on characterizations of some special curves of spacelike curves according to the type-2 bishop

16 Pages·2016·0.28 MB·English

Unluturk/ Kirklareli University Journal of Engineering and Science 1 (2015) 78-93

ON CHARACTERIZATIONS OF SOME SPECIAL CURVES OF
SPACELIKE CURVES ACCORDING TO THE TYPE-2 BISHOP
FRAME IN MINKOWSKI 3-SPACE

Yasin ÜNLÜTÜRK

Department of Mathematics, Kırklareli University,
39000 Kırklareli, Turkey,
E-mail:
Unluturk/ Kirklareli University Journal of Engineering and Science 1 (2015) 78-93

1 INTRODUCTION

One of the fundamental structures of differential geometry is curves. It is safe to
3
report that the many important results in the theory of the curves in E were initiated by G.
Monge; and G. Darboux pionnered the moving frame idea.

At the beginning of the twentieth century, A.Einstein’s theory opened a door of use of
new geometries. One of them, Minkowski space-time, which is simultaneously the geometry
of special relativity and the geometry induced on each fixed tangent space of an arbitrary
Lorentzian manifold, see [19], [20].

One of the interesting problems is the problem of characterization of a regular curve in
the theory of curves in the Euclidean and Minkowski spaces, see, [1], [3], [4]. In the solution
of the problem, the curvature functions of a regular curve have an effective role. It is known
that we can determine the shape and size of a regular curve by using its curvatures and .
Another approach to the solution of the problem is to consider the relationship between the
corresponding Frenet vectors of two curves, see, [8], [10], [11], [12]. For instance, Bertrand
curves and Mannheim curves arise from this relationship, see, [5], [14], [17]. They adapted
the geometrical models to relativistic motion of charged particles, see, [6].

Bishop Frame which is also called alternative or parallel frame of the curves was
introduced by L.R. Bishop in 1975 by means of parallel vector fields. Recently many
researches related to this subject have been made in Euclidean space, see [13],[16],
Minkowski spaces see [2], and in dual space, see [15].

Bishop and Serret-Frenet frames have a common vector field, i.e., the tangent vector
field of Serret-Frenet frame. The construction of the Bishop frame has some advantages when
comparing with the Frenet frame in Euclidean 3-space, see, [13]. That is why he defined this
frame that curvature may vanish at some points on the curve. That is, second derivative of the
curve may be zero. In this situation, an alternative frame is needed for non continously
differentiable curves on which Bishop (parallel transport frame) frame is well defined and
constructed in the Euclidean and its ambient spaces.

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On Characterizations of Some Special Curves of Spacelike Curves According to the Type-2 Bishop Frame in Minkowski 3-space
Unluturk/ Kirklareli University Journal of Engineering and Science 1 (2015) 78-93

In this work, using common vector field as the binormal vector of Serret-Frenet
frame, we define a rectifying curve of a spacelike curve according to type-2 Bishop frame in
⊥
Minkowski 3-space as its position vector always lies in the orthogonal complement Ω of its
2
principal vector field Ω . Moreover we characterize Bertrand curves in the same space. In
2
3
particular, we study Mannheim partner curves according to type-2 Bishop frame in E and
1
express such curves in temrs of their curvature functions. Finally we give a characterization
3
regarding to inclined curves according to type-2 Bishop frame in E ⋅
1

2 PRELIMINARIES

To meet the requirements in the next sections, here, the basic elements of the theory of
3
curves in the Minkowski 3--space E are briefly presented.
1
3 3
The three dimensional Minkowski space E is a real vector space E endowed with
1
the standard flat Lorentzian metric given by
2 2 2
, = −dx + dx + dx ,
L 1 2 3
3
where (x , x , x ) is a rectangular coordinate system of E . This metric is an indefinite one
1 2 3 1
3
[19]. Let u = (u ,u ,u ) and v = (v ,v ,v ) be arbitrary vectors in E , then, [20], the
1 2 3 1 2 3 1
Lorentzian cross product of u and v is defined as
− i j k 
 
u × v = −det u u u .
1 2 3
 
 v v v 
 1 2 3 

3
Recall that a vector v∈E has one of three Lorentzian characters such as: it is a
1
spacelike vector if v,v > 0 or v = 0 ; timelike v,v < 0 and null (lightlike) v,v = 0 for
3
v ≠ 0. Similarly, an arbitrary curve α =α(s) in E can locally be spacelike, timelike or null
1
(lightlike) if its velocity vector α ' are ,respectively, spacelike, timelike or null (lightlike), for
every s ∈ I ⊂ R [20].
3
The pseudo-norm of an arbitrary vector a∈E is given by a = a, a . The curve
1
' '
α = α(s) is called a unit speed curve if its velocity vector α is the unit one, i.e., α =1. For
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On Characterizations of Some Special Curves of Spacelike Curves According to the Type-2 Bishop Frame in Minkowski 3-space
Unluturk/ Kirklareli University Journal of Engineering and Science 1 (2015) 78-93

3
the vectors v, w∈E , they are said to be orthogonal eachother if and only if v, w = 0.
1
Denote by {T , N, B} the moving Serret-Frenet frame along the curve α =α(s) in the space
3
E [20].
1
3
For an arbitrary spacelike curve α =α(s) in E , the Serret-Frenet formulae are given as
1
follows
'
T   0 κ 0T 
 '    
N = γκ 0 τ N (1)
    
'
B   0 τ 0B
    

where γ = 1, and the functions κ and τ are, respectively, the first and second (torsion)
curvature and also

' ' '' '''
' T (s) det(α ,α ,α )
T (s) = α (s), N(s) = , B(s) = T (s)× N(s) and τ (s) = .
2
κ (s) κ (s)

• If γ = −1, then α(s) is a spacelike curve with spacelike principal normal N
and timelike binormal B, and its Serret-Frenet invariants are given as

' ' '
κ (s) = T (s),T (s) andτ (s) = − N (s), B(s) .
• If γ =1, then α(s) is a spacelike curve with timelike principal normal N and
spacelike binormal B, also we obtain its Serret-Frenet invariants as

' ' '
κ (s) = − T (s),T (s) andτ (s) = N (s), B(s) .

2
The Lorentzian sphere S of radius r > 0 and with the center in the origin of the
1
3
space E is defined, [19], by
1
2 3 2
S (r) = {p = ( p , p , p )∈E : p, p = r }.
1 1 2 3 1

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On Characterizations of Some Special Curves of Spacelike Curves According to the Type-2 Bishop Frame in Minkowski 3-space
Unluturk/ Kirklareli University Journal of Engineering and Science 1 (2015) 78-93

Theorem 2.1, [2], Let α =α(s) be a spacelike unit speed curve with a spacelike principal
normal. If {Ω1,Ω2, B} is an adapted frame, then we have

'
Ω1   0 0 ξ1  Ω1 
 '     

Ω2  =  0 0 −ξ2  Ω2  . (2 )

 B'  −ξ1 −ξ2 0   B 

Theorem 2.2 Let {T , N, B} and {Ω1,Ω2, B} be Frenet and Bishop frames, respectively. There
exists a relation between them as

T  sinhθ (s) coshθ (s) 0Ω1 
    

N = coshθ (s) sinhθ (s) 0Ω2  (3)
B  0 0 1 B 
where θ is the angle between the vectors N and Ω1 [2].
Using (2), we have

'
B =τN = −ξ1Ω1 −ξ2Ω2,

and taking the norm of both sides, we find

2 2
τ = ξ2 −ξ1 ,
and

ξ1 2 ξ2 2
( ) − ( ) = 1.
τ τ
By (5), we express

ξ1 = τ (s)coshθ (s),ξ2 = τ (s)sinhθ (s). (4)
s
The frame {Ω1,Ω2, B} is properly oriented, and τ and θ (s) =
∫κ(s)ds are polar
0
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On Characterizations of Some Special Curves of Spacelike Curves According to the Type-2 Bishop Frame in Minkowski 3-space
Unluturk/ Kirklareli University Journal of Engineering and Science 1 (2015) 78-93

coordinates for the curve α =α(s) . We shall call the set {Ω ,Ω , B,ξ ,ξ } as type-2 Bishop
1 2 1 2
3
invariants of the curve α =α(s) in E .
1
3
Definition 2.1. Let α =α(s) be a spacelike curve in E and V be the first Frenet vector field
1 1
of α . If
V , X = coshθ , (constant), (5)
1

3
for a constant unit vector field X ∈χ(E ) , then α is called a general helix (inclined curve) in
1
3
E [18].
1

∗ 3
Definition 2.2. Let {α , α } be Bertrand curves in E , these curves are said to be Bertrand
1
curves such that they satisfy the following relation:

* *
α = α + λN ,
* ∗
where N is the principal normal vector of the curve α and λ ∈R.

∗ 3
Definition 2.3. Let {α , α } be Mannheim curves in E , these curves are said to be
1
Mannheim curves such that they satisfy the following relation:

* *
α = α + λB ,
* ∗
where N is the principal normal vector of the curve α and λ ∈R.

3 ON CHARACTERİZATİONS OF SOME SPECIAL CURVES OF SPACELIKE
3
CURVES ACCORDING TO THE TYPE-2 BISHOP FRAME IN E
1

In this section, we study characterizations of some special curves such as inclined, rectifying,
Bertrand curves.

3
3.1 A characterization for inclined curves according to the type-2 Bishop frame in E
1
3 3
Definition 3.1. Let α ⊂ E be a curve in E , the function
1 1
ξ (s)
2
H (s) = (6)
ξ (s)
1
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On Characterizations of Some Special Curves of Spacelike Curves According to the Type-2 Bishop Frame in Minkowski 3-space
Unluturk/ Kirklareli University Journal of Engineering and Science 1 (2015) 78-93

is called harmonic curvature function of the curve α provided that ξ1 ≠ 0 and ξ2 ≠ 0
3
according to the type-2 Bishop frame in E1 .
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Theorem 3.1. If the curve α ⊂ E1 is an inclined curve, then the harmonic curvature H is
constant.
Proof. From (5), we write that
ξ2 (s)
= tanhϕ(s), (7)
ξ1(s)
and differentiating (7) with respect to s , we find
ξ2 '
( )
dϕ ξ1
= , (8)
ds ξ2 2
1− ( )
ξ1
ξ2 (s)
and using H (s) = in (6), we get
ξ1(s)
'
dϕ H
= , (9)
2
ds 1− H
and integrating (9), we reach out
'
H
ϕ = ds.
∫ 2
1− H
The solution of this integral is found as follows:
1 H −1
ϕ = ln( ) + c,
2 H +1
where c∈R. If the curve α is an inclined curve which satisfies (7), then ϕ is a constant. As
a result, we obtain that
2ϕ
1+ e
H = = cons tan t.
2ϕ
1− e
Hence, the proof is completed.

3
3.2 Rectifying curves according to the type-2 Bishop frame in E1

In the Euclidean space, rectifying curves were introduced by B.Y. Chen in [3] as space curves
whose position vectors always lie in its rectifying plane spanned by the tangent and the
binormal vector fields of the curve, i.e., T and B , respectively. Accordingly, the position
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On Characterizations of Some Special Curves of Spacelike Curves According to the Type-2 Bishop Frame in Minkowski 3-space
Unluturk/ Kirklareli University Journal of Engineering and Science 1 (2015) 78-93

3
vector with respect to a chosen origin of a rectifying curve α in E satisfies the equation

α (s) = λ(s)T (s) + µ(s)B(s),

where λ(s) and µ(s) are arbitrary differentiable functions in arc-length parameter s∈ I ⊂ E
[1]. In this section, we define rectifying curves according to the type-2 Bishop frame in
Minkowski 3-space in analogy to its Euclidean case. Position vector of the rectifying curves
⊥
always lies in the orthogonal complement Ω2 of its principal normal vector field Ω2.
⊥
Consequently, the complement Ω2 is given by
α(s) = λ(s)Ω1(s) + µ(s)B(s) (10)

for the differentiable functions λ(s) and µ(s) in arc-length parameter s .
Next, we characterize rectifying curves according to their curvature functions ξ1 , ξ2 and then
give the necessary and sufficient conditions for an arbitrary curve to be a rectifying curve in
3 3
E1 . Moreover, we obtain an explicit equation of rectifying curve in E1 .

Theorem 3.2. There is a rectifying curve of a spacelike curve according to the type-2 Bishop
3
frame in E 1 if and only if its position vector is as:

X (s) = α(s) + (
∫ξ1(s)µ(s)ds)Ω1(s) − (∫ξ1(s)λ(s)ds)B(s).

Proof. Assume that there is a rectifying curve of a spacelike curve according to the type-2
3
Bishop frame in E 1 , so the equation (10) is satisfied. Differentiating (10) with respect to s ,
and using Bishop equations (2) in it, we reach

' '
T (s) = (λ (s) −ξ1(s)µ(s))Ω1(s) −ξ2 (s)µ(s)Ω2 (s) + (ξ1(s)λ(s) + µ (s))B(s),

and it is followed by

'
λ (s) −ξ
1(s)µ(s) = 0,

ξ2 (s)µ(s) = 0, (11)
 '
ξ1(s)λ(s) + µ (s) = 0,
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On Characterizations of Some Special Curves of Spacelike Curves According to the Type-2 Bishop Frame in Minkowski 3-space
Unluturk/ Kirklareli University Journal of Engineering and Science 1 (2015) 78-93

so we easily get
 2 2
λ (s) + µ (s) = c,

λ(s) =
∫ξ1(s)µ(s)ds, (12)

µ(s) = −∫ξ1(s)λ(s)ds,

where c∈R. Thus, the functions λ(s) and µ(s) are expressed in terms of the curvature
functions ξ1(s) and ξ2 (s). Moreover, using the last equation in (11), and the relation (12),
we easily find that the curvatures ξ1(s) and ξ2 (s) satisfy the equation

2 '
(−
∫ξ1(s)µ(s)ds)ξ1(s) + (∫ξ1(s)λ(s)ds)ξ2 (s) = ( c −λ ) . (13)

Conversely, assume that the curvatures ξ1(s) and ξ2 (s) satisfy (13), then the position vector
X of rectifying curve is given by

X (s) = α(s) + (
∫ξ1(s)µ(s)ds)Ω1(s) − (∫ξ1(s)λ(s)ds)B(s). (14)

3
3.3 Bertrand curves of the spacelike curves according to the type-2 Bishop frame in E1

∗
Definition 3.2. Let α and α be regular curves according to the type-2 Bishop frame in E₁³,
these curves are said to be Bertrand curves such that they satisfy the following relation:

∗
α =α + lΩ2, (15)
where Ω2 is the vector and l ∈R

3
Theorem 3.3. A spacelike curve according to the type-2 Bishop frame in E1 does not admit a
Bertrand mate curve.
Proof. Taking the norm of (15), and then differentiating it, we have
d 2 d ∗ ∗ ∗' '
(l ) = (α −α) = 2(α −α)(α −α ). (16)
ds ds

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On Characterizations of Some Special Curves of Spacelike Curves According to the Type-2 Bishop Frame in Minkowski 3-space
Unluturk/ Kirklareli University Journal of Engineering and Science 1 (2015) 78-93

Using the following expressions
∗' ∗' ∗ ∗
α = T = sinhθΩ − coshθΩ ,
1 2
(17)
'
α = T = sinhθΩ − coshθΩ ,
1 2
which are obtained from in (16) through using (2), we get
d 2 ∗ ∗ ∗
(l ) = 2(α −α)(sinhθΩ −coshθΩ −sinhθΩ + coshθΩ ), (18)
1 2 1 2
ds

and rearranging (18), we find
d 2
(l ) = −λ(coshθ − coshθ ). (19)
ds
∗
If we take θ = θ in (19), then θ is constant, and also κ = κ.

∗
Let's continue to study by differentiating T .T with respect to the parameter s, then we have
d ∗ ∗' ∗ '
(T .T ) = T .T +T .T ,
ds
and using Frenet derivative formulae we get
d ∗ ∗ ∗
(T .T ) = κ [coshθ sinhθΩ Ω −coshθ sinhθ ]
1 1
ds (20)
∗
+κ sinhθ coshθΩ Ω −coshθ sinhθ ].
1 1

Using trigonometric transformation formulae, and rearranging (20), we find

d ∗
(T .T ) = κ sinh 2θ (coshϕ −1), (21)
ds
since
∗
Ω = coshϕΩ + sinhϕB,
1 1

where ϕ is the angle between the vectors Ω and B. In (21), if κ = 0, we get
1
π π
θ = 0 or ϕ = ,or if κ ≠ 0, we obtain θ = 0 or ϕ = . In this case, we can write that
2 2
∗
Ω = B.
1
Conversely, let's take
∗
T .T = 0 (22)
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On Characterizations of Some Special Curves of Spacelike Curves According to the Type-2 Bishop Frame in Minkowski 3-space