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

Abstract

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 𝜅₁ and 𝜏₁ of 𝛾₁ satisfy the following equation:𝜏̇=𝜅¹𝜆1+𝜆²𝜏₁₂,(1.1) 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 Ergü‌t [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 𝐶 a smooth null curve in 𝔼³¹, given by 𝛾(𝑡)=𝛾₁(𝑡),𝛾₂(𝑡),𝛾₃(𝑡),𝑡∈𝐼⊂ℝ.(2.1) Then, the tangent vector field 𝑙=𝛾′ in 𝔼³¹ 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 𝔼³¹∣_𝐶. Thus, we must find a complementary vector bundle to 𝑇𝐶 of 𝐶 in 𝔼³¹ 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^⟂=𝑇𝐶⟂𝑆𝑇𝐶^⟂,(2.3) where ⟂ means the orthogonal direct sum. It follows that 𝑆(𝑇𝐶^⟂) is a nondegenerate vector subbundle of 𝔼³¹, of rank of 1. We call 𝑆(𝑇𝐶^⟂) a screen vector bundle of 𝐶, which being non-degenerate, and we have𝔼³¹∣_𝐶=𝑆𝑇𝐶^⟂⟂𝑆𝑇𝐶^⟂^⟂,(2.4) where 𝑆(𝑇𝐶^⟂)^⟂ is a complementary orthogonal vector subbundle to 𝑆(𝑇𝐶^⟂) in 𝔼³¹|_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 𝔼³¹ 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 ⟨𝑙,𝑛⟩=1,⟨𝑛,𝑛⟩=⟨𝑛,𝑋⟩=0,∀𝑋∈Γ𝑆𝑇𝐶^⟂.(2.5)

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^⟂,(2.6) which from (2.5) is complementary but not orthogonal to 𝑇𝐶 in 𝔼³¹∣_𝐶.

More precisely, we have𝔼³¹∣_𝐶=𝑇𝐶⊕tr(𝐶)=(𝑇𝐶⊕ntr(𝐶))⟂𝑆𝑇𝐶^⟂.(2.7) 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(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 by𝑝=^𝑡∗_𝑡∗0exp^𝑠𝑠₀𝛾^,𝑛𝑑𝑡^∗𝑑𝑠,(2.8) 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]: 𝑑𝑑𝑝⎡⎢⎢⎢⎣𝑛𝑢𝑙⎤⎥⎥⎥⎦=⎡⎢⎢⎢⎣−00𝜏−00𝜅𝜅𝜏0⎤⎥⎥⎥⎦⎡⎢⎢⎢⎣𝑛𝑢𝑙⎤⎥⎥⎥⎦.(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 ⟨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⎡⎢⎢⎢⎣𝑁𝐵𝑇⎤⎥⎥⎥⎦=⎡⎢⎢⎢⎣𝜅0𝜅𝜏0𝜏00⎤⎥⎥⎥⎦⎡⎢⎢⎢⎣𝑁𝐵𝑇⎤⎥⎥⎥⎦.(2.10)

Case 2. 𝑇 is a time-like vector, and 𝑁 and 𝐵 are space-like vectors 𝑑𝑑𝑠̃⎡⎢⎢⎢⎣𝑁𝐵𝑇⎤⎥⎥⎥⎦=⎡⎢⎢⎢⎣−0𝜅−𝜅0𝜏𝜏00⎤⎥⎥⎥⎦⎡⎢⎢⎢⎣𝑁𝐵𝑇⎤⎥⎥⎥⎦,(2.11) where 𝜅 and 𝜏 are called the dual curvature and dual torsion of 𝛾, respectively [5].

3. Null Mannheim Curves in 𝔼³¹

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 𝔼³¹. 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^∗,(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 𝑙𝑑𝑝𝑑𝑝^∗=𝑇+𝜇′𝐵+𝜇−𝜏^∗𝑁.(3.2) Since 𝑢 coincides with 𝐵, we get 𝜇′=0,(3.3) which means that 𝜇 is a nonzero constant. Thus, we have 𝑙𝑑𝑝𝑑𝑝^∗=𝑇−𝜇𝜏^∗𝑁.(3.4) Since 𝑙 is null and from (3.4), we obtain −1+𝜇𝜏^∗²=0→𝜏^∗=∓1𝜇,(3.5) 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^∗,(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 𝑙𝑑𝑝𝑑𝑝^∗=𝑇−𝜇𝜏^∗𝑁.(3.7) If we use 𝜏^∗=∓1/𝜇 in (3.7), we obtain 𝑙𝑑𝑝𝑑𝑝^∗=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 𝔼³¹ 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 𝔼³¹. 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),(3.9) for some function 𝜇(𝑝)≠0. Differentiating (3.9) with respect to 𝑝, we obtain 𝑇𝑑𝑝^∗𝑑𝑝=(1−𝜇𝜏)𝑙−𝜇𝜅𝑛+𝜇′𝑢.(3.10) 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=−𝜇𝜏′𝑙−𝜇𝜅′𝑛+(1−2𝜇𝜏)𝜅𝑢.(3.11) Since 𝑢 is in the binormal direction of 𝛾^∗, we have (1−2𝜇𝜏)𝜅=0,(3.12) and hence 𝜏=12𝜇=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 𝛾^∗=𝛾^∗(𝑝^∗) of 𝛾=𝛾(𝑝) in 𝔼³¹. 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 𝑑𝑑𝑝𝑝^∗²=𝜇𝜅=const.(3.14) From (3.11), we have 𝜅^∗𝑁𝑑𝑝^∗𝑑𝑝²=0,(3.15) and thus, we get 𝜅^∗=0. This shows that 𝛾^∗=𝛾^∗(𝑝^∗) 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 𝛾^∗, the signs of 𝜅 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 𝛾^∗, the signs of 𝜅 and 𝜏 are opposite.

Proof. From (3.10) and (3.13), 𝑑𝑑𝑝𝑝^∗²=𝜅2𝜏ifMannheimpartnercurveistime-like,(3.16)𝑑𝑑𝑝𝑝^∗²=−𝜅2𝜏ifMannheimpartnercurveisspace-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.