Mannheim curves and their partner curves in Minkowski 3-space E13

Ayman Elsharkawy; Ahmed M. Elshenhab

doi:10.1515/dema-2022-0163

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Mannheim curves and their partner curves in Minkowski 3-space E₁³

Ayman Elsharkawy and Ahmed M. Elshenhab

Published/Copyright: November 16, 2022

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From the journal Demonstratio Mathematica Volume 55 Issue 1

Abstract

The modified orthogonal frame is an important tool to study analytic space curves whose curvatures have discrete zero points. In this article, by using the modified orthogonal frame, Mannheim curves and their partner curves are investigated in Minkowski 3-space. Some characterizations according to the curvatures and torsions of the curves are given. Finally, some relations under the conditions for Mannheim curves and their partner curves to be generalized helices are presented. All the possible cases for the partner curves to be spacelike and timelike are considered in the whole of the article.

Keywords: Mannheim curves; Mannheim partner curves; modified orthogonal frame; Minkowski 3-space; timelike curve; spacelike curve; curvature; torsion

MSC 2010: 53A35; 53C50

1 Introduction

In differential geometry, it is well known that the characterization of a regular curve is an important problem that has received a lot of attention to many mathematicians. The curvature κ ( s ) and the torsion τ ( s ) of a regular curve have a significant role to identify the size and shape of the curve. Moreover, the relationship between the Frenet vectors of the curves is another way to classification and characterization of curves. Some curves have been introduced and studied in the last two decades, especially, the partner curves, i.e., the curves which are related to each other at the corresponding points, have attracted the attention of many mathematicians so far. The well-known of the partner curves is Bertrand curves which are defined by the property that at the corresponding points of two space curves the principal normal vectors are common. Bertrand curves and their partner curves have been studied in [1,2, 3,4] and references therein. In 2007 and 2008, Liu and Wang in [5,6] defined a new curve pair for space curves. They called these new curves as Mannheim partner curves. If there exists a correspondence between two curves in the three-dimensional Euclidean space E 3 such that, at the corresponding points of the curves, the principal normal vectors of the first curve coincide with the binormal vectors of the other one, then the first curve is called a Mannheim curve, and the second one is called a Mannheim partner curve of the first, and the pair of these two curves is called a Mannheim pair. They showed that the curve α 1 ( s 1 ) is the Mannheim partner curve of the curve α ( s ) , where s , s 1 are the arc length for the curves α ( s ) and α 1 ( s 1 ) , respectively, if and only if the curvature κ 1 and the torsion τ 1 of α 1 ( s 1 ) satisfy an equation given by

τ 1 ′ = d τ 1 d s 1 = κ 1 λ ( 1 + λ 2 τ 1 2 ) ,

for some nonzero constant λ . They also studied the Mannheim partner curves in the Minkowski 3-space E 1 3 and obtained the necessary and sufficient conditions for the Mannheim partner curves in E 1 3 . In 2009, Orbay and Kasap in [7] gave new characterizations of Mannheim partner curves in Euclidean 3-space. In 2011, Kahraman et al. in [8] gave some characterizations of Mannheim partner curves in Minkowski 3-space E 1 3 . On the other hand, the moving Frénet frame is inappropriate for studying analytic space curves of which curvatures have discrete zero points because the principal normal and binormal vectors might be undefined at zero points. To solve this problem, in 1984, Sasai in [9] presented an orthogonal frame and obtained a formula, which corresponds to the Frenet-Serret equations. Recently, some authors in [3,10,11] have derived some characterizations of Mannheim curves, helices, spherical curves, and the Bertrand curves by using the modified orthogonal frame in Euclidean space E 3 . Furthermore, Bükcü and Karacan in [12] have described the modified orthogonal frame with nonzero curvature and torsion in Minkowski 3-space E 1 3 . More recently, in 2021 Elsayied et al. in [13,14] studied some curves according to the modified frame in Minkowski 3- space. Furthermore, there is another frame called quasi frame studied in [15,16,17, 18,19]. However, to the best of our knowledge, there are no results dealing with the characterizations of Mannheim curves according to modified orthogonal frame in Minkowski 3-space E 1 3 . Motivated by these papers, we will remove the condition of κ ( s ) ≠ 0 and consider a general set of curves with a discrete set of zeros of κ ( s ) to give some characterizations of Mannheim curves and their partner curves by using the modified orthogonal frame in E 1 3 . The article is organized as follows: In Section 2, we present some basic definitions about the Minkowski 3-space E 1 3 , curves including nonnull curves and null curves, and the angle θ between two vectors in E 1 3 . Furthermore, we give the definition of the modified orthogonal frame, Mannheim pairs, and Mannheim curves in the Minkowski 3-space E 1 3 . In Section 3, we give some characterizations of Mannheim partner curves with the modified orthogonal frame and establish necessary and sufficient conditions for the Mannheim partner curves in E 1 3 . In Section 4, we give some characterizations of Mannheim curves with the modified orthogonal frame and establish necessary and sufficient conditions for Mannheim curves in E 1 3 . Moreover, we derive the relationships between the curvatures and the torsions of the Mannheim pairs in E 1 3 .

2 Preliminaries

In this section, we present some preliminaries used in our subsequent discussions. The Lorentz-Minkowski space is the metric space E 1 3 = ( R 3 , ⟨ , ⟩ ) where the metric ⟨ , ⟩ is given by

⟨ ⋅ , ⋅ ⟩ = − d x 1 2 + d x 2 2 + d x 3 2 ,

with ( x 1 , x 2 , x 3 ) a rectangular coordinate system of E 1 3 . Based on this metric, in E 1 3 an arbitrary vector γ = ( γ 1 , γ 2 , γ 3 ) is said to be spacelike if ⟨ γ , γ ⟩ > 0 or γ = 0 , timelike if ⟨ γ , γ ⟩ < 0 , and null (lightlike) if ⟨ γ , γ ⟩ = 0 and γ ≠ 0 . Similarly, an arbitrary curve α = α ( s ) in E 1 3 can locally be spacelike, timelike, or null (lightlike), if all of its velocity vectors α ′ ( s ) are spacelike, timelike, or null (lightlike), respectively. Spacelike curve in E 1 3 is called pseudo null curve if its principal normal vector is null. The norm of a vector γ is given by ∥ γ ∥ = ∣ ⟨ γ , γ ⟩ ∣ . The distance between two vectors u and v is given by ∥ u − v ∥ = ∣ ⟨ u − v , u − v ⟩ ∣ . Two vectors β and γ are said to be orthogonal in E 1 3 , if ⟨ β , γ ⟩ = 0 . A nonnull curve α ( s ) is parameterized by arc-length s if ⟨ α ′ ( s ) , α ′ ( s ) ⟩ = ± 1 . We say that a timelike vector is future pointing or past pointing if the first component of the vector is positive or negative, respectively. For the vectors x = ( x 1 , x 2 , x 3 ) and y = ( y 1 , y 2 , y 3 ) in E 1 3 , the Lorentzian vector product of x and y is defined by

x × y = − e 1 e 2 e 3 x 1 x 2 x 3 y 1 y 2 y 3 = ( x 3 y 2 − x 2 y 3 , x 3 y 1 − x 1 y 3 , x 1 y 2 − x 2 y 1 ) ,

where { e 1 , e 2 , e 3 } is the natural basis of E 1 3 , i.e.,

e 1 = ( 1 , 0 , 0 ) , e 2 = ( 0 , 1 , 0 ) , e 3 = ( 0 , 0 , 1 ) .

Then we have

e 1 × e 2 = e 3 , e 2 × e 3 = − e 1 , e 3 × e 1 = e 2 .

The vector product operation × is related to the determinant function by det ( x , y , z ) = ⟨ x × y , z ⟩ [20,21].

Let α ( s ) be a space curve in Minkowski space E 1 3 , parameterized by arc-length s . Denote by { t , n , b } the moving Frénet frame along the curve α ( s ) . We also assume that its curvature κ ( s ) ≠ 0 anywhere. Then for an arbitrary spacelike curve α ( s ) in the space E 1 3 , the following Frénet formulae are given by

(2.1) t ′ ( s ) n ′ ( s ) b ′ ( s ) = 0 κ 0 − ε κ 0 τ 0 τ 0 t ( s ) n ( s ) b ( s ) ,

where

⟨ t , t ⟩ = 1 , ⟨ n , n ⟩ = ε = ± 1 , ⟨ b , b ⟩ = − ε , ⟨ t , n ⟩ = ⟨ n , b ⟩ = ⟨ b , t ⟩ = 0 ,

and t ( s ) is the unit tangent vector, n ( s ) is the unit principal normal vector, b ( s ) is the unit binormal vector, τ ( s ) is the torsion, and ε , here, demonstrates the type of a spacelike curve α ( s ) . If ε = 1 , then α ( s ) is a spacelike curve with spacelike principal normal n and timelike binormal b . If ε = − 1 , then α ( s ) is a spacelike curve with timelike principal normal n and spacelike binormal b . Furthermore, for a timelike curve α ( s ) in the space E 1 3 , the following Frenet formulae are given as follows:

(2.2) t ′ ( s ) n ′ ( s ) b ′ ( s ) = 0 κ 0 κ 0 τ 0 − τ 0 t ( s ) n ( s ) b ( s ) ,

where ⟨ t , t ⟩ = − 1 , ⟨ n , n ⟩ = ⟨ b , b ⟩ = 1 , ⟨ t , n ⟩ = ⟨ n , b ⟩ = ⟨ b , t ⟩ = 0 [20,22].

Now, we assume that the curvature κ ( s ) of α ( s ) has discrete points or κ ( s ) is not identically zero. Now we define an orthogonal frame { T , N , B } as follows:

T = d α d s , N = d T d s , B = T × N ,

where T × N is the vector product of T and N . The relations between those and the classical Frénet frame { t , n , b } at nonzero points of κ are

(2.3) T = t , N = κ n , B = κ b .

Thus, N ( s 0 ) = B ( s 0 ) = 0 when κ ( s 0 ) = 0 and squares of the length of N and B vary analytically in s . By the definition of { T , N , B } and equations (2.1), (2.2), and (2.3), we deduce the following modified orthogonal frames: In case that α ( s ) is an arbitrary spacelike curve in the space E 1 3 , then the orthogonal frame is

(2.4) T ′ ( s ) N ′ ( s ) B ′ ( s ) = 0 1 0 − ε κ 2 κ ′ κ τ 0 τ κ ′ κ T ( s ) N ( s ) B ( s ) ,

where

⟨ T , T ⟩ = 1 , ⟨ N , N ⟩ = ε κ 2 , ⟨ B , B ⟩ = − ε κ 2 , ⟨ T , N ⟩ = ⟨ N , B ⟩ = ⟨ B , T ⟩ = 0 , T × N = ε B , N × B = − κ 2 T , B × T = − ε N ,

and ε , here, demonstrates that α ( s ) is a spacelike or a timelike curve. If ε = 1 , then α ( s ) is a spacelike curve with spacelike principal normal N and a timelike binormal B . If ε = − 1 , then α ( s ) is a spacelike curve with timelike principal normal N and spacelike binormal B . Furthermore, in case that α ( s ) a timelike curve in the space E 1 3 , then the orthogonal frame is

(2.5) T ′ ( s ) N ′ ( s ) B ′ ( s ) = 0 1 0 κ 2 κ ′ κ τ 0 − τ κ ′ κ T ( s ) N ( s ) B ( s ) ,

where

⟨ T , T ⟩ = − 1 , ⟨ N , N ⟩ = ⟨ B , B ⟩ = κ 2 , ⟨ T , N ⟩ = ⟨ N , B ⟩ = ⟨ B , T ⟩ = 0 , T × N = − B , N × B = κ 2 T , B × T = − N .

We see that for κ = 1 , the Frenet-Serret frame coincides with the modified orthogonal frame in the space E 1 3 [12,13,14].

Let us consider a curve given by the parametric equation

α ( s ) = 1 2 ∫ 0 s cos π t 2 2 d t , 1 2 ∫ 0 s sin π t 2 2 d t , s 2 ,

which is a helical curve over clothoid (Cornu spiral or Euler spiral) and has various applications in real life such as the highway, railway route design, or roller coasters, etc. Here the components ∫ 0 s cos π t 2 2 d t and ∫ 0 s sin π t 2 2 d t are called Fresnel integrals. The elements of the Frenet trihedron of the curve α are obtained as

t ( s ) = 1 2 cos π s 2 2 , 1 2 sin π s 2 2 , 1 2 , n ( s ) = − s ∣ s ∣ sin π s 2 2 , s ∣ s ∣ cos π s 2 2 , 0 , b ( s ) = − s 2 ∣ s ∣ cos π s 2 2 , − s 2 ∣ s ∣ sin π s 2 2 , s 2 ∣ s ∣ ,

and the curvature is κ ( s ) = π ∣ s ∣ 2 . Besides the curvature is not differentiable, the principal normal and binormal vectors are discontinuous at s = 0 . Therefore, the curvature κ ( s ) must not be equal to zero. On the other hand, the modified orthogonal frame of the curve α is obtained as

T ( s ) = 1 2 cos π s 2 2 , 1 2 sin π s 2 2 , 1 2 , N ( s ) = − π s 2 sin π s 2 2 , π s 2 cos π s 2 2 , 0 , B ( s ) = − π s 2 cos π s 2 2 , − π s 2 sin π s 2 2 , π s 2 ,

which implies that the curvature κ ( s ) = 0 at s = 0 , the modified orthogonal frame { T , N , B } of the curve α with unique elements and continuous at s = 0 . Therefore, we can consider a general set of curves with a discrete set of zeros of κ ( s ) or κ ( s ) is not identically zero.

Definition 1

[21]

Let u and v be future pointing (or past pointing) timelike vectors in E 1 3 . Then there is a unique real number ϕ ≥ 0 such that
⟨ u , v ⟩ = − ∥ u ∥ ∥ v ∥ cosh ϕ .
Let u and v be spacelike vectors in E 1 3 that span a timelike vector subspace. Then there is a unique real number ϕ ≥ 0 such that
⟨ u , v ⟩ = ∥ u ∥ ∥ v ∥ cosh ϕ .
Let u and v be spacelike vectors in E 1 3 that span a spacelike vector subspace. Then there is a unique real number ϕ ≥ 0 such that
⟨ u , v ⟩ = ∥ u ∥ ∥ v ∥ cos ϕ .
Let u be a spacelike vector and v be a timelike vector in E 1 3 . Then there is a unique real number ϕ ≥ 0 such that
⟨ u , v ⟩ = ∥ u ∥ ∥ v ∥ sinh ϕ ,
where ϕ is the angle between the vectors u and v .

Definition 2

[13] A curve α ( s ) is called a general helix in Minkowski 3-space E 1 3 if and only if the ratio of curvature to torsion is constant.

Definition 3

[5] Let Γ and Γ 1 be two curves in Minkowski 3-space E 1 3 given by the parameterizations α ( s ) and α 1 ( s 1 ) , respectively. If there exists a correspondence between the space curves Γ and Γ 1 such that the principal normal vectors of Γ coincide with the binormal vectors of Γ 1 at the corresponding points of curves, then Γ is called a Mannheim curve and Γ 1 is called a Mannheim partner curve of Γ . The pair { Γ , Γ 1 } is said to be a Mannheim pair.

From Definition 3, we see that there are five different types of the Mannheim pair { Γ , Γ 1 } in Minkowski 3-space E 1 3 . Then we deduce the following propositions:

Proposition 1

[8] If the curve Γ 1 is timelike with a spacelike principal normal vector and a spacelike binormal vector, then there are two cases:

Γ is a spacelike curve with a spacelike principal normal and a timelike binormal vector. In this case, we say that the pair { Γ , Γ 1 } is a Mannheim pair of type 1.
Γ is a timelike curve with a spacelike principal normal and a spacelike binormal vector. In this case, we say that the pair { Γ , Γ 1 } is Mannheim pair of type 2.

Proposition 2

[8] If the curve Γ 1 is a spacelike curve, then there are three cases:

Γ 1 is a spacelike curve with a timelike binormal vector and the curve Γ is a spacelike curve with a timelike principal normal vector. In this case, we say that the pair { Γ , Γ 1 } is a Mannheim pair of type 3.
Γ 1 is a spacelike curve with a spacelike binormal vector and the curve Γ is a timelike curve with a spacelike principal normal. In this case, we say that the pair { Γ , Γ 1 } is a Mannheim pair of type 4.
Γ 1 is a spacelike curve with a spacelike binormal vector and the curve Γ is a spacelike curve with a spacelike principal normal and a timelike binormal vector. In this case, we say that the pair { Γ , Γ 1 } is a Mannheim pair of type 5.

3 Mannheim partner curves with modified orthogonal frame in E 1 3

In this section, we extend the main results of Mannheim partner curves in E 3 to the Minkowski 3-space E 1 3 according to the modified orthogonal frame.

Theorem 1

Let Γ : α ( s ) be a Mannheim curve in E 1 3 parameterized by its arc length s and let Γ 1 : α 1 ( s 1 ) be neither a null nor a pseudo null Mannheim partner curve of Γ with an arc length parameter s 1 . Then the distance between the corresponding points of the Mannheim partner curves in E 1 3 is constant.

Proof

Let us consider the pair { Γ , Γ 1 } a Mannheim pair. From Definition 3, we can write

(3.1) α ( s ) = α 1 ( s 1 ) + λ ( s 1 ) B 1 ( s 1 ) ,

for some function λ ( s 1 ) . By taking the derivative of (3.1) with respect to s 1 and using (2.4) and (2.5), we obtain

T d s d s 1 = T 1 + λ ′ B 1 + λ δ τ 1 N 1 + κ 1 ′ κ 1 B 1

(3.2) T d s d s 1 = T 1 + δ λ τ 1 N 1 + λ ′ + λ κ 1 ′ κ 1 B 1 ,

where δ = ± 1 , here, demonstrates that α 1 ( s 1 ) is a spacelike or a timelike curve. Since N and B 1 are linearly dependent, we have ⟨ T , B 1 ⟩ = 0 . By taking the inner product of (3.2) with B 1 , we obtain

(3.3) λ ′ λ = − κ 1 ′ κ 1 .

Integrating (3.3), we obtain

(3.4) λ ( s 1 ) = c κ 1 ( s 1 ) , c > 0 .

Thus from (3.1) and (3.4), we have

□ ∥ α ( s ) − α 1 ( s 1 ) ∥ = c κ 1 ( s 1 ) B 1 ( s 1 ) = c 1 κ 1 ∣ κ 1 ∣ = c .

Remark 1

The authors in [11] said that the distance between the corresponding points of the Mannheim curves according to the modified orthogonal frame is not constant but it is not true. There was an error in determining the norm of B ψ κ ψ in the proof of Theorem 3.1. Therefore, the distance between the corresponding points of the Mannheim partner curves according to the modified orthogonal frame is constant in E 3 and E 1 3 .

Theorem 2

Let a pair of curves { Γ , Γ 1 } in E 1 3 be with respect to the modified orthogonal frame. Then we have the following cases:

If the pair { Γ , Γ 1 } is a Mannheim pair of type 1 or 2, then he necessary and sufficient condition for Mannheim partner curves Γ 1 of Γ is
(3.5) τ 1 ′ = κ 1 c ( 1 − c 2 τ 1 2 ) .
If the pair { Γ , Γ 1 } is a Mannheim pair of type 3, then the necessary and sufficient condition for Mannheim partner curves Γ 1 of Γ is
(3.6) τ 1 ′ = − κ 1 c ( 1 + c 2 τ 1 2 ) .
If the pair { Γ , Γ 1 } is a Mannheim pair of types 4 or 5, then the necessary and sufficient condition for Mannheim partner curves Γ 1 of Γ is
(3.7) τ 1 ′ = κ 1 c ( c 2 τ 1 2 − 1 ) ,
where c is a nonzero constant.

Proof

Consider the pair { Γ , Γ 1 } is a Mannheim pair of type 1. From (3.2) and (3.3), we obtain

(3.8) T d s d s 1 = T 1 − λ τ 1 N 1 + λ ′ + λ κ 1 ′ κ 1 B 1

and

(3.9) λ ( s 1 ) = c κ 1 ( s 1 ) , c > 0 .

Inserting (3.9) into (3.8), we obtain

(3.10) T d s d s 1 = T 1 − c τ 1 κ 1 N 1 .

On the other hand, let θ be the angle between T and T 1 at the corresponding points of Γ and Γ 1 in (3.1). Then, if { Γ , Γ 1 } is a Mannheim pair of type 1, from Definition 1 and considering the equality cosh 2 θ − sinh 2 θ = 1 , we obtain

(3.11) T = sinh θ T 1 + 1 κ 1 cosh θ N 1 .

By taking the derivative of (3.11) with respect to s 1 , we obtain

(3.12) N d s d s 1 = ( θ ′ + κ 1 ) cosh θ T 1 + ( θ ′ + κ 1 ) sinh θ κ 1 N 1 + τ 1 κ 1 cosh θ B 1 .

By taking the inner product of (3.12) with T 1 , we obtain

( θ ′ + κ 1 ) cosh θ = 0 .

Therefore, we have

(3.13) θ ′ = − κ 1 .

From (3.10) and (3.11), we find that

(3.14) d s d s 1 = 1 sinh θ = − c τ 1 cosh θ .

Thus, we have

(3.15) coth θ = − c τ 1 .

By taking the derivative of this equation with respect to s 1 and applying (3.13) and (3.15), we obtain

τ 1 ′ = κ 1 c ( 1 − c 2 τ 1 2 ) .

Conversely, if the curvature κ 1 and the torsion τ 1 of Γ 1 satisfy (3.5) for some nonzero constant c , then we define a curve Γ by

(3.16) α ( s ) = α 1 ( s 1 ) + c κ 1 B 1 ( s 1 ) ,

and we will prove that Γ is the Mannheim curve and Γ 1 is the partner curve of Γ . By taking the derivative of (3.16) with respect to s 1 twice, we obtain

(3.17) T d s d s 1 = T 1 − c τ 1 κ 1 N 1

and

(3.18) N d s d s 1 2 + T d 2 s d s 1 2 = − c τ 1 κ 1 T 1 + 1 − c τ 1 ′ κ 1 N 1 − c τ 1 2 κ 1 B 1 .

Taking the cross product of (3.17) with (3.18) and noting that

κ 1 − c τ 1 ′ − c 2 κ 1 τ 1 2 = 0 ,

we obtain

(3.19) B d s d s 1 3 = c 2 τ 1 3 T 1 − c τ 1 2 κ 1 N 1 .

Again taking the cross product of (3.17) with (3.19), we obtain

N d s d s 1 4 = c τ 1 2 ( 1 − c 2 τ 1 2 ) B 1 κ 1 .

Since both N κ and B 1 κ 1 have unit length, we obtain

d s d s 1 4 = ± c τ 1 2 ( 1 − c 2 τ 1 2 ) κ .

Thus, we have

N = ± κ κ 1 B 1 .

This means that the principal normal direction N of Γ coincides with the binormal direction B 1 of Γ 1 . Hence, Γ is the Mannheim curve and Γ 1 is the partner curve of Γ . In (i) if { Γ , Γ 1 } is a Mannheim pair of type 2, we just replace (3.10) and (3.11) with

T d s d s 1 = T 1 − c τ 1 κ 1 N 1 and T = cosh θ T 1 + 1 κ 1 sinh θ N 1 ,

and the proof can be given by a similar way to (i). In (ii) if { Γ , Γ 1 } is a Mannheim pair of type 3, we just replace (3.10) and (3.11) with

T d s d s 1 = T 1 + c τ 1 κ 1 N 1 and T = cos θ T 1 + 1 κ 1 sin θ N 1 ,

and the proof can be given by a similar way to (i). In (iii) if { Γ , Γ 1 } is a Mannheim pair of types 4 or 5, we just replace (3.11) with

T d s d s 1 = T 1 + c τ 1 κ 1 N 1 and T = sinh θ T 1 + 1 κ 1 cosh θ N 1 ,

T d s d s 1 = T 1 + c τ 1 κ 1 N 1 and T = cosh θ T 1 + 1 κ 1 sinh θ N 1 ,

and the proof can be given by a similar way to (i). Therefore, the proof is completed.□

Remark 2

A simple parametric transformation reduces

condition
τ 1 ′ = κ 1 c ( 1 − c 2 τ 1 2 )
to
τ 1 = 1 c tanh ∫ κ 1 d s 1 + c 0 ;
condition
τ 1 ′ = − κ 1 c ( 1 + c 2 τ 1 2 )
to
τ 1 = 1 c tan − ∫ κ 1 d s 1 + c 0 ;
condition
τ 1 ′ = κ 1 c ( c 2 τ 1 2 − 1 )
to
τ 1 = 1 c tanh − ∫ κ 1 d s 1 + c 0 , if ∣ c τ ∣ < 1 ,
or
τ 1 = 1 c coth − ∫ κ 1 d s 1 + c 0 , if ∣ c τ ∣ > 1 .

Thus, the existence of a Mannheim partner curve to a Mannheim curve is unique.

Proposition 3

Let Γ : α ( s ) be a Mannheim curve in E 1 3 parameterized by its arc length s and let Γ 1 : α 1 ( s 1 ) be neither null a Mannheim nor pseudo null a Mannheim partner curve of Γ with an arc length parameter s 1 . If Γ : α ( s ) is a generalized helix according to the modified orthogonal frame in E 1 3 , then Γ 1 : α 1 ( s 1 ) is a straight line.

Proof

Let T , N , and B be the tangent, the principal normal, and the binormal vectors of α ( s ) , respectively. From the definition of the Mannheim curve and properties of generalized helices, we have

⟨ N , u ⟩ = ⟨ B 1 , u ⟩ = 0 ,

where u is some constant vector. By taking the derivative of the last equality with respect to s 1 , we obtain

⟨ B 1 ′ , u ⟩ = δ τ 1 N 1 + κ 1 ′ κ 1 B 1 , u = δ τ 1 ⟨ N 1 , u ⟩ + κ 1 ′ κ 1 ⟨ B 1 , u ⟩ = 0 ,

and δ = ± 1 , here, demonstrates the type of a curve α ( s ) . If δ = 1 , then α ( s ) is a spacelike curve. If δ = − 1 , then α ( s ) is a timelike curve. We have two cases.

If ⟨ N 1 , u ⟩ ≠ 0 . Therefore, from the last equality, we obtain
τ 1 = 0 .
Using the equalities in Theorem 2, we obtain
κ 1 = 0 .
.
If u = ± T 1 , then ⟨ N 1 , u ⟩ = 0 . Thus, ⟨ T , u ⟩ = ⟨ T , T 1 ⟩ = sinh θ = constant , θ is constant. So, we have κ 1 = θ ′ = 0 .

The two cases imply that κ 1 = 0 . Hence, Γ 1 : α 1 ( s 1 ) is a straight line.□

4 Mannheim curves with modified orthogonal frame in E 1 3

In this section, we give the characterizations of Mannheim curves using the modified orthogonal frame in the Minkowski 3-space E 1 3 .

Theorem 3

Let Γ : α ( s ) be a space curve in E 1 3 with respect to the modified orthogonal frame. Then there exist three cases:

If α ( s ) is a spacelike curve with spacelike principal normal N and timelike binormal B, then the necessary and sufficient condition for Mannheim curves is
(4.1) κ = c ( κ 2 − τ 2 ) .
If α ( s ) is a spacelike curve with timelike principal normal N and spacelike binormal B, then the necessary and sufficient condition for Mannheim curves is
(4.2) κ = − c ( κ 2 + τ 2 ) .
If α ( s ) is a timelike curve with spacelike principal normal N and spacelike binormal B, then the necessary and sufficient condition for Mannheim curves is
(4.3) κ = c ( τ 2 − κ 2 ) ,
where c is a nonzero constant.

Proof

(i) Let Γ : α ( s ) be a Mannheim curve in E 1 3 parameterized by its arc length s and let Γ 1 : α 1 ( s 1 ) be the Mannheim partner curve of Γ with an arc length parameter s 1 . Let us assume that the pair { Γ , Γ 1 } Mannheim pair. From Definition 3, we can write

(4.4) α 1 ( s 1 ) = α ( s ) + μ ( s ) N ( s ) ,

for some function μ ( s ) . By taking the derivative of (4.4) with respect to s and using (2.4), we obtain

(4.5) T 1 d s 1 d s = ( 1 − μ κ 2 ) T + μ ′ + μ κ ′ κ N + μ τ B .

By taking the inner product of (4.5) with B 1 , we obtain

(4.6) μ ′ μ = − κ ′ κ .

Integrating (4.6), we obtain

(4.7) μ ( s ) = c κ ( s ) , c > 0 .

Inserting (4.7) into (4.4), we obtain

(4.8) α 1 ( s 1 ) = α ( s ) + c κ ( s ) N ( s ) .

Inserting (4.6) and (4.7) into (4.5), we have

(4.9) T 1 d s 1 d s = ( 1 − c κ ) T + c τ κ B .

Differentiating (4.9) with respect to s and applying the modified orthogonal frame formulas in (2.4), we obtain

(4.10) N 1 d s 1 d s 2 + T 1 d 2 s 1 d s 2 = − c κ ′ T + 1 κ ( κ − c κ 2 + c τ 2 ) N + c τ ′ κ B .

Taking the inner product of (4.10) with B 1 , we obtain

(4.11) κ − c κ 2 + c τ 2 = 0 .

Thus, we deduce that

(4.12) κ = c ( κ 2 − τ 2 ) .

Conversely, if the curvature κ and the torsion τ of curve α ( s ) satisfy (4.1) for some nonzero constant c , then we define a curve Γ : α ( s ) by (4.8), and we will prove that Γ is the Mannheim curve and Γ 1 is the partner curve of Γ . We have already found the equality as follows:

T 1 d s 1 d s = ( 1 − c κ ) T + c τ κ B .

Differentiating the last equality with respect to s and applying the modified orthogonal frame formulas in (2.4) and (2.5), and condition (4.11), we obtain

(4.13) N 1 d s 1 d s 2 + T 1 d 2 s 1 d s 2 = − c κ ′ T + c τ ′ κ B .

Taking the cross product of (4.9) with (4.13), we have

d s 1 d s 3 B 1 = c ( c τ ′ κ − τ ′ − c τ κ ′ ) N κ .

Since both N κ and B 1 κ 1 have unit length, we obtain

d s 1 d s 3 = ± c ( c τ ′ κ − τ ′ − c τ κ ′ ) κ 1 .

Thus, we have

B 1 = ± κ 1 κ N .

Hence, N and B 1 are linearly dependent. The proof of (ii) and (iii) can be given in the same way.□

Theorem 4

If a generalized helix is the Mannheim curve Γ : α ( s ) according to the modified orthogonal frame in E 1 3 , then there re three cases:

If α ( s ) is a spacelike curve with spacelike principal normal N and timelike binormal B, then the curvature and torsion of α ( s ) are obtained as follows:
κ = 1 c ( 1 − λ 2 ) and τ = λ c ( 1 − λ 2 ) for λ ≠ ± 1 .
If α ( s ) is a spacelike curve with timelike principal normal N and spacelike binormal B, then the curvature and torsion of α ( s ) are obtained as follows:
κ = − 1 c ( 1 + λ 2 ) and τ = − λ c ( 1 + λ 2 ) ,
If α ( s ) is a timelike curve with spacelike principal normal N and spacelike binormal B, then the curvature and torsion of α ( s ) are obtained as follows:
κ = 1 c ( λ 2 − 1 ) and τ = λ c ( λ 2 − 1 ) , for λ ≠ ± 1 ,
where c is a nonzero constant and λ is a real constant.

Proof

(i) Let Γ : α ( s ) be a generalized helix. From the definition of helix, the curvature and torsion of α ( s ) satisfy the following equation:

τ κ = λ ,

where λ ∈ R . Applying Theorem 3(i), we obtain

κ = 1 c ( 1 − λ 2 ) and τ = λ c ( 1 − λ 2 ) .

Similarly, applying Theorem 3(ii) and (iii), we obtain

κ = − 1 c ( 1 + λ 2 ) and τ = − λ c ( 1 + λ 2 ) ,

and

κ = 1 c ( λ 2 − 1 ) and τ = λ c ( λ 2 − 1 ) ,

respectively. Therefore, the proof is completed.□

Theorem 5

Suppose a pair of curves { Γ , Γ 1 } in E 1 3 with respect to the modified orthogonal frame.

If { Γ , Γ 1 } is a Mannheim pair of types 1 or 2, then ⟨ T , T 1 ⟩ = sinh θ or ⟨ T , T 1 ⟩ = − cosh θ , where θ is the angle between T and T 1 and
( i ) cosh θ = − c τ 1 sinh θ , ( i i ) c τ sinh θ = ( c κ − 1 ) cosh θ , ( i i i ) sinh 2 θ = ( c κ − 1 ) , ( i v ) cosh 2 θ = − c 2 τ τ 1 ,
or
( i ) sinh θ = − c τ 1 cosh θ , ( i i ) c τ cosh θ = ( c κ + 1 ) sinh θ , ( i i i ) cosh 2 θ = ( c κ + 1 ) , ( i v ) sinh 2 θ = − c 2 τ τ 1 .
If { Γ , Γ 1 } is a Mannheim pair of type 3, then ⟨ T , T 1 ⟩ = cos θ , where θ is the angle between T and T 1 and
( i ) sin θ = c τ 1 cos θ , ( i i ) c τ cos θ = ( 1 + ε c κ ) sin θ , ( i i i ) cos 2 θ = ( 1 + ε c κ ) , ( i v ) sin 2 θ = c 2 τ τ 1 .
If { Γ , Γ 1 } is a Mannheim pair of types 4 or 5, then ⟨ T , T 1 ⟩ = sinh θ or ⟨ T , T 1 ⟩ = cosh θ , where θ is the angle between T and T 1 and
( i ) cosh θ = c τ 1 sinh θ , ( i i ) c τ sinh θ = ( ε c κ − 1 ) cosh θ , ( i i i ) sinh 2 θ = ( ε c κ − 1 ) , ( i v ) cosh 2 θ = c 2 τ τ 1 ,
or
( i ) sinh θ = c τ 1 cosh θ , ( i i ) c τ cosh θ = ( 1 + ε c κ ) sinh θ , ( i i i ) cosh 2 θ = ( 1 + ε c κ ) , ( i v ) sinh 2 θ = c 2 τ τ 1 ,
where c is a nonzero constant and ε = ± 1 .

Proof

Consider the pair { Γ , Γ 1 } is a Mannheim pair of type 1. We can prove (i) in the same way of Theorem 2(i). Now, we can write

(4.14) α 1 ( s 1 ) = α ( s ) + λ ( s ) N ( s ) ,

for some function λ ( s ) . By taking the derivative of (4.14) with respect to s and using (2.4), we obtain

(4.15) T 1 d s 1 d s = ( 1 − λ κ 2 ) T + λ ′ + λ κ ′ κ N + λ τ B .

By taking the inner product of (4.15) with B 1 , we obtain

(4.16) λ ′ λ = − κ ′ κ .

Integrating (4.16), we obtain

(4.17) λ ( s ) = c κ ( s ) , c > 0 .

It follows that

(4.18) T 1 d s 1 d s = ( 1 − c κ ) T + c τ κ B .

On the other hand, by taking the cross product of (3.11) with N κ = B 1 κ 1 , we obtain

(4.19) B = κ cosh θ T 1 + κ κ 1 sinh θ N 1 .

From (3.11) and (4.19), we obtain

(4.20) T 1 = − sinh θ T + cosh θ κ B .

From (4.18) and (4.20), we obtain

d s 1 d s = ( c κ − 1 ) − sinh θ = c τ cosh θ ,

which implies that

c τ sinh θ = − ( c κ − 1 ) cosh θ ,

which is (ii). Also, we obtain

(4.21) d s d s 1 ( 1 − c κ ) = − sinh θ

and

(4.22) d s d s 1 ( c τ ) = cosh θ .

Thus, inserting (3.14) into (4.21) and (4.22), and using (i), we obtain

sinh 2 θ = ( c κ − 1 ) ,

and

cosh 2 θ = − c 2 τ τ 1 ,

which are (iii) and (iv), respectively. Similarly, if the pair { Γ , Γ 1 } is a Mannheim pair of types 2, 3, 4, or 5, the proof can be given by a similar way to (i). Therefore, this completes the proof.□

5 Conclusion

In this article, by using the modified orthogonal frame in Minkowski 3-space, first, we defined Mannheim curves. Second, we gave some characterizations of Mannheim curves and their partner curves and established necessary and sufficient conditions for the Mannheim curves and their partner curves. Third, we gave some characterizations for general helices which have Mannheim curves and Mannheim partner curves. Finally, we derived relationships between the curvatures and the torsions of the Mannheim pairs.

Acknowledgment

The authors gratefully thank the editor and the anonymous referees for their helpful and useful suggestions.

Funding information: Not applicable.
Author contributions: The authors equally contributed to this study.
Conflict of interest: The authors declare that they have no competing interests.

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Received: 2021-04-29

Accepted: 2022-07-26

Published Online: 2022-11-16

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2022-0163

Keywords for this article

Mannheim curves; Mannheim partner curves; modified orthogonal frame; Minkowski 3-space; timelike curve; spacelike curve; curvature; torsion

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