute of MathematicsAcademia SinicaTaipei 10617Taiwan201121072011201129122010230520112011Copyright © 2011 Jae Won Lee.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study a null Mannheim curve with time-like or space-like Mannheim partner curve in the Minkowski 3-space 𝔼13. We get the characterization of a null Mannheim curve. Then, we investigate there is no null-helix Mannheim curve in 𝔼13.

1. 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 .

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 . 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:τ̇=κ1λ(1+λ2τ12), 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 . Another example is null Mannheim curves from Öztekin and Ergüt . Since a null vector and a nonnull vector are linearly independent in the Minkowski space 𝔼13, 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 𝔼13.

2. Preliminaries

Let 𝔼13 be a 3-dimensional Lorentzian space and C a smooth null curve in 𝔼13, given by γ(t)=(γ1(t),γ2(t),γ3(t)),tIR. Then, the tangent vector field l=γ in 𝔼13 satisfiesl,l=0. 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 𝔼13C. Thus, we must find a complementary vector bundle to TC of C in 𝔼13 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 haveTC=TCS(TC), where means the orthogonal direct sum. It follows that S(TC) is a nondegenerate vector subbundle of 𝔼13, of rank of 1. We call S(TC) a screen vector bundle of C, which being non-degenerate, and we haveE13C=S(TC)S(TC), where S(TC) is a complementary orthogonal vector subbundle to S(TC) in 𝔼13|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 [<xref ref-type="bibr" rid="B1">5</xref>, <xref ref-type="bibr" rid="B2">6</xref>]).

Let C be a null curve of a Lorentzian space 𝔼13 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 nΓ(ntr(C)) satisfying l,n=1,n,n=n,X=0,XΓ(S(TC)).

We call the vector bundle ntr(C) the null transversal bundle of C with respect to S(TC). Next consider the vector bundletr(C)=ntr(C)S(TC), which from (2.5) is complementary but not orthogonal to TC in 𝔼13C.

More precisely, we haveE13C=TCtr(C)=(TCntr(C))S(TC). One 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(TCntr(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 byp=t0*t*exp(s0sγ,ndt*)ds, 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=κ0 (). 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 : ddp[lnu]=[00κ00τ-τ-κ0][lnu].

Definition 2.2.

Let γ be a curve in the Minkowski 3-space 𝔼13 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.

Case 1.

T and B are space-like vectors, and N is a time-like vector dds[TNB]=[κ0κ0τ0τ0][TNB].

Case 2.

T is a time-like vector, and N and B are space-like vectors dds̃[TNB]=[κ0-κ0τ0-τ0][TNB], where κ and τ are called the dual curvature and dual torsion of γ, respectively .

3. Null Mannheim Curves in <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M116"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>𝔼</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula>Definition 3.1.

Let C:γ(p) be a Cartan framed null curve and C*:γ*(p*) a time-like or space-like curve in the Minkowski space 𝔼13. 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].

Theorem 3.2.

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*), for some function μ(p*). By taking the derivative of (3.1) with respect to p* and applying the Frenet formulae, we have ldpdp*=T+μB+μ(-τ*N). Since u coincides with B, we get μ=0, which means that μ is a nonzero constant. Thus, we have ldpdp*=T-μτ*N. Since l is null and from (3.4), we obtain -1+(μτ*)2=0τ*=1μ, which 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 γ(p*)=γ*(p*)+μ(p*)B(p*), 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 ldpdp*=T-μτ*N. If we use τ*=1/μ in (3.7), we obtain ldpdp*=TN,which means that l lies in the plane which is spanned by T and N, hence lB. The proof is complete.

Theorem 3.3.

A Cartan framed null curve γ in 𝔼13 is a null Mannheim curve with time-like Mannheim partner curve γ* if and only if the torsion τ of γ is nonzero constant.

Proof.

Let γ=γ(p) be a null Mannheim curve in 𝔼13. Suppose that γ*=γ*(p*) is a time-like curve whose binormal direction coincides with the principal normal of γ. Then, B(p*)=u(p). Therefore, we can write γ*(p)=γ(p)+μ(p)u(p), for some function μ(p)0. Differentiating (3.9) with respect to p, we obtain Tdp*dp=(1-μτ)l-μκn+μu. 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 κ*N(dp*dp)2+Td2p*dp2=-μτl-μκn+(1-2μτ)κu. Since u is in the binormal direction of γ*, we have (1-2μτ)κ=0, and hence τ=12μ=const. Conversely, similar to the proof of Theorem 3.2, we easily get a null Mannheim curve with time-like Mannheim partner curve.

Proposition 3.4.

If γ=γ(p) be a generalized null-helix in 𝔼13, then, the curve can not be a Mannheim curve.

Proof.

Suppose that γ=γ(p) is a Mannheim curve in 𝔼13. Then, there exists the Mannheim partner curve γ*=γ*(p*) of γ=γ(p) in 𝔼13. From Theorems 3.2 and 3.3, the torsions of the Mannheim pair {γ,γ*}, τ and τ*, are nonzero-constant. Since γ=γ(p) be a generalized null-helix, κ/τ is constant, and thus κ is constant. Using (3.10), (3.13), and the fact that T is time-like, we have (dp*dp)2=μκ=const. From (3.11), we have κ*N(dp*dp)2=0, and thus, we get κ*=0. This shows that γ*=γ*(p*) is a straight line with nonzero torsion in 𝔼13, which is impossible. Therefore, γ=γ(p) cannot be a dual Mannheim curve in 𝔼13.

Corollary 3.5.

(1) If a Cartan framed null curve γ in 𝔼13 is a null Mannheim curve with time-like Mannheim partner curve γ*, the signs of κ and τ are the same.

(2) If a Cartan framed null curve γ in 𝔼13 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), (dp*dp)2=κ2τif  Mannheim  partner  curve  is  time-like,(dp*dp)2=-κ2τif  Mannheimpartner  curve  is  space-like. The proof is complete.

Remarks.

(a) Theorems hold for null dual Mannheim curve with space-like dual Mannheim partner curve.

(b) Some results in  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  are related with a null-helix partner curve.

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