# On Pseudohyperbolical Smarandache Curves in Minkowski 3-Space

Authors: Esra Betul Koc Ozturk; Ufuk Ozturk; Kazim İlarslan; Emilija Nešović
Journal: International Journal of Mathematics and Mathematical Sciences
 (2013)

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## Abstract

We define pseudohyperbolical Smarandache curves according to the Sabban frame in Minkowski 3-space. We obtain the geodesic curvatures and the expression for the Sabban frame vectors of special pseudohyperbolic Smarandache curves. Finally, we give some examples of such curves.


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## Body

## 1. Introduction

In the theory of curves in the Euclidean and Minkowski spaces, one of the interesting problems is the problem of characterization of a regular curve. In the solution of the problem, the curvature functionsκ and τ 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 κ and τ. Another approach to the solution of the problem is to consider the relationship between the corresponding Frenet vectors of two curves. For instance, Bertrand curves and Mannheim curves arise from this relationship. Another example is the Smarandache curves. They are the objects of Smarandache geometry, that is, a geometry which has at least one Smarandachely denied axiom [1]. An axiom is said to be Smarandachely denied if it behaves in at least two different ways within the same space. Smarandache geometries are connected with the theory of relativity and the parallel universes.If the position vector of a regular curveα is composed by the Frenet frame vectors of another regular curve β, then the curve α is called a Smarandache curve [2]. Special Smarandache curves in Euclidean and Minkowski spaces are studied by some authors [3–8]. The curves lying on a pseudohyperbolic space H02 in Minkowski 3-space 𝔼13 are characterized in [9].In this paper, we define pseudohyperbolical Smarandache curves according to the Sabban frame{α,T,ξ} in Minkowski 3-space. We obtain the geodesic curvatures and the expressions for the Sabban frame's vectors of special pseudohyperbolical Smarandache curves. In particular, we prove that special Tξ-pseudohyperbolical Smarandache curves do not exist. Besides, we give some examples of special pseudohyperbolical Smarandache curves in Minkowski 3-space.


## 2. Basic Concepts

The Minkowski 3-spaceℝ13 is the Euclidean 3-space ℝ3 provided with the standard flat metric given by
(1)〈·,·〉=-dx12+dx22+dx32,
where (x1,x2,x3) is a rectangular Cartesian coordinate system of ℝ13. Since g is an indefinite metric, recall that a nonzero vector x→∈ℝ13 can have one of the three Lorentzian causal characters: it can be spacelike if 〈x→,x→〉>0, timelike if 〈x→,x→〉<0, and null (lightlike) if 〈x→,x→〉=0. In particular, the norm (length) of a vector x→∈ℝ13 is given by ∥x→∥=|〈x→,x→〉| and two vectors x→ and y→ are said to be orthogonal if 〈x→,y→〉=0. Next, recall that an arbitrary curve α=α(s) in 𝔼13 can locally be spacelike, timelike, or null (lightlike) if all of its velocity vectors α′(s) are, respectively, spacelike, timelike, or null (lightlike) for every s∈I[10]. If ∥α′(s)∥≠0 for every s∈I, then α is a regular curve in ℝ13. A spacelike (timelike) regular curve α is parameterized by pseudo-arclength parameter s which is given by α:I⊂ℝ→ℝ13, and then the tangent vector α′(s) along α has unit length, that is, 〈α′(s),α′(s)〉=1(〈α′(s),α′(s)〉=-1) for all s∈I, respectively.For anyx→=(x1,x2,x3) and y→=(y1,y2,y3) in the space ℝ13, the pseudovector product of x→ and y→ is defined by
(2)x→×y→=(-x2y3+x3y2,x3y1-x1y3,x1y2-x2y1).Remark 1.
Letx→=(x1,x2,x3), y→=(y1,y2,y3), and z→=(z1,z2,z3) be vectors in ℝ13. Then, (i)
〈x→×y→,z→〉=|x1x2x3y1y2y3z1z2z3|,(ii)
x→×(y→×z→)=-〈x→,z→〉y→+〈x→,y→〉z→,(iii)
〈x→×y→,x→×y→〉=-〈x→,x→〉〈y→,y→〉+〈x→,y→〉2,where × is the pseudovector product in the space ℝ13.Lemma 2.
In the Minkowski 3-spaceℝ13, the following properties are satisfied [10]:(i)
two timelike vectors are never orthogonal;(ii)
two null vectors are orthogonal if and only if they are linearly dependent;(iii)
timelike vector is never orthogonal to a null vector.Pseudohyperbolic spacein the Minkowski 3-space ℝ13 is a quadric defined by
(3)H02={x→∈ℝ13∣-x12+x22+x32=-1}.Letα:I⊂ℝ→H02 be a regular unit speed curve lying fully in H02 in ℝ13. Then its position vector α is timelike vector, which implies that tangent vector T→=α′ is the unit spacelike vector for all s∈I. Hence we have orthonormal Sabban frame {α(s),T(s),ξ(s)} along the curve α, where ξ(s)=α(s)×T(s) is the unit spacelike vector. The corresponding Frenet formulae of α, according to the Sabban frame, read
(4)[α′T′ξ′]=[01010kg(s)0-kg(s)0][αTξ],
where kg(s)=det(α(s),T(s),T′(s)) is the geodesic curvature of α on H02 in ℝ13 and s is arc length parameter ofalpha. In particular, the following relations hold:
(5)α×T=ξ,T×ξ=-α,ξ×α=T.


## 3. Pseudohyperbolical Smarandache Curves in Minkowski 3-Space

In this section, we define and investigate pseudohyperbolical Smarandache curves in Minkowski 3-space according to the Sabban frame.Letα=α(s) and β=β(s*) be two regular unit speed curves lying fully in pseudohyperbolic space H02 in ℝ13 and let {α,T,ξ} and {β,Tβ,ξβ} be the moving Sabban frames of these curves, respectively. Then we have the following definitions of pseudohyperbolical Smarandache curves.Definition 3.
Letα:I⊂ℝ↦H02 be a regular unit speed curve lying fully in H02. Then αξ-pseudohyperbolical Smarandache curve β:I⊂ℝ↦H02 of α is defined by
(6)β(s⋆(s))=12(aα(s)+bξ(s)),
where a,b∈ℝ0 and a2-b2=2.Definition 4.
Letα:I⊂ℝ↦H02 be a regular unit speed curve lying fully in H02. Then αT-pseudohyperbolical Smarandache curve β:I⊂ℝ↦H02 of α is defined by
(7)β(s⋆(s))=12(aα(s)+bT(s)),
where a,b∈R0 and a2-b2=2.Definition 5.
Letα:I⊂R↦H02 be a regular unit speed curve lying fully in H02. Then αTξ -pseudohyperbolical Smarandache curve β:I⊂R↦H02 of α is defined by
(8)β(s⋆(s))=13(aα(s)+bT(s)+cξ(s)),
where a,b,c∈ℝ0 and a2-b2-c2=3.Theorem 6.
Letα:I⊂ℝ↦H02 be a regular unit speed curve lying fully in H02. Then Tξ-pseudohyperbolical Smarandache curve β:I⊂ℝ↦H02 of α does not exist.Proof.
Assume that there existsTξ-pseudohyperbolical Smarandache curve of α. Then it can be written as
(9)β(s⋆(s))=12(aT(s)+bξ(s)),
where a2+b2=-2, which is a contradiction.In the theorems which follow, we obtain Sabban frame{β,Tβ,ξβ} and geodesic curvature κgβ of pseudohyperbolical Smarandache curve β.Theorem 7.
Letα:I⊂ℝ↦H02 be a regular unit speed curve lying fully in H02 with the Sabban frame {α,T,ξ} and the geodesic curvature kg. If β:I⊂ℝ↦H02 is αξ-pseudohyperbolical Smarandache curve of α, then its frame {β,Tβ,ξβ} is given by
(10)[βTβξβ]=[a20b20ϵ0ϵb20ϵa2][αTξ],
and the corresponding geodesic curvature kgβ reads
(11)kgβ=akg-b|a-bkg|,a,b∈ℝ0,
where a2-b2=2 and ϵ=±1.Proof.
Differentiating (6) with respect to s and using (4), we obtain
(12)β′(s)=dβds*ds*ds=a-bkg2T,Tβds*ds=a-bkg2T,
where
(13)ds*ds=|a-bkg|2.
Therefore, the unit spacelike tangent vector of the curve β is given by
(14)Tβ=ϵT,
where ϵ=+1 if a-bkg>0 for all s and ϵ=-1 if a-bkg<0 for all s.
Differentiating (14) with respect to s, we find
(15)dTβds*ds*ds=ϵ(α+kgξ),
and from (13) and (15) we get
(16)Tβ′=2ϵ|a-bkg|(α+kgξ).
On the other hand, from (6) and (14) it can be easily seen that
(17)ξβ=β×Tβ=ϵb2α+ϵa2ξ
is a unit spacelike vector.
Consequently, the geodesic curvaturekgβ of the curve β=β(s*) is given by
(18)kgβ=det(β,Tβ,Tβ′)=akg-b|a-bkg|.Theorem 8.
Letα:I⊂ℝ↦H02 be a regular unit speed curve lying fully in H02 with the Sabban frame {α,T,ξ} and the geodesic curvature kg. If β:I⊂ℝ↦H02 is αT-pseudohyperbolical Smarandache curve of α, then its frame {β,Tβ,ξβ} is given by
(19)[βTβξβ]=[a2b20b2+(bkg)2a2+(bkg)2bkg2+(bkg)2-b2kg4+2(bkg)2-abkg4+2(bkg)222+(bkg)2]×[αTξ],
and the corresponding geodesic curvature kgβ reads
(20)kgβ=b2kgε1-abkgε2+2ε3(2+b2kg2)5/2,a,b∈ℝ0,
where
(21)ε1=-b3kgkg′+a(2+(bkg)2),ε2=-ab2kgkg′+(b-bkg2)(2+(bkg)2),ε3=-b3kg2kg′+(akg+bkg′)(2+(bkg)2),
and a2-b2=2.Proof.
Differentiating (7) with respect to s and using (4), we obtain
(22)β′(s)=dβds*ds*ds=12(bα+aT+bkgξ),Tβds*ds=12(bα+aT+bkgξ),
where
(23)ds*ds=2+(bkg)22.
Therefore, the unit spacelike tangent vector of the curveβ is given by
(24)Tβ=12+(bkg)2(bα+aT+bkgξ).
Differentiating (24) with respect to s, it follows that
(25)dTβds*ds*ds=1(2+(bkg)2)3/2(ε1α+ε2T+ε3ξ).
From (4) and (23), we get
(26)Tβ′=2(2+(bkg)2)2(ε1α+ε2T+ε3ξ),
where
(27)ε1=-b3kgkg′+a(2+(bkg)2),ε2=-ab2kgkg′+(b-bkg2)(2+(bkg)2),ε3=-b3kg2kg′+(akg+bkg′)(2+(bkg)2).
On the other hand, from (7) and (24), it can be easily seen that
(28)ξβ=β×Tβ=-b2kg4+2(bkg)2α-abkg4+2(bkg)2T+22+(bkg)2ξ.
Hence ξβ is a unit spacelike vector.
Therefore, the geodesic curvaturekgβ of the curve β=β(s*) is given by
(29)kgβ=det(β,Tβ,Tβ′)=b2kgε1-abkgε2+2ε3(2+b2kg2)5/2.Theorem 9.
Letα:I⊂ℝ↦H02 be a regular unit speed curve lying fully in H02 with the Sabban frame {α,T,ξ} and the geodesic curvature kg. If β:I⊂ℝ↦H02 is αTξ-pseudohyperbolical Smarandache curve of α, then its frame {β,Tβ,ξβ} is given by(30)[βTβξβ]=[a3b3c3b|(akg-c)2-3kg2+3|a-ckg|(akg-c)2-3kg2+3|bkg|(akg-c)2-3kg2+3|-(a2-3)kg+ac|3(akg-c)2-9kg2+9|bc-abkg|3(akg-c)2-9kg2+9|3+c2-ackg|3(akg-c)2-9kg2+9|][αTξ]and the corresponding geodesic curvature kgβ reads
(31)kgβ=(-(ac-(a2-3)kg)ε1kgβ+(bc-abkg)ε2+(3+c2-ackg)ε3)×(((akg-c)2-3kg2+3)5/2)-1,ma,b,c∈ℝ0,
where
(32)ε1=-((akg-c)akg′-3kg′kg)b+((akg-c)2-3kg2+3)(a-ckg),ε2=-((akg-c)akg′-3kg′kg)(a-ckg)+((akg-c)2-3kg2+3)(b-ckg′-bkg2),ε3=-((akg-c)akg′-3kg′kg)bkg+((akg-c)2-3kg2+3)((a-ckg)kg+bkg′)
and a2-b2-c2=3.Proof.
Differentiating (8) with respect to s and by using (4), we find
(33)β′(s)=dβds*ds*ds=13(bα+(a-ckg)T+bkgξ),
and thus
(34)Tβds*ds=13(bα+(a-ckg)T+bkgξ),
where
(35)ds*ds=|(akg-c)2-3kg2+3|3.
The geodesic curvaturekg≠(ac±a2c2-(a2-3)(c2+3))/(a2-3) for all s, since β=β(s*) is a unit speed regular curve in ℝ13.
Therefore, the unit spacelike tangent vector of the curveβ is given by
(36)Tβ=1|(akg-c)2-3kg2+3|(bα+(a-ckg)T+bkgξ).
Differentiating (36) with respect to s and from (4) and (35), it follows that
(37)Tβ′=3((akg-c)2-3kg2+3)2(ε1α+ε2T+ε3ξ),
where
(38)ε1=-((akg-c)akg′-3kg′kg)b+((akg-c)2-3kg2+3)(a-ckg),ε2=-((akg-c)akg′-3kg′kg)(a-ckg)+((akg-c)2-3kg2+3)(b-ckg′-bkg2),ε3=-((akg-c)akg′-3kg′kg)bkg+((akg-c)2-3kg2+3)((a-ckg)kg+bkg′).
On the other hand, from (8) and (36) it can be easily seen that (39)ξβ=β×Tβ=-(a2-3)kg+ac|3(akg-c)2-9kg2+9|α+bc-abkg|3(akg-c)2-9kg2+9|T+3+c2-ackg|3(akg-c)2-9kg2+9|ξ.
Hence ξβ is a unit spacelike vector.
Therefore, the geodesic curvaturekgβ of the curve β=β(s*) is given by
(40)kgβ=det(β,Tβ,Tβ′)=(-(ac-(a2-3)kg)ε1+(bc-abkg)ε2+(3+c2-ackg)ε3)×(((akg-c)2-3kg2+3)5/2)-1.Corollary 10.
Ifα:I⊂ℝ↦H02 is a geodesic curve on H02 in Minkowski 3-space 𝔼13, then(1)
αT-pseudohyperbolic Smarandache curve is also geodesic on H02;(2)
αξ-pseudohyperbolic and αTξ-pseudohyperbolic Smarandache curves have constant geodesic curvatures on H02.


## 4. Examples

Example 1.
Letα be a unit speed curve lying in pseudohyperbolic space H02(1) in the Minkowski 3-space 𝔼13 with parameter equation (see Figure 1)
(41)α(s)=(s22+1,s22,s).
The orthonormal Sabban frame {α(s),T(s),ξ(s)} along the curve α is given by
(42)α(s)=(s22+1,s22,s),T(s)=α′(s)=(s,s,1),ξ(s)=α(s)×T(s)=(s22,s22-1,s).
In particular, the geodesic curvature kg of the curve α has the form
(43)kg(s)=-1.Figure 1
The curveα on H02(1).Case 1.
If we takea=2, b=2, then from (6) the αξ-pseudohyperbolical Smarandache curve β is given by (see Figure 2)
(44)β(s*(s))=12((2+1)s2+22,(2+1)s2-2,(22+2)s).
From Theorem 7, its frame {β,Tβ,ξβ} is given by
(45)[βTβξβ]=[201010-102][αTξ],
and the corresponding geodesic curvature kgβ reads
(46)kgβ=-1.Figure 2
Theαξ-pseudohyperbolical Smarandache curve β and the curve α on H02(1).Case 2.
If we takea=3, b=-1, then from (7) the αT-pseudohyperbolical Smarandache curve is given by (see Figure 3)
(47)β(s*(s))=24(3s2-2s+23,3s2-2s,23s-2).
According to Theorem 8, its frame {β,Tβ,ξβ} is given by
(48)[βTβξβ]=[62-220-3313366-2263][αTξ],
and the corresponding geodesic curvature kgβ reads
(49)kgβ=-1.Figure 3
TheαT-pseudohyperbolical Smarandache curve β and the curve α on H02(1).Case 3.
Ifa=3, b=3, and c=3, then from (7) the αTξ-pseudohyperbolical Smarandache curve β is given by (see Figure 4)
(50)β(s*(s))=12((3+1)s2+2s+23,(3+1)s2+2s-2,(23+2)s+2).
By using Theorem 9, it follows that the frame {β,Tβ,ξβ} is given by (see Figure 4)
(51)[βTβξβ]=[3113-1211-323+1213+12][αTξ],
and the corresponding geodesic curvature kgβ (Figure 4) reads
(52)kgβ=-6(2+3)(3+3)1/2.Figure 4
TheαTξ-pseudohyperbolical Smarandache curve β  and the curve α on H02(1).Also, Pseudohyperbolical Smarandache curves ofα and the curveα on H02 (1) with Figure 5 are shown.Figure 5
Pseudohyperbolical Smarandache curves ofα and the curve α on H02(1).



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