On Mannheim partner curves in dual Lorentzian space

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Hacettepe Journal of Mathematics and Statistics Volume 40 (5) (2011), 649 – 661

ON MANNHEIM PARTNER CURVES IN DUAL LORENTZIAN SPACE

Sıddıka ¨Ozkaldı^∗^†‡, Kazım ˙Ilarslan^§and Yusuf Yaylı^∗

Received 28 : 12 : 2009 : Accepted 09 : 03 : 2011

Abstract

In this paper we define non-null Mannheim partner curves in three dimensional dual Lorentzian space D³1, and obtain necessary and sufficient conditions for the existence of non-null Mannheim partner curves in dual Lorentzian space D³1.

Keywords: Mannheim partner curve, Dual Lorentzian space, Dual Lorentzian space curve.

2000 AMS Classification: 53 A 25, 53 A 40, 53 B 30.

1. Introduction

In the differential geometry of a regular curve in Euclidean 3-space E³, it is well known that one of the important problems is the characterization of a regular curve.

The curvature functions k¹ (curvature κ) and k² (torsion τ ) of a regular curve play an important role in determining the shape and size of the curve [4, 8]. For example: If k¹= k²= 0, then the curve is a geodesic. If k¹6= 0 (constant) k² = 0, then the curve is a circle with radius 1/k¹. If k¹6= 0 (constant) and k²6= 0 (constant), then the curve is a helix in the space, etc.

Another route to the classification and characterization of curves is the relationship between the Frenet vectors of the curves. For example Saint Venant, in 1845, proposed the question whether upon the surface generated by the principal normal of a curve, a second curve can exist which has for its principal normal the principal normal of the given curve. This question was answered by Bertrand in 1850, when he showed that a necessary and sufficient condition for the existence of such a second curve is that a linear

∗Department of Mathematics, Faculty of Science, Ankara University, Tando˘gan, Ankara, Turkey. E-mail:

(S. ¨Ozkaldı) sozkaldi@science.ankara.edu.tr (Y. Yaylı) yayli@science.ankara.edu.tr

†Current Address: Department of Mathematics, Faculty of Science and Arts, Bilecik Univer- sity, Bilecik, Turkey. E-mail: sozkaldi98@yahoo.com

‡Corresponding Author.

§Department of Mathematics, Faculty of Science and Arts, Kırıkkale University, Kırıkkale, Turkey. E-mail: kilarslan@yahoo.com

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relationship with constant coefficients shall exist between the first and second curvatures of the given original curve. The pairs of curves of this kind have been called Conjugate Bertrand Curves, or more commonly Bertrand Curves [4, 8, 13]. There are many works related with Bertrand curves in Euclidean space and Minkowski space. Associated curves of another kind, called Mannheim curves and Mannheim partner curves occur if there exists a relationship between the space curves α and β such that, at the corresponding points of the curves, the principal normal lines of α coincide with the binormal lines of β.

Then, α is called a Mannheim curve, and β the Mannheim partner curve of α. Mannheim partner curves were studied by Liu and Wang [9] in Euclidean 3-space and in Minkowski 3-space.

Dual numbers had been introduced by William Kingdon Clifford (1845–1879) as a tool for his geometrical investigations. After him Eduard Study (1862–1930) used dual numbers and dual vectors in his research on line geometry and kinematics. He devoted special attention to the representation of oriented lines by dual unit vectors and defined the famous mapping: The set of oriented lines in a Euclidean three-dimensional space E³ is one to one correspondence with the points of a dual space D³ of triples of dual numbers [5].

Differential Geometric properties of regular curves in a dual space D³, as well as in a dual Lorentzian space D³1 have been studied in many papers. For example see [16, 3, 7, 17, 18, 1, 2, 11, 15, 14]. Mannheim curves in a dual space were studied by the authors in [12]

In this paper we study non-null Mannheim partner curves in a dual Lorentzian space D³₁.

2. Preliminary

Dual numbers were introduced by W. K. Clifford (1845–1879) as a tool for his geometrical investigations. After him, E. Study used dual numbers and dual vectors in his research on the geometry of lines and kinematics. He devoted special attention to the representation of directed lines by dual unit vectors, and defined the mapping that is known by his name. Namely, there exists one-to-one correspondence between the points of dual unit sphere S² and the directed lines in R³ [5].

If we take the Minkowski 3-space R³1 instead of R³ the E. Study (1862–1930) mapping can be stated as follows: The dual timelike and spacelike unit vectors of the dual hyperbolic and Lorentzian unit spheres H²0and S²1 in the dual Lorentzian space D³1are in one-to-one correspondence with the directed timelike and spacelike lines in R³1, respectively. Then a differentiable curve on H²0 corresponds to a timelike ruled surface at R³1. Similarly, the timelike (resp. spacelike) curve on S²1 corresponds to any spacelike (resp.

timelike) ruled surface in R³1.

We will survey briefly various fundamental concepts and properties in Lorentzian space. We refer mainly to O’Neill [10].

Let R³1 be the 3-dimensional Lorentzian space with Lorentzian metric h · , · i : −dx²1+ dx²2+dx²3. It is known that in R³1there are three categories of curves and vectors, namely, spacelike, timelike and null, depending on their causal character. Let −→x be a tangent vector of Lorentzian space. Then −→x is said to be spacelike if h−→x , −→x i > 0 or −→x =−→

0 , timelike if h−→x , −→x i < 0, null (lightlike) if h−→x , −→x i = 0 and −→x 6=−→

0 .

Let α : I ⊂ R −→ R³1 be a regular curve in R³1. Then, the curve α is spacelike if all its velocity vectors are spacelike. Similarly, it is called timelike and a null curve if all its velocity vectors are timelike and null vectors, respectively.

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A dual number bx has the form x + εx^∗with the properties ε 6= 0, 0ε = ε0 = 0, 1ε = ε1 = ε, ε²= 0,

where x and x^∗ are real numbers and ε is the dual unit (for the properties of dual numbers, see [16]). An ordered triple of dual numbers (bx1, bx2, bx3) is called a dual vector and the set of dual vectors is denoted by

D³= D × D × D

=−→ x |b −→

bx = (x¹+ εx^∗1, x²+ εx^∗2, x³+ εx^∗3)

= {−→ x |b −→

bx = (x¹, x², x³) + ε (x^∗1, x^∗2, x^∗3)}

=−→ x |b −→

bx = −→x + ε−→ x^∗, −→x , −→

x^∗∈ R³ For any−→

x = −b →x + ε−→ x^∗,−→

y = −b →y + ε−→

y^∗∈ D³, if the Lorentzian inner product of the dual vectors−→

bx and−→

y is defined byb −→

bx ,−→ yb

:= −→x , −→y

+ ε −→x ,−→ y^∗

+ −→ x^∗, −→y

,

then the dual space D³ together with this Lorentzian inner product is called the dual Lorentzian space, and it is shown by D³1. A dual vector−→

bx in D³ is said to be spacelike, timelike and lightlike (null) if the vector −→

bx is spacelike, timelike and lightlike (null), respectively. The Lorentzian vector product of dual vectors−→

bx = (bx1, bx2, bx3) and−→ x =b (bx¹, bx², bx³) in D³1is defined by

−

→x ∧b −→

y := (bb x³yb²− bx²yb³, bx³yb¹− bx¹yb³, bx¹yb²− bx²by¹) . If −→x 6= 0, the norm −→

xb of−→

bx = −→x + ε−→

x^∗is defined by

−→ bx

:=

r −→

x ,b −→ xb

.

A dual vector −→x with norm 1 is called a dual unit vector. Let−→

bx = (bx1, bx2, bx3) ∈ D³1. Then,

i) The set S²1=n−→

bx = −→x + ε−→ x^∗

−→

xb = (1, 0); −→x ,−→

x^∗∈ R³1 and the vector −→x is spacelikeo is called the pseudo dual sphere with center−→

O in D³1. ii) The set

H²0=n−→

bx = −→x + ε−→ x^∗

−→

xb = (1, 0); −→x ,−→

x^∗∈ R³1and the vector −→x is timelikeo is called the pseudo dual hyperbolic space in D³1 [15]

If all the real valued functions x¹(t) and x^∗1, 1 ≤ i ≤ 3, are differentiable, the dual Lorentzian curve

bx : I ⊂ R → D³1

t 7→−→

x (t) = (xb 1(t) + εx^∗1(t), x2(t) + εx^∗2(t), x3(t) + εx^∗3(t))

= −→x (t) + ε−→ x^∗(t)

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in D³1 is differentiable. We call the real part −→x (t) the indicatrix of−→

x (t). The dual arcb lengthof the curve−→

bx (t) from t1to t is defined as

bs :=

Zt

t1

−→

bx (t)^′ dt =

Zt

t1

−→x (t)^′ dt + ε

Zt

t1

D−→ t ,−→

x^∗′E

= s + εs^∗,

where−→

t is a unit tangent vector of −→x (t). From now on we will take the arc length s of

−

→x (t) as the parameter instead of t.

The equalities relative to the derivatives of the dual Frenet vectors−→ b t ,−→

bn ,−→

bb through- out the dual space curve are written in the matrix form as

(2.1) d dbs







−

→bt

−

→nb

−

→bb





=





0 bκ 0

−ε1ε2bκ 0 bτ 0 −ε²ε³τb 0











−

→bt

−

→bn

−

→bb





,

where bκ = κ + εκ^∗is the nowhere pure dual curvature, bτ = τ + ετ^∗is the nowhere pure dual torsion and

D−→ b t ,−→

b tE

= ε1, D−→ bn ,−→

bnE

= ε2, D−→ bb ,−→

bbE

= ε3, D−→ b t ,−→

bnE

=D−→ b t ,−→

bbE

=D−→ bn ,−→

bbE

= 0.

The formulae (2.1) are called the Frenet formulae of the dual curve in dual Lorentzian space. The planes spanned byn−→

b t ,−→

bbo ,n−→

b t ,−→

bno

andn−→ n ,b −→

bbo

at each point of the dual Lorentzian curve are called the rectifying plane, the osculating plane, and the normal plane, respectively [1, 2, 11, 14].

3. Mannheim partner curves in D

³₁

In this section, we define Mannheim partner curves in the dual space D³1 and we give characterizations for Mannheim partner curves in the same space.

3.1. Definition. Let D³1 be dual Lorentzian space with the Lorentzian inner product h·, ·i. If there exists a correspondence between the dual Lorentzian space curves bα and β such that, at the corresponding points of the dual Lorentzian curves, the principalb normal lines of bα coincides with the binormal lines of bβ, then bα is called a dual Lorentzian Mannheim curve, and bβ a dual Lorentzian Mannheim partner curve of bα. The pair {bα, bβ}

is said to be a dual Lorentzian Mannheim pair.

Let bα : bx(bs) be a dual Lorentzian Mannheim curve in D³1 parameterized by its arc length bs and bβ : bx1(bs1) the dual Lorentzian Mannheim partner curve with an arc length parameter bs¹. Denote by −→

bt (bs),−→ bn (bs),−→

bb (bs)

the dual Lorentzian Frenet frame field along bα : bx(bs).

In the following theorems, we give necessary and sufficient conditions for a dual space curve to be a dual Lorentzian Mannheim curve.

3.2. Theorem. Let bα : bx(bs) be a Lorentzian curve in D³1 and−→ bt (bs),−→

bn (bs) and−→ bb (bs) the tangent, principal normal and binormal vector fields of bα : bx(bs), respectively.

i) In the case when−→

bt (bs) is a timelike vector,−→ bn (bs) and

−

→bb (bs) are spacelike vectors, then bα : bx(bs) is a dual Lorentzian Mannheim curve if and only if its curvature bκ and torsion bτ satisfy the formula bκ = bλ(bτ²− bκ²), where bλ is a never pure dual constant.

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ii) In the case when −→

bt (bs) and −→

n (bb s) are spacelike vectors, −→

bb (bs) a timelike vector, then bα : bx(bs) is a dual Lorentzian Mannheim curve if and only if its curvature bκ and torsion bτ satisfy the formula bκ = bλ(bκ²− bτ²), where bλ is a never pure dual constant.

iii) In the case when−→

bt (bs) and −→

bb (bs) are spacelike vectors, −→

n (bb s) a timelike vector, then bα : bx(bs) is a dual Lorentzian Mannheim curve if and only if its curvature b

κ and torsion bτ satisfy the formula bκ = −bλ(bκ²+ bτ²), where bλ is a never pure dual constant.

Proof. i) Let bα : bx(bs) be a dual Lorentzian Mannheim curve in D³1 with arc length parameter bs, and bβ : bx¹(bs¹) the dual Lorentzian Mannheim partner curve with arc length parameter bs1. Then by the definition we can assume that

(3.1) bx¹(bs) = bx(bs) + bλ(bs)−→ b n (bs)

for some never pure dual constant bλ(bs). By taking the derivative of (3.1) with respect to bs and applying the Frenet formulas we have

dbx¹(bs) dbs =

1 + bλbκ −→ b t + dbλ

dbs

−

→n + bb λbτ−→ bb.

Since−→ b

t¹ is coincident with

−

→bb¹ in direction, we get

dbλ(bs) dbs = 0.

This means that bλ is a never pure dual constant. Thus we have dbx¹(bs)

dbs =

1 + bλbκ −→ b t + bλbτ−→

bb . On the other hand, we have

−

→bt¹ =dbx1

dbs dbs dbs1

=

bb dbs dbs1

.

By taking the derivative of this equation with respect to bs¹ and applying the Frenet formulas we obtain

d−→ b t1

dbs dbs dbs¹ =

bλdbκ

dbs

−

→bt +

bκ + bλbκ²− bλbτ² −→ n + bb λdbτ

dbs

−

→bb dbs dbs¹ +

1 + bλbκ −→ b t + bλbτ

−

→bb dbs dbs¹

² . From this equation we get

bκ + bλbκ²− bλbτ² dbs dbs1

= 0, bκ = bλ(bτ²− bκ²).

This completes the proof.

ii) Let bα : bx(bs) be a dual Lorentzian Mannheim curve in D³1with arc length parameter bs and bβ : bx¹(bs¹) the dual Lorentzian Mannheim partner curve with arc length parameter bs1. Then by the definition we can assume that

(3.2) bx¹(bs) = bx(bs) + bλ(bs)−→ bn (bs)

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dbx¹(bs) dbs =

1 − bλbκ −→ b t + dbλ

dbs

−

→n + bb λbτ−→ bb.

Since−→ b

t1 is coincident with−→

bb1 in direction, we get dbλ(bs)

dbs = 0.

This means that bλ is a never pure dual constant. Thus we have dbx1(bs)

dbs =

1 − bλbκ −→ b t + bλbτ−→

−

→bt¹ =dbx¹ dbs

dbs dbs¹ =

1 − bλbκ −→ b t + bλbτ

−

→bb dbs dbs¹.

By taking the derivative of this equation with respect to bs¹ and applying the Frenet formulas we obtain

d−→ b t¹ dbs

dbs dbs1

=

−bλdbκ dbs

−

→bt +

bκ − bλbκ²+ bλbτ² −→ n + bb λdbτ

dbs

−

→bb

dbs dbs1

+

1 − bλbκ −→ b t + bλbτ−→

bb

dbs dbs1

bκ − bλbκ²+ bλbτ² dbs dbs1

= 0, bκ = bλ(bκ²− bτ²).

iii) Let bα : bx(bs) be a dual Lorentzian Mannheim curve in D³1with arc length parameter bs, and bβ : bx¹(bs¹) the dual Lorentzian Mannheim partner curve with arc length parameter bs1. Then by the definition we can assume that

(3.3) bx1(bs) = bx(bs) + bλ(bs)−→ bn (bs)

dbx1(bs) dbs =

1 + bλbκ −→ b t + dbλ

dbs

−

→n + bb λbτ

−

→bb .

Since−→ b

t¹ is coincident with

−

→bb¹ in direction, we get dbλ(bs)

dbs = 0.

This means that bλ is a never pure dual constant. Thus we have dbx¹(bs)

dbs =

−

→bt¹ =dbx1

dbs dbs dbs¹ =

bb dbs dbs¹.

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By taking the derivative of this equation with respect to bs¹ and applying the Frenet formulas, we obtain

d−→ b t¹ dbs

dbs dbs¹ =

b λdbκ

dbs

−

→bt +

bκ + bλbκ²+ bλbτ² −→ n + bb λdbτ

dbs

−

→bb dbs dbs¹ +

bb

dbs dbs¹

bκ + bλbκ²+ bλbτ² dbs dbs1

= 0, bκ = −bλ(bτ²+ bκ²).

3.3. Theorem. Let bα : bx(bs) be a dual Lorentzian Mannheim curve in D³1with arc length parameter bs.

i) In the case where−→

bt (bs) is a timelike vector,−→

bn (bs) and−→

bb (bs) are spacelike vectors, then bβ : bx1(bs1) is the dual Mannheim partner curve of bα if and only if the curvature bκ1 and the torsion bτ1 of bβ satisfy the following equation

dbτ1

dbs1

= bκ1

bµ 1 − bµ²τb1²

for some never pure dual constant bµ.

ii) In the case where −→

bt (bs) and−→

n (bb s) are spacelike vectors,−→

bb (bs) a timelike vector, then bβ : bx¹(bs¹) is the dual Mannheim partner curve of bα if and only if the curvature bκ¹ and the torsion bτ¹ of bβ satisfy the following equation

dbτ¹ dbs¹ = − bκ¹

b

µ 1 + bµ²bτ1²

iii) In the case where−→

bt (bs) and−→

bb (bs) are spacelike vectors,−→

bn (bs) a timelike vector, then bβ : bx1(bs1) is the dual Mannheim partner curve of bα if and only if the curvature bκ1 and the torsion bτ1 of bβ satisfy the following equation

dbτ1

dbs¹ = −κb1

µb 1 − bµ²bτ1²

Proof. i) Suppose that bα : bx(bs) is a dual Lorentzian Mannheim curve. Then by the definition we can assume that

(3.4) bx(bs¹) = bx¹(bs¹) + bµ(bs¹)−→ bb¹(bs¹)

for some function bµ(bs¹). By taking the derivative of (3.4) with respect to bs¹and applying the Frenet formulas, we have

(3.5) −→ b t dbs

dbs¹ =−→ b t¹+ dbµ

dbs¹

−

→bb − bµbτ¹−→ bn¹. Since−→

bb is coincident with−→

n in direction, we getb dbµ

dbs1

= 0.

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This means that bµ is a never pure dual constant. Thus we have (3.6) −→

b t dbs

dbs¹ =−→ b

t¹− bµbτ¹−→ b n¹. On the other hand, we have (3.7) −→

b t =−→

bt1cosh bθ +−→ bn1sinh bθ,

where bθ is the dual hyberbolic angle between−→ bt and−→

b

t¹ at the corresponding points of α and bb β. By taking the derivative of this equation with respect to bs¹, we obtain

bκ−→ bn dbs

dbs¹ = bκ¹+ dbθ dbs¹

! sinh bθ−→

b

t¹+ bκ¹+ dbθ dbs¹

!

cosh bθ−→

bn¹+ bτ¹sinh bθ

−

→bb¹.

From this equation and the fact that the direction of −→

n is coincident with that ofb −→ bb¹, we get

bκ1+ dbθ dbs¹

!

cosh bθ = 0.

Therefore we have (3.8) dbθ

dbs¹ = −bκ¹.

From (3.6) and (3.7) and noting that−→

bt1 is orthogonal to−→

bb¹, we find that dbs

dbs¹ = 1

cosh bθ = − µbbτ¹ sinh bθ. Then we have

µbbτ¹= − tanh bθ.

By taking the derivative of this equation and applying (3.8), we get µbdbτ1

dbs¹ = bκ¹ 1 − bµ²bτ1²

, that is

dbτ¹ dbs1

= bκ¹

bµ 1 − bµ²τb1²

.

Conversely, if the curvature bκ¹ and torsion bτ¹ of the dual Lorentzian curve bβ satisfy dbτ¹

dbs¹ = bκ¹ b

µ 1 − bµ²τb1²

for some never pure dual constant bµ(bs), then we define a dual curve by (3.9) bx(bs1) = bx1(bs1) + bµ−→

bb¹(bs1) ,

and we will prove that bα is a dual Lorentzian Mannheim curve and that bβ is the dual Lorentzian partner curve of bα.

By taking the derivative of (3.9) with respect to bs¹ twice, we get

−

→bt dbs dbs1

=−→ b t1− bµbτ1

−

→bn1, (3.10)

bκ−→ bn

dbs dbs¹

² +−→

b t d²bs

dbs²1

= −bµbκ¹bτ¹−→ bt¹+

bκ¹− bµdbτ1

dbs¹

−→ nb¹− bµbτ1²

−

→bb¹, (3.11)

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respectively. Taking the cross product of (3.10) with (3.11) and noticing that bκ1− bµdbτ¹

dbs1 − bµ²bκ1τb1²= 0, we have

(3.12) bκ−→ bb

dbs dbs¹

³

= −bµ²bτ1³

−

→bt¹+ bµbτ1²

−

→bn¹

By taking the cross product of (3.12) with (3.10), we obtain also bκ−→

bn

dbs dbs¹

⁴

= −bµbτ1² 1 − bµ²τb1²

−→ bb¹.

This means that the principal normal direction−→

n of bb α : bx(bs) coincides with the binormal direction −→

bb¹ of bβ : bx¹(bs¹). Hence bα : bx(bs) is a dual Lorentzian Mannheim curve and β : bb x1(bs1) is its dual Lorentzian Mannheim partner curve.

ii) Suppose that bα : bx(bs) is a dual Lorentzian Mannheim curve. Then by the definition we can assume that

for some function bµ(bs1). By taking the derivative of (3.13) with respect to bs1and applying the Frenet formulas, we have

(3.14) −→ b t dbs

dbs1

=−→ b t1+ dbµ

dbs1

−

→bb + bµbτ¹−→ bn1. Since−→

dbs1

= 0.

b t dbs

dbs¹ =−→ b

t¹+ bµbτ¹−→ bn¹. On the other hand, we have (3.16) −→

b t =−→

bt¹cos bθ +−→ bn¹sin bθ, where bθ is the dual angle between −→

b t and−→

b

t¹ at the corresponding points of bα and bβ.

By taking the derivative of this equation with respect to bs1, we obtain bκ−→

bn dbs dbs1

= − bκ1+ dbθ dbs1

! sin bθ−→

b

t1+ κb1+ dbθ dbs1

! cos bθ−→

bn1+ bτ1sin bθ−→ bb1.

n is coincident with that ofb

−

→bb¹, we get





κb¹+_db^dbs^θ

1

sin bθ = 0

κb¹+_db^dbs^θ

1

cos bθ = 0.

dbs1 = −bκ1.

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From (3.15), (3.16) and noticing that−→ b

t1is orthogonal to−→

bb¹, we find that dbs

dbs1

= 1

cos bθ = µbbτ¹ sin bθ. Then we have

µbbτ1= tan bθ.

By taking the derivative of this equation and applying (3.17), we get µbdbτ1

dbs1 = −bκ¹ 1 + bµ²bτ1²

, that is

dbτ1

dbs1

= − bκ1

µb 1 + bµ²bτ1²

.

Conversely, if the curvature bκ1 and torsion bτ1 of the dual Lorentzian curve bβ satisfy dbτ1

dbs¹ = − bκ1

µb 1 + bµ²bτ1²

for some never pure dual constant bµ(bs), then we define a dual curve by (3.18) bx(bs1) = bx1(bs1) + bµ−→

bb1(bs1) ,

−

→bt dbs dbs1

=−→ b t1+ bµbτ1

−

→bn1, (3.19)

bκ−→ bn

dbs dbs¹

² +−→

b t d²bs

dbs²1

= −bµbκ¹bτ¹−→ bt¹+

bκ¹+ bµdbτ1

dbs¹

−→ nb¹+ bµbτ1²

−

→bb¹, (3.20)

respectively. Taking the cross product of (3.19) with (3.20) and noticing that

−bκ1− bµdbτ¹

dbs1 − bµ²bκ1τb1²= 0, we have

(3.21) bκ

−

→bb dbs dbs¹

³

= −bµ²bτ1³

−

→bt¹+ bµbτ1²

−

→bn¹.

bn

dbs dbs1

⁴

= bµbτ1² 1 + bµ²τb1²

−→ bb¹. This means that the principal normal direction−→

n of bb α : bx(bs) coincides with the binormal direction −→

bb¹ of bβ : bx1(bs1). Hence, bα : bx(bs) is a dual Lorentzian Mannheim curve and β : bb x1(bs1) is its dual Lorentzian Mannheim partner curve.

iii) Suppose that bα : bx(bs) is a dual Lorentzian Mannheim curve. Then by the definition we can assume that

for some function bµ(bs1). By taking the derivative of (3.22) with respect to bs1and applying the Frenet formulas, we have

(3.23) −→ b t dbs

dbs1

=−→ b t1+ dbµ

dbs1

−

→bb + bµbτ¹−→ bn1.

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Since−→

dbs¹ = 0.

b t dbs

dbs1

=−→ b t1+ bµbτ1

−

→bn1.

On the other hand, we have (3.25) −→

b t =−→

bt1cosh bθ +−→ bn1sinh bθ,

where bθ is the dual hyberbolic angle between−→ bt and−→

b

t1 at the corresponding points of b

α and bβ. By taking the derivative of this equation with respect to bs¹, we obtain bκ−→

bn dbs dbs1

= bκ¹+ dbθ dbs1

! sinh bθ−→

b

t¹+ bκ¹+ dbθ dbs1

!

cosh bθ−→

bn¹+ bτ¹sinh bθ−→ bb¹.

n is coincident with that ofb −→ bb¹, we get

bκ¹+ dbθ dbs1

!

cosh bθ = 0.

dbs¹ = −bκ¹.

From (3.24), (3.25) and noticing that−→ b

t¹is orthogonal to

−

→bb¹, we find that dbs

dbs¹ = 1

cosh bθ = bµbτ¹ sinh bθ. Then we have

µbbτ1= tanh bθ.

By taking the derivative of this equation and applying (3.26), we get µbdbτ¹

dbs1 = −bκ1 1 − bµ²bτ1²

,

that is dbτ1

dbs1

= − bκ1

.

Conversely, if the curvature bκ1 and torsion bτ1 of the dual curve bβ satisfy dbτ¹

dbs¹ = − bκ¹

for some never pure dual constant bµ(bs), then we define a dual Lorentzian curve by (3.27) bx(bs1) = bx1(bs1) + bµ−→

bb¹(bs1) ,

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−

→bt dbs dbs¹ =−→

b

t¹+ bµbτ¹−→ bn¹, (3.28)

bκ−→ bn

dbs dbs¹

² +−→

b t d²bs

dbs²1

= bµbκ¹bτ¹−→ b t¹+

bκ¹+ bµdbτ¹ dbs¹

−→ bn¹+ bµbτ1²

−

→bb¹, (3.29)

respectively. Taking the cross product of (3.28) with (3.29) and noticing that bκ¹+ bµdbτ¹

dbs¹ − bµ²bκ¹τb1²= 0, we have

(3.30) bκ

−

→bb dbs dbs¹

³

= bµ²bτ1³

−

→bt¹+ bµbτ1²

−

→bn¹

bn

dbs dbs¹

⁴

= −bµbτ1² 1 − bµ²τb1²

−→ bb¹. This means that the principal normal direction−→

n of bb α : bx(bs) coincides with the binormal direction

−

→bb¹ of bβ : bx¹(bs¹). Hence, bα : bx(bs) is a dual Lorentzian Mannheim curve and β : bb x¹(bs¹) is its dual Lorentzian Mannheim partner curve. 3.4. Remark. Let bα = α + εα^∗ and bβ = β + εβ^∗ be non-null dual curves in the dual Lorentzian space D³1. As a special case, if we consider the dual part of the given curves as α^∗= 0 and β^∗= 0, then bα and bβ are non-null Lorentzian curves in R³1. In this case all the results of this paper are similar to the results given in [9].

Acknowledgement

We wish to thank the referee for the careful reading of the manuscript and for the constructive comments that have substantially improved the presentation of the paper.

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