We study a null Mannheim curve with time-like or space-like Mannheim partner curve in the Minkowski 3-space 𝔼31. We get the characterization of a null Mannheim curve. Then, we investigate there is no null-helix Mannheim curve in 𝔼31.

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 [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 πœ…1 and 𝜏1 of 𝛾1 satisfy the following equation:πœ…Μ‡πœ=1πœ†ξ€·1+πœ†2𝜏12ξ€Έ,(1.1) for some nonzero constant πœ†. 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 Μˆβ€ŒOztekin and ErgΜˆβ€Œut [4]. Since a null vector and a nonnull vector are linearly independent in the Minkowski space 𝔼31, 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 𝔼31.

2. Preliminaries

Let 𝔼31 be a 3-dimensional Lorentzian space and 𝐢 a smooth null curve in 𝔼31, given by 𝛾𝛾(𝑑)=1(𝑑),𝛾2(𝑑),𝛾3ξ€Έ(𝑑),π‘‘βˆˆπΌβŠ‚β„.(2.1) Then, the tangent vector field 𝑙=𝛾′ in 𝔼31 satisfiesβŸ¨π‘™,π‘™βŸ©=0.(2.2) Denote by 𝑇𝐢 the tangent bundle of 𝐢 and π‘‡πΆβŸ‚ the 𝑇𝐢 perpendicular. Clearly, π‘‡πΆβŸ‚ is a vector bundle over 𝐢 of rank 2. Since πœ‰ is null, the tangent bundle 𝑇𝐢 of 𝐢 is a subbundle of π‘‡πΆβŸ‚ of rank 1. This implies that π‘‡πΆβŸ‚ is not complementary of 𝑇𝐢 in 𝔼31∣𝐢. Thus, we must find a complementary vector bundle to 𝑇𝐢 of 𝐢 in 𝔼31 which will play the role of the normal bundle π‘‡πΆβŸ‚ consistent with the classical non-degenerate theory.

Suppose 𝑆(π‘‡πΆβŸ‚) denotes the complementary vector subbundle to 𝑇𝐢 in π‘‡πΆβŸ‚; that is, we haveπ‘‡πΆβŸ‚ξ€·=π‘‡πΆβŸ‚π‘†π‘‡πΆβŸ‚ξ€Έ,(2.3) where βŸ‚ means the orthogonal direct sum. It follows that 𝑆(π‘‡πΆβŸ‚) is a nondegenerate vector subbundle of 𝔼31, of rank of 1. We call 𝑆(π‘‡πΆβŸ‚) a screen vector bundle of 𝐢, which being non-degenerate, and we have𝔼31βˆ£πΆξ€·=π‘†π‘‡πΆβŸ‚ξ€Έξ€·βŸ‚π‘†π‘‡πΆβŸ‚ξ€ΈβŸ‚,(2.4) where 𝑆(π‘‡πΆβŸ‚)βŸ‚ is a complementary orthogonal vector subbundle to 𝑆(π‘‡πΆβŸ‚) in 𝔼31|𝐢 of rank 2.

We denote by 𝐹(𝐢) the algebra of smooth functions on 𝐢 and by Ξ“(𝐸) the 𝐹(𝐢) module of smooth sections of a vector bundle 𝐹 over 𝐢. We use the same notation for any other vector bundle.

Theorem 2.1 (see [5, 6]). Let 𝐢 be a null curve of a Lorentzian space 𝔼31 and 𝑆(π‘‡πΆβŸ‚) a screen vector bundle of 𝐢. Then, there exists a unique vector bundle ntr(𝐢) over 𝐢 of rank 1 such that there is a unique section π‘›βˆˆΞ“(ntr(𝐢)) satisfying ξ€·π‘†ξ€·βŸ¨π‘™,π‘›βŸ©=1,βŸ¨π‘›,π‘›βŸ©=βŸ¨π‘›,π‘‹βŸ©=0,βˆ€π‘‹βˆˆΞ“π‘‡πΆβŸ‚.ξ€Έξ€Έ(2.5)

We call the vector bundle ntr(𝐢) the null transversal bundle of 𝐢 with respect to 𝑆(π‘‡πΆβŸ‚). Next consider the vector bundletr(𝐢)=ntrξ€·(𝐢)βŸ‚π‘†π‘‡πΆβŸ‚ξ€Έ,(2.6) which from (2.5) is complementary but not orthogonal to 𝑇𝐢 in 𝔼31∣𝐢.

More precisely, we have𝔼31∣𝐢=π‘‡πΆβŠ•tr(𝐢)=(π‘‡πΆβŠ•ntrξ€·(𝐢))βŸ‚π‘†π‘‡πΆβŸ‚ξ€Έ.(2.7) One calls tr(𝐢) the transversal vector bundle of 𝐢 with respect to 𝑆(π‘‡πΆβŸ‚). The vector field 𝑛 in Theorem 2.1 is called the null transversal vector field of 𝐢 with respect to πœ‰. As {πœ‰,𝑛} is a null basis of Ξ“(π‘‡πΆβŠ•ntr(𝐢)) satisfying (2.5), any screen vector bundle 𝑆(π‘‡πΆβŸ‚) of 𝐢 is Euclidean.

Note that for any arbitrary parameter 𝑑 on 𝐢 and a screen vector bundle 𝑆(π‘‡πΆβŸ‚) one finds a distinguished parameter given byξ€œπ‘=π‘‘βˆ—π‘‘βˆ—0ξ‚΅ξ€œexp𝑠𝑠0ξ«π›Ύξ…žξ…žξ¬,π‘›π‘‘π‘‘βˆ—ξ‚Άπ‘‘π‘ ,(2.8) where 𝑛 is the null transversal vector field with respect to 𝑆(π‘‡πΆβŸ‚) and 𝛾′.

Let 𝐢=𝐢(𝑝) be a smooth null curve, parametrized by the distinguished parameter 𝑝 instead of 𝑑 such that β€–π›Ύξ…žξ…žβ€–=πœ…β‰ 0 ([6]). Using (2.5) and (2.7) and taking into account that the screen vector bundle 𝑆(π‘‡πΆβŸ‚) is Euclidean of rank 1, one obtains the following Frenet equations [1]: π‘‘βŽ‘βŽ’βŽ’βŽ’βŽ£π‘™π‘›π‘’βŽ€βŽ₯βŽ₯βŽ₯⎦=⎑⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ£π‘™π‘›π‘’βŽ€βŽ₯βŽ₯βŽ₯⎦.𝑑𝑝00πœ…00πœβˆ’πœβˆ’πœ…0(2.9)

Definition 2.2. Let 𝛾 be a curve in the Minkowski 3-space 𝔼31 and 𝛾′ a velocity of vector of 𝛾. The curve 𝛾 is called time-like (or space-like) if βŸ¨π›Ύβ€²,π›Ύβ€²βŸ©<0 (or if βŸ¨π›Ύβ€²,π›Ύβ€²βŸ©>0).

Let 𝑇,𝑁,𝐡 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 𝐡 are space-like vectors, and 𝑁 is a time-like vector π‘‘βŽ‘βŽ’βŽ’βŽ’βŽ£π‘‡π‘π΅βŽ€βŽ₯βŽ₯βŽ₯⎦=⎑⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ£π‘‡π‘π΅βŽ€βŽ₯βŽ₯βŽ₯⎦.π‘‘π‘ πœ…0πœ…0𝜏0𝜏0(2.10)

Case 2. 𝑇 is a time-like vector, and 𝑁 and 𝐡 are space-like vectors π‘‘βŽ‘βŽ’βŽ’βŽ’βŽ£π‘‡π‘π΅βŽ€βŽ₯βŽ₯βŽ₯⎦=⎑⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ¦βŽ‘βŽ’βŽ’βŽ’βŽ£π‘‡π‘π΅βŽ€βŽ₯βŽ₯βŽ₯⎦,π‘‘Μƒπ‘ πœ…0βˆ’πœ…0𝜏0βˆ’πœ0(2.11) where πœ… and 𝜏 are called the dual curvature and dual torsion of 𝛾, respectively [5].

3. Null Mannheim Curves in 𝔼31

Definition 3.1. Let πΆβˆΆπ›Ύ(𝑝) be a Cartan framed null curve and πΆβˆ—βˆΆπ›Ύβˆ—(π‘βˆ—) a time-like or space-like curve in the Minkowski space 𝔼31. If there exists a corresponding relationship between the space curves 𝐢 and πΆβˆ— such that the principal normal lines of 𝐢 coincides with the binormal lines of πΆβˆ— at the corresponding points of the curves, then 𝐢 called a null Mannheim curve and πΆβˆ— is called a time-like or space-like Mannheim partner curve of 𝐢. The pair of {𝐢,πΆβˆ—} is said to be a null Mannheim pair [2, 4].

Theorem 3.2. Let πΆβˆΆπ›Ύ(𝑝) be a null Mannheim curve with time-like Mannheim partner curve πΆβˆ—βˆΆπ›Ύβˆ—(π‘βˆ—), and let {𝑙(𝑝),𝑛(𝑝),𝑒(𝑝)} be the Cartan frame field along 𝐢 and {𝑇(π‘βˆ—),𝑁(π‘βˆ—),𝐡(π‘βˆ—)} the Frenet frame field along πΆβˆ—. Then, πΆβˆ— is the time-like Mannheim partner curve of 𝛾 if and only if its torsion πœβˆ— is constant such that πœβˆ—=βˆ“(1/πœ‡), where πœ‡ is nonzero constant.

Proof. Assume that 𝛾 is a null Mannheim curve with time-like Mannheim partner curve π›Ύβˆ—. Then, by Definition 3.1, we can write π›Ύξ€·π‘ξ€·π‘βˆ—ξ€Έξ€Έ=π›Ύβˆ—ξ€·π‘βˆ—ξ€Έξ€·π‘+πœ‡βˆ—ξ€Έπ΅ξ€·π‘βˆ—ξ€Έ,(3.1) for some function πœ‡(π‘βˆ—). By taking the derivative of (3.1) with respect to π‘βˆ— and applying the Frenet formulae, we have π‘™π‘‘π‘π‘‘π‘βˆ—ξ€·=𝑇+πœ‡β€²π΅+πœ‡βˆ’πœβˆ—π‘ξ€Έ.(3.2) Since 𝑒 coincides with 𝐡, we get πœ‡β€²=0,(3.3) which means that πœ‡ is a nonzero constant. Thus, we have π‘™π‘‘π‘π‘‘π‘βˆ—=π‘‡βˆ’πœ‡πœβˆ—π‘.(3.4) Since 𝑙 is null and from (3.4), we obtain ξ€·βˆ’1+πœ‡πœβˆ—ξ€Έ2=0β†’πœβˆ—1=βˆ“πœ‡,(3.5) which means that π›Ύβˆ— is a time-like curve with constant torsion.
Conversely, let the torsion πœβˆ—of the time-like curve πΆβˆ— be a constant with πœβˆ—=βˆ“(1/πœ‡) for some nonzero constant πœ‡. By considering a null curve πΆβˆΆπ›Ύ(𝑝) defined by π›Ύξ€·π‘βˆ—ξ€Έ=π›Ύβˆ—ξ€·π‘βˆ—ξ€Έξ€·π‘+πœ‡βˆ—ξ€Έπ΅ξ€·π‘βˆ—ξ€Έ,(3.6) we prove that 𝛾 is a null Mannheim and π›Ύβˆ— is the time-like Mannheim partner curve of 𝛾. By differentiating (3.6) with respect to π‘βˆ—, we get π‘™π‘‘π‘π‘‘π‘βˆ—=π‘‡βˆ’πœ‡πœβˆ—π‘.(3.7) If we use πœβˆ—=βˆ“1/πœ‡ in (3.7), we obtain π‘™π‘‘π‘π‘‘π‘βˆ—=π‘‡βˆ“π‘,(3.8)which means that 𝑙 lies in the plane which is spanned by 𝑇 and 𝑁, hence π‘™βŸ‚π΅. The proof is complete.

Theorem 3.3. A Cartan framed null curve 𝛾 in 𝔼31 is a null Mannheim curve with time-like Mannheim partner curve π›Ύβˆ— if and only if the torsion 𝜏 of 𝛾 is nonzero constant.

Proof. Let 𝛾=𝛾(𝑝) be a null Mannheim curve in 𝔼31. Suppose that π›Ύβˆ—=π›Ύβˆ—(π‘βˆ—) is a time-like curve whose binormal direction coincides with the principal normal of 𝛾. Then, 𝐡(π‘βˆ—)=βˆ“π‘’(𝑝). Therefore, we can write π›Ύβˆ—(𝑝)=𝛾(𝑝)+πœ‡(𝑝)𝑒(𝑝),(3.9) for some function πœ‡(𝑝)β‰ 0. Differentiating (3.9) with respect to 𝑝, we obtain π‘‡π‘‘π‘βˆ—=𝑑𝑝(1βˆ’πœ‡πœ)π‘™βˆ’πœ‡πœ…π‘›+πœ‡β€²π‘’.(3.10) Since the binormal direction of π›Ύβˆ— coincides with the principal normal of 𝛾, we get βŸ¨π‘‡,π‘’βŸ©=0. Therefore, we have πœ‡β€²=0 and πœ‡ is constant. By taking the derivative of (3.10), we get πœ…βˆ—π‘ξ‚΅π‘‘π‘βˆ—ξ‚Άπ‘‘π‘2𝑑+𝑇2π‘βˆ—π‘‘π‘2=βˆ’πœ‡πœβ€²π‘™βˆ’πœ‡πœ…β€²π‘›+(1βˆ’2πœ‡πœ)πœ…π‘’.(3.11) Since 𝑒 is in the binormal direction of π›Ύβˆ—, we have (1βˆ’2πœ‡πœ)πœ…=0,(3.12) and hence 1𝜏==2πœ‡const.(3.13) 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 𝔼31, then, the curve can not be a Mannheim curve.

Proof. Suppose that 𝛾=𝛾(𝑝) is a Mannheim curve in 𝔼31. Then, there exists the Mannheim partner curve π›Ύβˆ—=π›Ύβˆ—(π‘βˆ—) of 𝛾=𝛾(𝑝) in 𝔼31. From Theorems 3.2 and 3.3, the torsions of the Mannheim pair {𝛾,π›Ύβˆ—}, 𝜏 and πœβˆ—, are nonzero-constant. Since 𝛾=𝛾(𝑝) be a generalized null-helix, πœ…/𝜏 is constant, and thus πœ… is constant. Using (3.10), (3.13), and the fact that 𝑇 is time-like, we have ξ‚΅π‘‘π‘βˆ—ξ‚Άπ‘‘π‘2=πœ‡πœ…=const.(3.14) From (3.11), we have πœ…βˆ—π‘ξ‚΅π‘‘π‘βˆ—ξ‚Άπ‘‘π‘2=0,(3.15) and thus, we get πœ…βˆ—=0. This shows that π›Ύβˆ—=π›Ύβˆ—(π‘βˆ—) is a straight line with nonzero torsion in 𝔼31, which is impossible. Therefore, 𝛾=𝛾(𝑝) cannot be a dual Mannheim curve in 𝔼31.

Corollary 3.5. (1) If a Cartan framed null curve 𝛾 in 𝔼31 is a null Mannheim curve with time-like Mannheim partner curve π›Ύβˆ—, the signs of πœ… and 𝜏 are the same.
(2) If a Cartan framed null curve 𝛾 in 𝔼31 is a null Mannheim curve with space-like Mannheim partner curve π›Ύβˆ—, the signs of πœ… and 𝜏 are opposite.

Proof. From (3.10) and (3.13), ξ‚΅π‘‘π‘βˆ—ξ‚Άπ‘‘π‘2=πœ…2𝜏ifMannheimpartnercurveistime-like,(3.16)ξ‚΅π‘‘π‘βˆ—ξ‚Άπ‘‘π‘2πœ…=βˆ’2𝜏ifMannheimpartnercurveisspace-like.(3.17) The proof is complete.

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.