# The Ruled Surfaces According to Bishop Frame in Minkowski 3-Space

Authors: Nural Yüksel
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We investigate the ruled surfaces generated by a straight line in Bishop frame moving along a spacelike curve in Minkowski 3-space. We obtain the distribution parameters, mean curvatures. We give some results and theorems related to be developable and minimal of them. Furthermore, we show that, if the base curve of the ruled surface is also an asymtotic curve and striction line, then the ruled surface is developable.


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## Body

## 1. Introduction

Recently, the theory of surfaces and their transformations has been studied extensively in differential geometry. The ruled surfaces have been a powerful subject in the Minkowski spaceℝ13 for line geometry for a long time. In the literature, Kobayashi [1] was the first author to address this problem and examined minimal spacelike ruled surfaces in the Minkowski ℝ13. Kim and Yoon [2] have classified the Lorentz surfaces.Izumiya and Takeuchi [3] obtained some characterizations for ruled surfaces. Turgut and Hacısalihoğlu [4, 5] defined spacelike ruled surfaces and obtained some characterizations in the three-dimensional Minkowski space. Yaylı [6] obtained the distribution parameter of a spacelike ruled surface generated by a spacelike straight line in Frenet frame along a spacelike curve. Yaylı and Saracoglu [7, 8] studied timelike and spacelike developable ruled surfaces in Minkowski space. Orbay and Aydemir [9] obtained the distrubition parameter, mean curvature, and Gaussian curvature, and some new results and theorems were given for developable and minimal spacelike ruled surfaces.In this paper, making use of the method in a paper of Yaylı [6], we obtained some characterizations for spacelike Ruled surfaces according to Bishop frame in Minkowski 3-space.


## 2. Preliminaries

Letℝ13 be a Minkowski 3-space with the metric tensor I=〈·,·〉=dx12-dx22+dx32. The norm of v∈ℝ13 is defined by ∥v∥=|〈v,v〉|. A vector v∈ℝ13 is said to be spacelike if 〈v,v〉>0 or v=0,timelike if 〈v,v〉<0, and lightlike (or null)if 〈v,v〉=0 and v≠0.Letα:I→ℝ13,α(s)=(α1(s),α2(s),α3(s)) be a smooth regular curve in ℝ13. We say that α is a spacelike (resp. timelike, lightlike) if α′(t), a spacelike (resp. timelike, lightlike) vector for all s∈I⊂ℝ.A surface in the Minkowski 3-space is called a spacelike surface if the Lorentz metric on the surface is a positive definite [10]. A ruled surface is a surface swept out by a straight line X moving along a curve α. The various positions of the generating line X are called the rullings of the surface. Such a surface has a parametrization in the ruled form as follows:
(1)ϕ(s,v)=α(s)+vX(s),
where α is the base curve and X is the director vector along α. If the tangent plane is constant along a fixed rulling, then the ruled surface is called a developable surface. The remaining ruled surfaces are called skew surfaces [4]. The spacelike ruled surface M in ℝ13 is given by the parametrization
(2)ϕ:I×ℝ⟶ℝ13(s,v)⟶ϕ(s,v)=α(s)+vX(s),
where α:I→ℝ13 is a differentiable spacelike curve parametrized by its arc length in ℝ13 and X(s) is the director vector of the director curve such that X is ortogonal the tangent vector field T of the base curve α.Denote by{T,N,B} the moving Frenet frame along the regular curve with arc-lenght parameter s. The Frenet trihedron consists of the tangent vector T, the principal normal vector N, and the binormal vector B. If α is a spacelike curve with a spacelike binormal, then the Frenet frame has the following properties:
(3)T′(s)=ϰ(s)N,N′(s)=ϰ(s)T(s)+τ(s)B(s),B′(s)=τ(s)N(s),
where
(4)〈T,T〉=1,〈N,N〉=-1,〈B,B〉=1.The Bishop frame or parallel transport frame is an alternative approach to defining a moving frame that is well defined even when the curve has a vanishing second derivative. One can express parallel transport of an orthonormal frame along a curve simply by parallel transporting each component of the frame. The tangent vector and any convenient arbitrary basis for the remainder of the frame are used.Let us consider theBishop frame{T,N1,N2} of the spacelike curve α(s) such that T(s) the spacelike unit tangent vector, N1(s) is timelike unit normal vector, and N2(s) the spacelike unit binormal vector. So scalar product and cross product of the vectors {T,N1,N2} are given by
(5)〈T,T〉=-〈N1,N1〉=〈N2,N2〉=1,〈T,N1〉=〈T,N2〉=〈N1,N2〉=0,(6)T∧N1=N2,N1∧N2=T,N2∧T=-N1.
The Bishop frame {T,N1,N2} is expressed as
(7)T′(s)=k1N1(s)-k2N2(s),N1′(s)=k1T(s),N2′(s)=k2T(s).
One can show that
(8)κ=|k22-k12|,(9)τ=dθds,θ(s)=argtanhk2k1
so that k1 and k2 effectively correspond to a cartesian coordinate system for the polar coordinates κ, θ with θ=∫‍τ(s)ds [11].Remark 1.
From the definition of the argtanh function we assume that|k2/k1|<1.The distribution parameter, the mean curvature, and the Gaussian curvature of the ruled surfaceϕ(s,v) are given by
(10)Px=det(T,X,DTX)〈DTX,DTX〉,(11)H=12[Gl+En-2FmEG-F2],
where D is the Levi-Civita connection on ℝ13.Theorem 2.
A spacelike ruled surface is a developable surface if and only if the distrubition parameter of the spacelike ruled surface is zero [4].The foot on the main rulling of the common perpendicular of two constructive rullings in the ruled surface is called a central point. The locus of the central point is called the striction curve. The parametrization of the striction curve on the ruled surface is given by(12)α¯(s)=α(s)-〈T,DTX〉〈DTX,DTX〉X(s).


## 3. One Parameter Spatial Motion inℝ13

Letα:I→ℝ13 be a spacelike curve and {T,N1,N2} be its Bishopframe where T,N1, and N2 are the tangent, principal normal, and binormal vectors of the curve α, respectively. T and N2 are spacelike vectors and N1 is a timelike vector.The two coordinate systems{O;T,N1,N2} and {O′;e1,e2,e3} are orthogonal coordinate systems in ℝ13 which represent the moving space H and the fixed space H′, respectively.LetX be a unit spacelike vector
(13)X∈Sp{T(s),N1(s),N2(s)},X=x1T(s)+x2N1(s)+x3N2(s),
such that
(14)〈X,X〉=1.We can obtain the distrubition parameter of the spacelike ruled surface generated by a straight lineX of the moving space H. Differentiating (13) with respect to s, we get
(15)DTX=x1T′(s)+x2N1′(s)+x3N2′(s),x12-x22+x32=1.
By using the Bishop frame in (15), we obtain
(16)DTX=(x2k1+x3k2)T(s)+x1k1N1(s)-x1k2N2(s).
From (10) we get
(17)Px=-x1(x2k2+x3k1)(x2k1+x3k2)2+x12(k22-k12),x12-x22+x32=1.Theorem 3.
LetM be a spacelike ruled surface given by the parametrization (2). M is developable if and only if either the director vector X lies in the plane generated by N1(s) and N2(s) or the base curve α is a planar curve such that the curvatures of α, k1 and k2 satisfy (18)k1k2=-x2x3.Proof.
LetM be a ruled surface. By using (17) and Theorem 2,
(19)x1(x2k2+x3k1)=0
is obtained. In that case, we have
(20)x1=0or(x2k2+x3k1)=0.
Thus
(21)X(s)∈Sp{N1(s),N2(s)}ork1k2=-x2x3=constant.
From (9), we get
(22)τ=0.
So, α is a planar curve. This completes the proof.


## 4. Special Cases

LetM be a spacelike ruled surface given by the parametrization (2), and, X be the director vector of the base curve α.


### 4.1.  The CaseX=T (Spacelike)

In this case,x1=1, x2=x3=0 thus from (17)
(23)PT=0.
Hence the following theorem is hold.Theorem 4.
During the one-parameter spatial motionH/H′ the spacelike ruled surface in the fixed space H′ generated by the tangent line T of the curve α(s) in the moving space H is developable.


### 4.2. The CaseX=N2 (Spacelike)

From Theorem3, it is obvious that PN2=0.


### 4.3.  The CaseX∈Sp{T(s),N1(s)}

In this case,x3 is zero. So, the director vector X is given by
(24)X=x1T+x2N1,x12-x22=1.
The distribution parameter of the ruled surface is given by
(25)Px=-x1x2k2x22k12+x12(k22-k12)=±x21+x22k2x22k12+(1+x22)(k22-k12).
The ruled surface is developable if and only if Px=0. Thus
(26)x2=0ork2=0.Ifx2=0, this is case 4.1. If the second curvature k2 is zero, then we can say that the base curve α is a planar curve.


### 4.4.  The CaseX∈Sp{T(s),N2(s)}

In this case,x2 is zero. So, the director vector X is given by
(27)X=x1T+x3N2,x12+x32=1.From (17) the distribution parameter is obtained as
(28)Px=-x1x3k1k22-x12k12Px=0 if and only if
(29)x1=0orx3=0.
If x1 is zero, this is case 4.3. If x3 is zero, this is The case 4.1.From Theorem3, M is a developable spacelike ruled surface.


### 4.5.  The CaseX∈Sp{N1(s),N2(s)}

From Theorem3 it is obvious that the spacelike ruled surface is developable.By using (11) we compute the mean curvatures of the spacelike ruled surfaces generated by spacelike vectors T(s),N2(s), and X.Proposition 5.
LetMT be a spacelike ruled surface generated by the tangent line T of the curve α. From (11) the mean curvature is obtained as follows:
(30)HT=12ɛ(k1k2′-k2k1′)v|k12-k22|,ϵ=±1.
Thus from (6) we have
(31)HT=-12ɛvτκ.Corollary 6.
The surfaceMT is minimal if and only if α is a planar curve.Proof.
LetMT be minimal. In this case, from (31), we get
(32)τ=0.
Conversely, letα be a planar curve. Then τ=0 implies that HT=0. This completes the proof.Proposition 7.
LetMN2 be a spacelike ruled surface generated by the binormal line N2 of the base curve α. From (11) the mean curvature is obtained as follows: (33)HN2=-ɛk12(1+vk2),ɛ=±1.
So, the following result may be given.Corollary 8.
According to the Bishop frame, there is no minimal spacelike ruled surface generated by the binormal lineN2 in ℝ13.Proposition 9.
LetM be a spacelike ruled surface which is given by the parametrization (2):
(34)Hx=12[Gl+En-2FmEG-F2],
where
(35)E=〈ϕs,ϕs〉=(1+v(k1x2+k2x3))2+v2k12(k22-k12),F=〈ϕs,ϕv〉=x1,G=〈ϕv,ϕv〉=1,(36)l=1∥ϕs×ϕv∥〈ϕss,ϕs×ϕv〉=1∥ϕs×ϕv∥×[-(1+v(x2k1+x3k2))2(x3k1+x2k2)-(1+v(x2k1+x3k2))vx1(x3k1′+x2k2′)+v2x13(k1k2′-k1′k2)+v2x1(x2k1+x3k2)(x2k1′+x3k2′)+v2x12(x3k1+x2k2)(k12-k22)](37)m=1∥ϕs×ϕv∥〈ϕsv,ϕs×ϕv〉=-x1(k1x3+k2x2)∥ϕs×ϕv∥,(38)n=1∥ϕs×ϕv∥〈ϕvv,ϕs×ϕv〉=0,
where
(39)N=ϕs×ϕv∥ϕs×ϕv∥=1∥ϕs×ϕv∥{[vx1(x3k1+x2k2)]T+[x3(1+v(x2k1+x3k2))+vx12k2]N1+[x2(1+v(x2k1+x3k2))-vx12k1]N2}
is a unit normal vector of the spacelike ruled surface M.Proposition 10.
LetM be a spacelike ruled surface given by the parametrization (2). If the base curve of M is also a striction curve, then the curvature functions k1 and k2 of the base curve α satisfy the following equation:
(40)x2k1+x3k2=0.Proof.
Let the base curveα be the striction curve. Thus, from (12),
(41)〈T,DTX〉=0.
Then we have
(42)x2k1+x3k2=0.Hence the following result holds.Corollary 11.
LetM be a spacelike ruled surface given by the parametrization (2). If the base curve of M is also striction curve, then α is a planar curve.Proof.
Let the base curveα be also striction curve. Thus from (42)
(43)x2k1+x3k2=0.
Hence we get
(44)k1k2=-x3x2=constant.
From (9), α is a planar curve.Proposition 12.
LetM be a spacelike ruled surface given by the parametrization (2). If the base curve of M is also asymtotic curve, then
(45)(1+v(x2k1+x3k2))(x2k2+x3k1)=0.Proof.
We assume that the base curve of the surfaceM is the asymtotic curve. In that case,
(46)〈α′′,N〉=0.
From (46), we have
(47)(1+v(x2k1+x3k2))(x2k2+x3k1)=0.Theorem 13.
Let the base curve of the surfaceM be an asymtotic curve. If the base curve of M is also a striction curve, the spacelike ruled surface M is developable.Proof.
Let the base curve of the surfaceM be both an asymtotic curve and striction curve. By using (42) and (45) we obtain
(48)(x2k2+x3k1)=0.
From (17), the surface is developable.Proposition 14.
LetM be a spacelike ruled surface given by the parametrization (2). We obtain the following results for the spacelike ruled surfaces.(i)
Thes-parameter curve of M is also an asymtotic curve if and only if
(49)[-(1+v(x2k1+x3k2))2(x3k1+x2k2)-(1+v(x2k1+x3k2))vx1(x3k1′+x2k2′)+v2x13(k1k2′-k1′k2)+v2x1(x2k1+x3k2)(x2k1′+x3k2′)+v2x12(x3k1+x2k2)(k12-k22)]=0.(ii)
Thev-parameter curve of M is also an asymtotic curve.Proof.
(i) If thes-parameter curve of M is also an asymtotic curve, then
(50)〈ϕss,N〉=0.From (36), we obtain (46).(ii)
If thev-parameter curve of M is also an asymtotic curve, then
(51)〈ϕvv,N〉=0.v-parameter curve of M is an asymtotic curve.Theorem 15.
LetM be a developable spacelike ruled surface given by the parametrization (2). The s-parameter curve of M is also asymtotic curve if and only if M is a minimal surface.Proof.
Assume thats-parameter curve of the surface M an asymtotic curve. Then
(52)l=〈ϕss,N〉=0,
where N is a unit normal vector field of the surface M. Since M is a developable ruled surface,
(53)x1(x2k1+x3k2)=0.
Thus from (34)
(54)Hx=0
is obtained.
Conversely, letM be a minimal surface. From (14), we get
(55)l-2Fm=0
since M is a developable ruled surface, we obtain
(56)l=0.
This completes the proof.


## 4.1.  The CaseX=T (Spacelike)

In this case,x1=1, x2=x3=0 thus from (17)
(23)PT=0.
Hence the following theorem is hold.Theorem 4.
During the one-parameter spatial motionH/H′ the spacelike ruled surface in the fixed space H′ generated by the tangent line T of the curve α(s) in the moving space H is developable.


## 4.2. The CaseX=N2 (Spacelike)

From Theorem3, it is obvious that PN2=0.


## 4.3.  The CaseX∈Sp{T(s),N1(s)}

In this case,x3 is zero. So, the director vector X is given by
(24)X=x1T+x2N1,x12-x22=1.
The distribution parameter of the ruled surface is given by
(25)Px=-x1x2k2x22k12+x12(k22-k12)=±x21+x22k2x22k12+(1+x22)(k22-k12).
The ruled surface is developable if and only if Px=0. Thus
(26)x2=0ork2=0.Ifx2=0, this is case 4.1. If the second curvature k2 is zero, then we can say that the base curve α is a planar curve.


## 4.4.  The CaseX∈Sp{T(s),N2(s)}

In this case,x2 is zero. So, the director vector X is given by
(27)X=x1T+x3N2,x12+x32=1.From (17) the distribution parameter is obtained as
(28)Px=-x1x3k1k22-x12k12Px=0 if and only if
(29)x1=0orx3=0.
If x1 is zero, this is case 4.3. If x3 is zero, this is The case 4.1.From Theorem3, M is a developable spacelike ruled surface.


## 4.5.  The CaseX∈Sp{N1(s),N2(s)}

From Theorem3 it is obvious that the spacelike ruled surface is developable.By using (11) we compute the mean curvatures of the spacelike ruled surfaces generated by spacelike vectors T(s),N2(s), and X.Proposition 5.
LetMT be a spacelike ruled surface generated by the tangent line T of the curve α. From (11) the mean curvature is obtained as follows:
(30)HT=12ɛ(k1k2′-k2k1′)v|k12-k22|,ϵ=±1.
Thus from (6) we have
(31)HT=-12ɛvτκ.Corollary 6.
The surfaceMT is minimal if and only if α is a planar curve.Proof.
LetMT be minimal. In this case, from (31), we get
(32)τ=0.
Conversely, letα be a planar curve. Then τ=0 implies that HT=0. This completes the proof.Proposition 7.
LetMN2 be a spacelike ruled surface generated by the binormal line N2 of the base curve α. From (11) the mean curvature is obtained as follows: (33)HN2=-ɛk12(1+vk2),ɛ=±1.
So, the following result may be given.Corollary 8.
According to the Bishop frame, there is no minimal spacelike ruled surface generated by the binormal lineN2 in ℝ13.Proposition 9.
LetM be a spacelike ruled surface which is given by the parametrization (2):
(34)Hx=12[Gl+En-2FmEG-F2],
where
(35)E=〈ϕs,ϕs〉=(1+v(k1x2+k2x3))2+v2k12(k22-k12),F=〈ϕs,ϕv〉=x1,G=〈ϕv,ϕv〉=1,(36)l=1∥ϕs×ϕv∥〈ϕss,ϕs×ϕv〉=1∥ϕs×ϕv∥×[-(1+v(x2k1+x3k2))2(x3k1+x2k2)-(1+v(x2k1+x3k2))vx1(x3k1′+x2k2′)+v2x13(k1k2′-k1′k2)+v2x1(x2k1+x3k2)(x2k1′+x3k2′)+v2x12(x3k1+x2k2)(k12-k22)](37)m=1∥ϕs×ϕv∥〈ϕsv,ϕs×ϕv〉=-x1(k1x3+k2x2)∥ϕs×ϕv∥,(38)n=1∥ϕs×ϕv∥〈ϕvv,ϕs×ϕv〉=0,
where
(39)N=ϕs×ϕv∥ϕs×ϕv∥=1∥ϕs×ϕv∥{[vx1(x3k1+x2k2)]T+[x3(1+v(x2k1+x3k2))+vx12k2]N1+[x2(1+v(x2k1+x3k2))-vx12k1]N2}
is a unit normal vector of the spacelike ruled surface M.Proposition 10.
LetM be a spacelike ruled surface given by the parametrization (2). If the base curve of M is also a striction curve, then the curvature functions k1 and k2 of the base curve α satisfy the following equation:
(40)x2k1+x3k2=0.Proof.
Let the base curveα be the striction curve. Thus, from (12),
(41)〈T,DTX〉=0.
Then we have
(42)x2k1+x3k2=0.Hence the following result holds.Corollary 11.
LetM be a spacelike ruled surface given by the parametrization (2). If the base curve of M is also striction curve, then α is a planar curve.Proof.
Let the base curveα be also striction curve. Thus from (42)
(43)x2k1+x3k2=0.
Hence we get
(44)k1k2=-x3x2=constant.
From (9), α is a planar curve.Proposition 12.
LetM be a spacelike ruled surface given by the parametrization (2). If the base curve of M is also asymtotic curve, then
(45)(1+v(x2k1+x3k2))(x2k2+x3k1)=0.Proof.
We assume that the base curve of the surfaceM is the asymtotic curve. In that case,
(46)〈α′′,N〉=0.
From (46), we have
(47)(1+v(x2k1+x3k2))(x2k2+x3k1)=0.Theorem 13.
Let the base curve of the surfaceM be an asymtotic curve. If the base curve of M is also a striction curve, the spacelike ruled surface M is developable.Proof.
Let the base curve of the surfaceM be both an asymtotic curve and striction curve. By using (42) and (45) we obtain
(48)(x2k2+x3k1)=0.
From (17), the surface is developable.Proposition 14.
LetM be a spacelike ruled surface given by the parametrization (2). We obtain the following results for the spacelike ruled surfaces.(i)
Thes-parameter curve of M is also an asymtotic curve if and only if
(49)[-(1+v(x2k1+x3k2))2(x3k1+x2k2)-(1+v(x2k1+x3k2))vx1(x3k1′+x2k2′)+v2x13(k1k2′-k1′k2)+v2x1(x2k1+x3k2)(x2k1′+x3k2′)+v2x12(x3k1+x2k2)(k12-k22)]=0.(ii)
Thev-parameter curve of M is also an asymtotic curve.Proof.
(i) If thes-parameter curve of M is also an asymtotic curve, then
(50)〈ϕss,N〉=0.From (36), we obtain (46).(ii)
If thev-parameter curve of M is also an asymtotic curve, then
(51)〈ϕvv,N〉=0.v-parameter curve of M is an asymtotic curve.Theorem 15.
LetM be a developable spacelike ruled surface given by the parametrization (2). The s-parameter curve of M is also asymtotic curve if and only if M is a minimal surface.Proof.
Assume thats-parameter curve of the surface M an asymtotic curve. Then
(52)l=〈ϕss,N〉=0,
where N is a unit normal vector field of the surface M. Since M is a developable ruled surface,
(53)x1(x2k1+x3k2)=0.
Thus from (34)
(54)Hx=0
is obtained.
Conversely, letM be a minimal surface. From (14), we get
(55)l-2Fm=0
since M is a developable ruled surface, we obtain
(56)l=0.
This completes the proof.



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