We introduce Smarandache curves according to the Lorentzian Darboux frame of a curve on spacelike surface in Minkowski 3-space R13. Also, we obtain the Sabban frame and the geodesic curvature of the Smarandache curves and give some characterizations on the curves when the curve α is an asymptotic curve or a principal curve. And we give an example to illustrate these curves.

1. Introduction

In the theory of curves in the Euclidean and Minkowski spaces, one of the interesting problems is the characterization of a regular curve. In the solution of the problem, the curvature functions κ and τ of a regular curve have an effective role. It is known that the shape and size of a regular curve can be determined 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]. The 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.

By definition, if the position vector of a curve β is composed by the Frenet frame’s vectors of another curve α, then the curve β is called a Smarandache curve [2]. Special Smarandache curves in the Euclidean and Minkowski spaces are studied by some authors [3–8]. For instance, the special Smarandache curves according to Darboux frame in E3 are characterized in [9].

In this paper, we define Smarandache curves according to the Lorentzian Darboux frame of a curve on spacelike surface in Minkowski 3-space R13. Inspired by the previous papers we investigate the geodesic curvature and the Sabban frame’s vectors of Smarandache curves. In Section 2, we explain the basic concepts of Minkowski 3-space and give Lorentzian Darboux frame that will be used throughout the paper. Section 3 is devoted to the study of four Smarandache curves, Tη-Smarandache curve, Tξ-Smarandache curve, ηξ-Smarandache curve, and Tηξ-Smarandache curve by considering the relationship with invariants kn, kg(s), and τg(s) of curve on spacelike surface in Minkowski 3-space R13. Also, we give some characterizations on the curves when the curve α is an asymptotic curve or a principal curve. Finally, we illustrate these curves with an example.

2. Basic Concepts

The Minkowski 3-space R13 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 〈·,·〉 is an indefinite metric, recall that a nonzero vector x∈R13 can have one of 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. For any x=(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).

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 pseudoarclength parameter s which is given by α:I⊂R→R13; 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.

Remark 1.

Let x=(x1,x2,x3), y=(y1,y2,y3), and z=(z1,z2,z3) be vectors in R13. Then
(3)(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-space R13, the following properties are satisfied [10]:

two timelike vectors are never orthogonal;

two null vectors are orthogonal if and only if they are linearly dependent;

timelike vector is never orthogonal to a null vector.

Let ϕ:U⊂R2→R13, ϕ(U)=M and γ:I⊂R→U be a spacelike embedding and a regular curve, respectively. Then we have a curve α on the surface M which is defined by α(s)=ϕ(γ(s)) and since ϕ is a spacelike embedding, we have a unit timelike normal vector field η along the surface M which is defined by
(4)η≡ϕx×ϕy∥ϕx×ϕy∥.
Since M is a spacelike surface, we can choose a future directed unit timelike normal vector field η along the surface M. Hence we have a pseudoorthonormal frame {T,η,ξ} which is called the Lorentzian Darboux frame along the curve α where ξ(s)=T(s)×η(s) is a unit spacelike vector. The corresponding Frenet formulae of α read
(5)[T′η′ξ′]=[0knkgkn0τg-kgτg0][Tηξ],
where kn(s)=-〈T′(s),η(s)〉, kg(s)=〈T′(s),ξ(s)〉, and τg(s)=-〈ξ′(s),η(s)〉 are the asymptotic curvature, the geodesic curvature, and the principal curvature of α on the surface M in R13, respectively, and s is arclength parameter of α. In particular, the following relations hold:
(6)T×η=ξ,η×ξ=-T,ξ×T=η.
Both kn and kg may be positive or negative. Specifically, kn is positive if α curves towards the normal vector η, and kg is positive if α curves towards the tangent normal vector ξ. Also, the curve α is characterized by kn, kg, and τg as follows:
(7)αis{anasymptoticcurveiffkn≡0,ageodesiccurveiffkg≡0,aprincipalcurveiffτg≡0.
Since α is a unit-speed curve, α¨ is perpendicular to T, but α¨ may have components in the normal and tangent normal directions:
(8)α¨=knη+kgξ.
These are related to the total curvature κ of α by the formula
(9)κ2=∥α¨∥2=kg2-kn2.
From (9) we can give the following proposition.

Proposition 3.

Let M be a spacelike surface in R13. Let α=α(s) be regular unit speed curves lying fully with the Lorentzian Darboux frame {T,η,ξ} on the surface M in R13. There is not a geodesic curve on M.

The pseudosphere with center at the origin and of radius r=1 in the Minkowski 3-space R13 is a quadric defined by
(10)S12={x→∈R13∣-x12+x22+x32=1}.
Let β:I⊂R→S12 be a curve lying fully in pseudosphere S12 in R13. Then its position vector β is a spacelike, which means that the tangent vector Tβ=β′ can be spacelike, timelike, or null. Depending on the causal character of Tβ, we distinguish the following three cases [5].

Case 1 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M132"><mml:mrow><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mi>β</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> is a unit spacelike vector).

Then we have orthonormal Sabban frame {β(s),Tβ(s),ξβ(s)} along the curve α, where ξβ(s)=-β(s)×Tβ(s) is the unit timelike vector. The corresponding Frenet formulae of β according to the Sabban frame read
(11)[β′Tβ′ξβ′]=[010-10-k¯g(s)0-k¯g(s)0][βTβξβ],
where k¯g(s)=det(β(s),Tβ(s),Tβ′(s)) is the geodesic curvature of β and s is the arclength parameter of β. In particular, the following relations hold:
(12)β×Tβ=-ξβ,Tβ×ξβ=β,ξβ×β=Tβ.

Case 2 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M143"><mml:mrow><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mi>β</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> is a unit timelike vector).

Hence we have orthonormal Sabban frame {β(s),Tβ(s),ξβ(s)} along the curve β, where ξβ(s)=β(s)×Tβ(s) is the unit spacelike vector. The corresponding Frenet formulae of β according to the Sabban frame read
(13)[β′Tβ′ξβ′]=[01010k¯g(s)0k¯g(s)0][βTβξβ],
where k¯g(s)=det(β(s),Tβ(s),Tβ′(s)) is the geodesic curvature of β and s is the arclength parameter of β. In particular, the following relations hold:
(14)β×Tβ=ξβ,Tβ×ξβ=β,ξβ×β=-Tβ.

Case 3 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M154"><mml:mrow><mml:msub><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mi>β</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula> is a null vector).

It is known that the only null curves lying on pseudosphere S12 are the null straight lines, which are the null geodesics.

3. Smarandache Curves according to Curves on a Spacelike Surface in Minkowski 3-Space <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M156"><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msubsup></mml:mrow></mml:math></inline-formula>

In the following section, we define the Smarandache curves according to the Lorentzian Darboux frame in Minkowski 3-space. Also, we obtain the Sabban frame and the geodesic curvature of the Smarandache curves lying on pseudosphere S12 and give some characterizations on the curves when the curve α is an asymptotic curve or a principal curve.

Definition 4.

Let α=α(s) be a spacelike curve lying fully on the spacelike surface M in R13 with the moving Lorentzian Darboux frame{T,η,ξ}. Then Tη-Smarandache curve of α is defined by
(15)β(s⋆(s))=12(aT(s)+bη(s)),
where a,b∈R0 and a2-b2=2.

Definition 5.

Let α=α(s) be a spacelike curve lying fully on the spacelike surface M in R13 with the moving Lorentzian Darboux frame{T,η,ξ}. Then Tξ-Smarandache curve of α is defined by
(16)β(s⋆(s))=12(aT(s)+bξ(s)),
where a,b∈R0 and a2+b2=2.

Definition 6.

Let α=α(s) be a spacelike curve lying fully on the spacelike surface M in R13 with the moving Lorentzian Darboux frame{T,η,ξ}. Then ηξ-Smarandache curve of α is defined by
(17)β(s⋆(s))=12(aη(s)+bξ(s)),
where a,b∈R0 and b2-a2=2.

Definition 7.

Let α=α(s) be a spacelike curve lying fully on the spacelike surface M in R13 with the moving Lorentzian Darboux frame{T,η,ξ}. Then Tηξ-Smarandache curve of α is defined by
(18)β(s⋆(s))=13(aT(s)+bη(s)+cξ(s)),
where a,b∈R0 and a2-b2+c2=3.

Thus, there are two following cases.

Case 4 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M196"><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:math></inline-formula> is an asymptotic curve).

Then, we have the following theorems.

Theorem 8.

Let α=α(s) be an asymptotic spacelike curve lying fully on the spacelike surface M in R13 with the moving Lorentzian Darboux frame{T,η,ξ}. Then

if akg+bτg≠0 for all s, then the Sabban frame {β,Tβ,ξβ} of the Tη -Smarandache curve β is given by
(19)[βTβξβ]=[a2b2000ϵϵb2ϵa20][Tηξ],
and the geodesic curvature k¯g of the curve β reads
(20)k¯g=aτg-bkg2,
where ϵ=sign(akg+bτg) for all s and
(21)ds*ds=ϵakg+bτg2;

if akg+bτg=0 for all s, then the Sabban frame {β,Tβ,ξβ} of the Tη-Smarandache curve β is a null geodesic.

Proof.

We assume that the curve α is an asymptotic curve. Differentiating (15) with respect to s and using (5) we obtain
(22)β′=dβds=12(akg+bτg)ξ,〈β′,β′〉=(akg+bτg)22.
Then, there are two following cases.

If akg+bτg≠0 for all s, since β′=(dβ/ds*)(ds*/ds), then the tangent vector Tβ of the curve β is a spacelike vector such that
(23)Tβ=ϵξ,
where
(24)ds*ds=ϵakg+bτg2.
On the other hand, from (15) and (23) it can be easily seen that
(25)ξβ=-β×Tβ=ϵb2T+ϵa2η
is a unit timelike vector.

Consequently, the geodesic curvature k¯g of the curve β=β(s*) is given by
(26)k¯g=det(β,Tβ,Tβ′)=aτg-bkg2.
From (15), (23), and (25) we obtain the Sabban frame {β,Tβ,ξβ} of β.

If akg+bτg=0 for all s, then β′ is null. So, the tangent vector Tβ of the curve β is a null vector. It is known that the only null curves lying on pseudosphere S12 are the null straight lines, which are the null geodesics.

In the theorems which follow, in a similar way as in Theorem 8, we obtain the Sabban frame {β,Tβ,ξβ} and the geodesic curvature k¯g of a spacelike Smarandache curve. We omit the proofs of Theorems 9, 10, and 11, since they are analogous to the proof of Theorem 8.

Theorem 9.

Let α=α(s) be an asymptotic spacelike curve lying fully on the spacelike surface M in R13 with the moving Lorentzian Darboux frame{T,η,ξ}. Then

if 2kg2-(bτg)2≠0 for all s, then the Sabban frame {β,Tβ,ξβ} of the Tξ-Smarandache curve β is given by
(27)[βTβξβ]=[a20b2-bkgϵ(2kg2-(bτg)2)bτgϵ(2kg2-(bτg)2)akgϵ(2kg2-(bτg)2)bτgϵ(2kg2-(bτg)2)-2kgϵ(2kg2-(bτg)2)abτgϵ(2kg2-(bτg)2)]×[Tηξ],
and the geodesic curvature k¯g of the curve β reads
(28)k¯g=((a2b3τg3-b4τg3+4bτgkg2)kg′iiiiiii+(b4τg2kg-a2b3τg2kg-4bkg3)τg′iiiiiii+(2ab2+2ab-4a)τgkg4iiiiiii-(ab4+ab3)τg3kg2+ab4τg5)×((4kg2-2(bτg)2)2)-1,
where ϵ=sign(2kg2-(bτg)2) for all s and
(29)ds*ds=ϵ2kg2-(bτg)22;

if akg+bτg=0 for all s, then the Sabban frame {β,Tβ,ξβ} of the Tη-Smarandache curve β is a null geodesic.

Theorem 10.

Let α=α(s) be an asymptotic spacelike curve lying fully on the spacelike surface M in R13 with the moving Lorentzian Darboux frame{T,η,ξ}. Then

if (bkg)2-2τg2≠0 for all s, then the Sabban frame {β,Tβ,ξβ} of the ηξ-Smarandache curve β is given by
(30)[βTβξβ]=[0a2b2-bkgϵ((bkg)2-2τg2)bτgϵ((bkg)2-2τg2)aτgϵ((bkg)2-2τg2)-2τg2ϵ((bkg)2-2τg2)b2kg2ϵ((bkg)2-2τg2)-abkg2ϵ((bkg)2-2τg2)]×[Tηξ],
and the geodesic curvature k¯g of the curve β reads
(31)k¯g=((42bτg3-(2a2b3+2b5)τgkg2)kg′+((2a2b3+2b5)kg3-42bτg2kg)τg′-2ab4kg5+22ab4τg2kg3+(42a-42ab2)τg4kg)×(2(b2kg2-2τg2)2)-1,
where ϵ=sign((bkg)2-2τg2) for all s and
(32)ds*ds=ϵ(bkg)2-2τg22;

if (bkg)2-2τg2=0 for all s, then the Sabban frame {β,Tβ,ξβ} of the Tη-Smarandache curve β is a null geodesic.

Theorem 11.

Let α=α(s) be an asymptotic spacelike curve lying fully on the spacelike surface M in R13 with the moving Lorentzian Darboux frame{T,η,ξ}. Then

if 3(kg2-τg2)+(bkg+aτg)2≠0 for all s, then the Sabban frame {β,Tβ,ξβ} of the Tηξ-Smarandache curve β is given by
(33)β=aT+bη+cξ3,Tβ=-ckgT+cτgη+(akg+bτg)ξϵ(3(kg2-τg2)+(bkg+aτg)2),ξβ=((b(akg+bτg)-c2τg)Tiiiii+(c2kg+a(akg+bτg))η-(acτg+bckg)ξ)×(ϵ(9(kg2-τg2)+3(bkg+aτg)2))-1,
and the geodesic curvature k¯g of the curve β reads
(34)k¯g=((b(akg+bτg)-c2τg)f1-(c2kg+a(akg+bτg))f2-(acτg+bckg)f3)×(3(3(kg2-τg2)+(bkg+aτg)2)2)-1,
where ϵ=sign(3(kg2-τg2)+(bkg+aτg)2) for all s and
(35)ds*ds=ϵ(bkg)2-2τg22,f1=((a2c-3c)τg2+abcτgkg)kg′+((3c-a2c)τgkg-abckg2)τg′+(3a+ab2)kg4+(a3+2ab2-3a)τg2kg2+(b3+2a2b+3b)τgkg3+(a2b-3b)τg3kg,f2=(abcτg2+(b2c+3c)τgkg)kg′-(abcτgkg+(3c+b2c)kg2)τg′+(3b-a2b)τg4-(b3+2a2b+3b)τg2kg2-(ab2+3a)τgkg3+(3a-2ab2-a3)τg3kg,f3=((-a3+3a+ab2)τg2+(b3+3b-a2b)τgkg)kg′+((-b3-3b+a2b)kg2+(a3-ab2-3a)τgkg)τg′+(3c+b2c)kg4+(3c-a2c)τg4+2abcτgkg3-2abcτg3kg+(a2c-6c-b2c)τg2kg2;

if 3(kg2-τg2)+(bkg+aτg)2=0 for all s, then the Sabban frame {β,Tβ,ξβ} of the Tη-Smarandache curve β is a null geodesic.

Case 5 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M305"><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:math></inline-formula> is a principal curve).

Then, we have the following theorems.

Theorem 12.

Let α=α(s) be a principal spacelike curve lying fully on the spacelike surface M in R13 with the moving Lorentzian Darboux frame{T,η,ξ}. Then the Tη-Smarandache curve β is spacelike and the Sabban frame {β,Tβ,ξβ} is given by
(36)[βTβξβ]=[a2b20bkn(akg)2-2kn2akn(akg)2-2kn2akg(akg)2-2kn2abkg2(akg)2-4kn2a2kg2(akg)2-4kn2-2kn2(akg)2-4kn2]×[Tηξ],
and the geodesic curvature k¯g of the curve β reads
(37)k¯g=abkgg1-a2kgg2-2kng32(a2kg2-2kn2)2,
where
(38)ds*ds=(akg)2-2kn22,g1=ba2kgknkg′-ba2kg2kn′+a3kg4-(a3+2a)kg2kn2+2akn4,g2=-a3kg2kn′+a3kgknkg′-ba2kg2kn2+2bkn4,g3=2akn2kg′-2akgknkn′-ba2kg3kn+2bkgkn3.

Proof.

We assume that the curve α is a principal curve. Differentiating (15) with respect to s and using (5) we obtain
(39)β′=dβds=12(bknT+aknη+akgξ),〈β′,β′〉=(akg)2-2kn22,
where from (9) (akg)2-2kn2>0 for all s. Since β′=(dβ/ds*)(ds*/ds), the tangent vector Tβ of the curve β is a spacelike vector such that
(40)Tβ=1(akg)2-2kn2(bknT+aknη+akgξ),
where
(41)ds*ds=(akg)2-2kn22.
On the other hand, from (15) and (40) it can be easily seen that
(42)ξβ=-β×Tβ=12(akg)2-4kn2(abkgT+a2kgη-2knξ)
is a unit timelike vector.

Consequently, the geodesic curvature k¯g of the curve β=β(s*) is given by
(43)k¯g=det(β,Tβ,Tβ′)=abkgg1-a2kgg2-2kng32(a2kg2-2kn2)2,
where
(44)g1=ba2kgknkg′-ba2kg2kn′+a3kg4-(a3+2a)kg2kn2+2akn4,g2=-a3kg2kn′+a3kgknkg′-ba2kg2kn2+2bkn4,g3=2akn2kg′-2akgknkn′-ba2kg3kn+2bkgkn3.
From (15), (40), and (42) we obtain the Sabban frame {β,Tβ,ξβ} of β.

In the theorems which follow, in a similar way as in Theorem 12, we obtain the Sabban frame {β,Tβ,ξβ} and the geodesic curvature k¯g of a spacelike Smarandache curve. We omit the proofs of Theorems 13 and 15, since they are analogous to the proof of Theorem 12.

Theorem 13.

Let α=α(s) be a principal spacelike curve lying fully on the spacelike surface M in R13 with the moving Lorentzian Darboux frame{T,η,ξ}. Then the Tξ-Smarandache curve β is spacelike and the Sabban frame {β,Tβ,ξβ} is given by
(45)[βTβξβ]=[a20b2-bkg2kg2-(akn)2akn2kg2-(akn)2akg2kg2-(akn)2-abkn4kg2-2(akn)22kg4kg2-2(akn)2-a2kn4kg2-2(akn)2]×[Tηξ],
and the geodesic curvature k¯g of the curve β reads
(46)k¯g=-abknh1+2kgh2+a2knh32(2kg2-(akn)2)2,
where
(47)ds*ds=2kg2-(akn)22,h1=-ba2kn2kg′+ba2kgknkn′+2akg4-a3kg2kn2-2akg2kn2+a3kn4,h2=2akgknkg′-2akg2kn′-2bkg3kn+ba2kgkn3,h3=a3kn2kg′-a3kgknkn′-2bkg4+ba2kg2kn2.

Theorem 14.

Let α=α(s) be a principal spacelike curve lying fully on the spacelike surface M in R13 with the moving Lorentzian Darboux frame {T,η,ξ}. Then

if akn-bkg≠0 for all s, the ηξ-Smarandache curve β is spacelike and the Sabban frame {β,Tβ,ξβ} is given by
(48)[βTβξβ]=[0a2b2ϵ000-ϵb2ϵa2][Tηξ],
and the geodesic curvature k¯g of the curve β reads
(49)k¯g=-akg+bkn2,
where ϵ=sign(akn-bkg) for all s and
(50)ds*ds=ϵakn-bkg2;

if akn-bkg=0 for all s, then the Sabban frame {β,Tβ,ξβ} of the ηξ-Smarandache curve β is a null geodesic.

Proof.

We assume that the curve α is a principal curve. Differentiating (17) with respect to s and using (5) we obtain
(51)β′=dβds*ds*ds=12(akn-bkg)T,〈β′,β′〉=(akn-bkg)22.
Then, there are two following cases.

If akn-bkg≠0 for all s, since β′=(dβ/ds*)(ds*/ds), then we obtain that the unit tangent vector Tβ of the curve β is a spacelike vector such that
(52)Tβ=ϵT,
where
(53)ds*ds=ϵakn-bkg2,
and ϵ=sign(akn-bkg).

On the other hand, from (17) and (52) it can be easily seen that
(54)ξβ=-β×Tβ=-ϵb2η+ϵa2ξ
is a unit timelike vector.

Consequently, the geodesic curvature k¯g of the curve β=β(s*) is given by
(55)k¯g=det(β,Tβ,Tβ′)=-akg+bkn2.
From (17), (52), and (54) we obtain the Sabban frame {β,Tβ,ξβ} of β.

If akn-bkg=0 for all s, then β′ is null. So, the tangent vector Tβ of the curve β is a null vector. It is known that the only null curves lying on pseudosphere S12 are the null straight lines, which are the null geodesics.

Theorem 15.

Let α=α(s) be a principal spacelike curve lying fully on the spacelike surface M in R13 with the moving Lorentzian Darboux frame{T,η,ξ}. Then the Tηξ-Smarandache curve β is spacelike and the Sabban frame {β,Tβ,ξβ} is given by
(56)β=13(aT+bη+cξ),Tβ=(bkn-ckg)T+aknη+akgξ3(kn2-kg2)+(bkg-ckn)2,ξβ=((abkg-ackn)T+((3+b2)kg-bckn)ηiiiiiiiiiii-((3-c2)kn+bckg)ξ)×(9(kn2-kg2)+3(bkg-ckn)2)-1,
and the geodesic curvature k¯g of the curve β reads
(57)k¯g=-((abkg-ackn)l1-((3+b2)kg-bckn)l2iiiiiiiii-((3-c2)kn+bckg)l3)×(2(a2kg2-2kn2)2)-1,
where
(58)ds*ds=3(kn2-kg2)+(bkg-ckn)23,l1=(3abkn2-3ackgkn)kg′+(3ackg2-3abkgkn)kn′+(bkn-ckg)(bkg-ckn)3+3bckg4-3bckn4+3(b2+c2)kgkn3-3(b2+c2)kg3kn,l2=((3b2+3c2+9)kn2-6bckgkn)kg′-((3b2+3c2+9)kgkn-6bckg2)kn′-(ac3+3ac)kn4+(3ab+3abc2)kgkn3+(ab3-3ab)kg3kn+(3ac-3ab2c)kg2kn2,l3=((3b2+3c2-9)kgkn-6bckn2)kg′+((9-3b2-3c2)kg2+6bckgkn)kn′+(ab3-3ab)kg4-(3ac+ac3)kgkn3+(3ac-3ab2c)kg3kn+(3ab+3abc2)kg2kn2.

Example 16.

Let us define a spacelike ruled surface (see Figure 1) in the Minkowski 3-space such as
(59)ϕ:U⊂R2⟶R13(s,u)⟶ϕ(s,u)=α(s)+ue(s),ϕ(s,u)=(-usinhs,s,-ucoshs),
where u∈(-1,1).

Then we get the Lorentzian Darboux frame {T,η,ξ} along the curve α as follows:
(60)T(s)=(0,1,0),η(s)≡11-u2(coshs,-u,sinhs),ξ(s)=11-u2(-sinhs,0,-coshs),
where ξ(s) is spacelike vectors and η(s) is a unit timelike vector.

Moreover, the geodesic curvature kg(s), the asymptotic curvature kn(s), and the principal curvature τg(s) of the curve α have the form
(61)kg(s)=〈T′(s),ξ(s)〉=0,kn(s)=-〈T′(s),η(s)〉=0,τg(s)=-〈ξ′(s),η(s)〉=-11-u2.
Taking a=3, b=1 and using (15), we obtain that the Tη-Smarandache curve β of the curve α is given by (see Figure 2(a))
(62)β(s⋆(s))=(coshs1-u2,3-u1-u2,sinhs1-u2).
Taking a=b=1 and using (16), we obtain that the Tξ-Smarandache curve β of the curve α is given by (see Figure 2(b))
(63)β(s⋆(s))=(-sinhs1-u2,1,-coshs1-u2).
Taking a=3, b=1 and using (17), we obtain that the ηξ-Smarandache curve β of the curve α is given by (see Figure 3(a))
(64)β(s⋆(s))=-11-u2×(sinhs-3coshs,3u,coshs-3sinhs).
Taking a=3, b=1, and c=1 and using (18), we obtain that the Tηξ-Smarandache curve β of the curve α is given by (see Figure 3(b))
(65)β(s⋆(s))=11-u2×(coshs-sinhs,3-3u2iiiiiiiii3-3u2-u,sinhs-coshs).
Also, the Smarandache curves on S2 of α for u=1/3 with Figure 4 are shown.

The spacelike surface ϕ(s,u).

(a) The Tη-Smarandache curve β on S12 for u=1/3. (b) The Tξ-Smarandache curve β on S12 for u=1/3.

(a) The ηξ-Smarandache curve β on S12 for u=1/3. (b) The Tηξ-Smarandache curve β on S12 for u=1/3.

The Smarandache curves on S12 for u=1/3.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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