# The Smarandache Curves onS12 and Its Duality on H02

Authors: Atakan Tuğkan Yakut; Murat Savaş; Tuğba Tamirci
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

We introduce special Smarandache curves based on Sabban frame onS12 and we investigate geodesic curvatures of Smarandache curves on de Sitter and hyperbolic spaces. The existence of duality between Smarandache curves on de Sitter space and Smarandache curves on hyperbolic space is shown. Furthermore, we give examples of our main results.


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## Body

## 1. Introduction

Curves as a subject of differential geometry have been intriguing for researchers throughout mathematical history and so they have been one of the interesting research fields. Regular curves play a central role in the theory of curves in differential geometry. In the theory of curves, there are some special curves such as Bertrand curves, Mannheim curves, involute and evolute curves, and pedal curves in which differential geometers are interested. A common approach to characterization of curves is to consider the relationship between the corresponding Frenet vectors of two curves. Bertrand and Mannheim curves are excellent examples for such cases. In the study of fundamental theory and the characterizations of space curves, the corresponding relations between the curves are a very fascinating problem. Recently, a new special curve is named according to the Sabban frame in the Euclidean unit sphere; Smarandache curve has been defined by Turgut and Yilmaz in Minkowski space-time [1]. Ali studied Smarandache curves with respect to the Sabban frame in Euclidean 3-space [2]. Then Taşköprü and Tosun studied Smarandache curves on S2 [3]. Smarandache curves have also been studied by many researchers [1, 4–7]. Smarandache curves are one of the most important tools in Smarandache geometry. Smarandache geometry has an important role in the theory of relativity and parallel universes. There are many results related to Smarandache curves in Euclidean and Minkowski spaces, but Smarandache curves are getting more tedious and complicated when de Sitter space is concerned. A regular curve in Minkowski space-time, whose position vector is associated with Frenet frame vectors on another regular curve, is called a Smarandache curve [1].In this paper, we define Smarandache curves on de Sitter surface according to the Sabban frame{α,t,η} in Minkowski 3-space. We obtain the geodesic curvatures and the expressions for the Sabban frame’s vectors of special Smarandache curves on de Sitter surface. Furthermore, we give some examples of special de Sitter and hyperbolic Smarandache curves in Minkowski 3-space.


## 2. Preliminaries

In this section, we prepare some definitions and basic facts. For basic concepts and details of properties, see [8, 9]. Consider R3 as a three-dimensional vector space. For any vectors x→=(x0,x1,x2) and y→=(y0,y1,y2) in R3 the pseudoscalar product of x→ and y→ is defined by 〈·,·〉L=-x0y0+x1y1+x2y2. It is called E13=(R3,〈·,·〉L) Minkowski 3-space. Recall that a nonzero vector x→∈E13 is spacelike if 〈x→,x→〉L>0, timelike if 〈x→,x→〉L<0, and null (lightlike) if 〈x→,x→〉L=0. The norm (length) of a vector x→∈E13 is given by ∥x→∥L=|〈x→,x→〉L| and two vectors x→ and y→ are said to be orthogonal if 〈x→,y→〉L=0. Next, we say 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 all s∈I. If ∥α′(s)∥L≠0 for every s∈I, then α is a regular curve in E13. A spacelike (timelike) regular curve α is parameterized by a pseudoarclength parameter s which is given by α:I⊂R→E13, and then the tangent vector α′(s) along α has unit length; that is, 〈α′(s),α′(s)〉L=1(〈α′(s),α′(s)〉L=-1) for all s∈I.Letx→=(x0,x1,x2), y→=(y0,y1,y2)∈E13. The Lorentzian vector cross-product is defined as follows:
(1)x→∧y→=|-e0e1e2x0x1x2y0y1y2|,
and also the following relations hold:(i)
〈x→∧y→,z→〉L=|x0x1x2y0y1y2z0z1z2|,(ii)
x→∧(y→∧z→)=〈x→,y→〉Lz→-〈x→,z→〉Ly→,where x→=(x0,x1,x2), y→=(y0,y1,y2), z→=(z0,z1,z2)∈E13.We now define de Sitter 2-space by(2)S12={x→∈R13:-x02+x12+x22=1}
and hyperbolic space in Minkowski 3-space by
(3)H02={x→∈R13:-x02+x12+x22=-1,x0>0}.
We can express a new frame different from the Frenet frame for a regular curve. Let α:I⊂R→S12 be a regular unit speed curve lying fully on S12. Then its position vector α is spacelike, which implies that the tangent vector α′=t is the unit timelike, spacelike, or null vector for all s∈I.In our work, we are concerned with the vectorα′=t which may be the unit timelike or spacelike.Letα:I⊂R→S12 be a regular unit speed curve lying fully on S12 for all s∈I and its position vector α a unit spacelike vector; then α′=t is a unit timelike and so η is a unit spacelike vector. In this case, the curve α is called a timelike curve. If α′=t is a unit spacelike vector, then η is a unit timelike vector. In this case, the curve α is called a spacelike curve and we have an orthonormal Sabban frame {α(s),t(s),η(s)} along the curve α, where η(s)=α(s)∧t(s) is the unit spacelike or timelike vector. Then Frenet formulas of α are given by
(4)[α′t′η′]=[010ε0εκg0εκg0][αtη],
where ε=±1, the curve α is timelike for ε=1 and spacelike for ε=-1, and κg(s) is the geodesic curvature of α on S12, which is given by κg(s)=det⁡(α(s),t(s),t′(s)), where s is the arc length parameter of α. This relation is also given by [10, 11] for ε=1. In particular, by using equation (ii), the following relations hold:
(5)εα=t∧η,t=α∧η,η=α∧t.Definition 1.
A unit speed regular curveβ(s¯(s)) lying fully in Minkowski 3-space, whose position vector is associated with Sabban frame vectors on another regular curve α(s), is called a Smarandache curve [1].Based on this definition, if a regular unit speed curveα:I⊂R→S12 is lying fully on S12 for all s∈I and its position vector α is a unit spacelike, then the Smarandache curve β=β(s¯(s)) of curve α is a regular unit speed curve lying fully on S12 or H02. In this case we have the following:(a)
the Smarandache curveβ(s¯(s)) may be a timelike curve on S12,(b)
the Smarandache curveβ(s¯(s)) may be a spacelike curve on S12, or(c)
the Smarandache curveβ(s¯(s)) is in H02 for all s∈I.Let {α,t,η} and {β,tβ,ηβ} be the moving Sabban frames of α and β, respectively. Then we have the following definitions and theorems of Smarandache curves β=β(s¯(s)) given in Section 3. In Section 3, we deal with Smarandache curves on de Sitter and hyperbolic spaces for timelike curves. Similar results are given for spacelike curves in the Appendix.


## 3. De Sitter and Hyperbolic Smarandache Curves for Timelike Curves

In this section we give different Smarandache curves on de Sitter and hyperbolic spaces in Minkowski-space. Letα be a timelike curve on S12; then the Smarandache partner curve of α is either timelike/spacelike or hyperbolic curve. We refer to the hyperbolic Smarandache curve of a timelike curve α as the hyperbolic duality of α.To avoid repetition we useε=±1 in the following theorems in this section. If we take ε=1, then the Smarandache curve β is timelike or spacelike, and if we take ε=-1, then β is hyperbolic.Definition 2.
Letα=α(s) be a unit speed regular timelike curve lying fully on S12. The curve β:I⊂R→S12(β:I⊂R→H02) of α defined by
(6)β(s¯(s))=12(c1α(s)+c2η(s))
is called the αη-Smarandache curve of α and fully lies on S12, where c1,c2∈R∖{0} and ε(c12+c22)=2. If ε=-1; then the hyperbolic αη-Smarandache curve is undefined since the equation (c12+c22)=-2 has no solution in R.Theorem 3.
Letα:I⊂R→S12 be a regular unit speed timelike curve lying fully on S12 with the Sabban frame {α,t,η} and geodesic curvature κg. If β:I⊂R→S12 is the αη-timelike Smarandache curve of α, then the relationships between the Sabban frames of α and its αη-Smarandache curve are given by
(7)[βtβηβ]=[c120c220ε0c2ε20c1ε2][αtη],
where ε=±1 and its geodesic curvature κgβ is given by
(8)κgβ=c1κg-c2|c1+c2κg|.Proof.
By taking the derivative of (6) with respect to s and by using (4), we get
(9)β′(s¯(s))=dβds¯ds¯ds=12(c1+c2κg)t
or equivalently
(10)tβds¯ds=12(c1+c2κg)t,
where
(11)ds¯ds=|c1+c2κg|2.
Hence, the unit timelike tangent vector of the curve β is given by
(12)tβ=εt,
where ε=1 if c1+c2κg>0 for all s and ε=-1 if c1+c2κg<0 for all s. From (6) and (12) we get
(13)ηβ=β∧tβ=ε2(c2α+c1η).
It is easily seen that ηβ is a unit spacelike vector. On the other hand, differentiating (12) with respect to s, we find
(14)dtβds¯ds¯ds=ε(α+κgη)
and by combining (11) and (14) we have
(15)tβ′=2ε|c1+c2κg|(α+κgη).
Consequently, the geodesic curvature κgβ of the curve β=β(s¯) is given by
(16)κgβ=det⁡(β,tβ,tβ′)=c1κg-c2|c1+c2κg|.Corollary 4.
Letα:I⊂R→S12 be a regular unit speed timelike curve lying fully on S12. Then the αη-spacelike Smarandache curve α:I⊂R→S12 of α does not exist.Definition 5.
Letα=α(s) be a regular unit speed timelike curve lying fully on S12. Then the αt-Smarandache curve β:I⊂R→S12(β:I⊂R→H02) of α is defined by
(17)β(s¯(s))=12(c1α(s)+c2t(s)),
where c1,c2∈R∖{0} and ε(c12-c22)=2.Theorem 6.
Letα:I⊂R→S12 be a regular unit speed timelike curve lying fully on S12 with the Sabban frame {α,t,η} and geodesic curvature κg. If β:I⊂R→S12(β:I⊂R→H02) is the αt-timelike (hyperbolic) Smarandache curve of α, then its frame {β,tβ,ηβ} is given by(18)[βtβηβ]=[c12c220c22-ε(c22κg2)c12-ε(c22κg2)c2κg2-ε(c22κg2)-c22κg2(2-ε(c22κg2))-c1c2κg2(2-ε(c22κg2))22(2-ε(c22κg2))][αtη].The geodesic curvature κgβ of curve β is given by
(19)κgβ=1(2-ε(c22κg2))5/2(c22κgλ1-c1c2κgλ2+2λ3),
where
(20)λ1=εc23κgκg′+c1(2-εc22κg2),λ2=εc1c22κgκg′+(c2+c2κg2)(2-εc22κg2),λ3=εc23κg2κg′+(c1κg+c2κg′)(2-εc22κg2).Proof.
We takeε=1. By taking the derivative of (17) with respect to s and by using (4), we get
(21)β′(s¯(s))=dβds¯ds¯ds=12(c2α+c1t+c2κgη)
or equivalently
(22)tβds¯ds=12(c2α+c1t+c2κgη).
Taking the Lorentzian inner product in (22) we have
(23)〈tβ,tβ〉L(ds¯ds)2=12(c22κg2-2).
If c22κg2<2, then tβ is a timelike vector. So
(24)ds¯ds=2-c22κg22.
Therefore, the unit timelike tangent vector of the curve β is given by
(25)tβ=12-c22κg2(c2α+c1t+c2κgη).
On the other hand, from (17) and (25) it can be easily seen that
(26)ηβ=β∧tβ=14-2c22κg2(-c22κgα-c1c2κgt+2η)
is a unit spacelike vector. Differentiating (25) with respect to s, we obtain
(27)dtβds¯ds¯ds=1(2-c22κg2)3/2(λ1α+λ2t+λ3η),
where
(28)λ1=c23κgκg′+c1(2-c22κg2),λ2=c1c22κgκg′+(c2+c2κg2)(2-c22κg2),λ3=c23κg2κg′+(c1κg+c2κg′)(2-c22κg2),
and by combining (24) and (26) we get
(29)tβ′=2(2-c22κg2)2(λ1α+λ2t+λ3η).
Consequently, we have
(30)κgβ=det⁡(β,tβ,tβ′)=1(2-c22κg2)5/2(c22κgλ1-c1c2κgλ2+2λ3).
The proof of ε=-1 case is similar.The following corollary is proved by the same methods as the above theorem.Corollary 7.
Letα:I⊂R→S12 be a regular unit speed timelike curve lying fully on S12 with the Sabban frame {α,t,η} and geodesic curvature κg. If β:I⊂R→S12 is the αt-spacelike Smarandache curve of α, then its frame {β,tβ,ηβ} is given by
(31)[βtβηβ]=[c12c220c2c22κg2-2c1c22κg2-2c2κgc22κg2-2-c22κg2(c22κg2-2)-c1c2κg2(c22κg2-2)22(c22κg2-2)]×[αtη].
The geodesic curvature κgβ of curve β is given by
(32)κgβ=1(c22κg2-2)5/2(c22κgλ1-c1c2κgλ2+2λ3),
where λ1,λ2, and λ3 can be calculated as in Theorem 6.Definition 8.
Letα:I⊂R→S12 be a regular unit speed timelike curve lying fully on S12. Then the tη-Smarandache curve β:I⊂R→S12(β:I⊂R→H02) of α is defined by
(33)β(s¯(s))=12(c1t(s)+c2η(s)),
where c1,c2∈R∖{0} and ε(-c12+c22)=2.Theorem 9.
Letα:I⊂R→S12 be a regular unit speed timelike curve lying fully on S12 with the Sabban frame {α,t,η} and geodesic curvature κg. If β:I⊂R→S12(β:I⊂R→H02) is the tη-timelike (hyperbolic) Smarandache curve of α, then its frame {β,tβ,ηβ} is given by
(34)[βtβηβ]=[0c12c22c12κg2-εc12c2κg2κg2-εc12c1κg2κg2-εc12ε2κg2(2κg2-εc12)c1c22(2κg2-εc12)-c122(2κg2-εc12)]×[αtη].
The geodesic curvature κgβ of curve β is given by
(35)κgβ=1(2κg2-εc12)5/2(-ε2κgλ1+c1c2λ2-c12λ3),
where
(36)λ1=-2c1κgκg′+c2κg(2κg2-εc12),λ2=-2c2κg2κg′+(c1+c2κg′+c1κg2)(2κg2-εc12),λ3=-2c1κg2κg′+(c2κg2+c1κg′)(2κg2-εc12).Proof.
Letε=1. By taking the derivative of (33) with respect to s and by using (4), we get
(37)β′(s¯(s))=dβds¯ds¯ds=12(c1α+c2κgt+c1κgη)
or equivalently
(38)tβds¯ds=12(c1α+c2κgt+c1κgη).
Taking the Lorentzian inner product in (38) we have
(39)〈tβ,tβ〉L(ds¯ds)2=12(c12-2κg2)
and tβ is a unit timelike vector for 2κg2>c12. It follows that
(40)ds¯ds=2κg2-c122.
Therefore, the unit timelike tangent vector of the curve β is given by
(41)tβ=12κg2-c12(c1α+c2κgt+c1κgη).
On the other hand, from (33) and (41) it can be easily seen that
(42)ηβ=β∧tβ=14κg2-2c12(2κgα+c1c2t-c12η)
is a unit spacelike vector. Differentiating (41) with respect to s, we find
(43)dtβds¯ds¯ds=1(2κg2-c12)3/2(λ1α+λ2t+λ3η),
where
(44)λ1=-2c1κgκg′+c2κg(2κg2-c12),λ2=-2c2κg2κg′+(c1+c2κg′+c1κg2)(2κg2-c12),λ3=-2c1κg2κg′+(c2κg2+c1κg′)(2κg2-c12),
and by combining (40) and (43) we get
(45)tβ′=2(2κg2-c12)2(λ1α+λ2t+λ3η).
As a result, we have
(46)κgβ=det⁡(β,tβ,tβ′)=1(2κg2-c12)5/2(-2κgλ1+c1c2λ2-c12λ3).
The proof of ε=-1 case is similar.Corollary 10.
Letα:I⊂R→S12 be a regular unit speed timelike curve lying fully on S12 with the Sabban frame {α,t,η} and geodesic curvature κg. If β:I⊂R→S12 is the tη-spacelike Smarandache curve of α, then its frame {β,tβ,ηβ} is given by
(47)[βtβηβ]=[0c12c22c1c12-2κg2c2κgc12-2κg2c1κgc12-2κg2ε2κg2(c12-2κg2)c1c22(c12-2κg2)-c122(c12-2κg2)]×[αtη].
The geodesic curvature κgβ of curve β is given by
(48)κgβ=1(c12-2κg2)5/2(-2κgλ1+c1c2λ2-c12λ3),
where λ1,λ2, and λ3 can be calculated as in Theorem 9.Definition 11.
Letα:I⊂R→S12 be a regular unit speed timelike curve lying fully on S12. Then the αtη-Smarandache curve β:I⊂R→S12(β:I⊂R→H02) of α is defined by
(49)β(s¯(s))=13(c1α(s)+c2t(s)+c3η(s)),
where c1,c2,c3∈R∖{0} and ε(c12-c22+c32)=3.Theorem 12.
Letα:I⊂R→S12 be a regular unit speed timelike curve lying fully on S12 with the Sabban frame {α,t,η} and geodesic curvature κg. If β:I⊂R→S12  (β:I⊂R→H02) is the αtη-timelike (hyperbolic) Smarandache curve of α, then its frame {β,tβ, ηβ} is given by(50)[βtβηβ]=[c13c23c33c2εAc1+c3κgεAc2κgεA-c22κg+c3(c1+c3κg)3εA-c1c2κg+c2c33εAc1(c1+c3κg)-c223εA][αtη],where A=(c1+c3κg)2-c22-c22κg2. If we take ε=1or-1, then the Smarandache curve β is timelike or hyperbolic, respectively. Furthermore, the geodesic curvature κgβ of curve β is given by(51)κgβ=((c22κg-c32κg-c1c3)λ1+(-c1c2κg+c2c3)λ2+(c12+c1c3κg-c22)λ3)×((ε((c1+c3κg)2-c22-c22κg2))5/2)-1,
where
(52)λ1=ε((c1+c3κg)2c2(-c3κg′(c1+c3κg)+c22κgκg′)+(c1+c3κg)((c1+c3κg)2-c22-c22κg2)),λ2=ε((c1+c3κg)2(c1+c3κg)(-c3κg′(c1+c3κg)+c22κgκg′)+(c2+c3κg′+c2κg2)((c1+c3κg)2-c22-c22κg2)),λ3=ε((c1+c3κg)2c2κg(-c3κg′(c1+c3κg)+c22κgκg′)+(κg(c1+c3κg)+c2κg′)×((c1+c3κg)2-c22-c22κg2)).Proof.
We takeε=1. By taking the derivative of (49) with respect to s and using (4), we get
(53)β′(s¯(s))=dβds¯ds¯ds=13(c2α+(c1+c3κg)t+c2κgη)
or equivalently
(54)tβds¯ds=13(c2α+(c1+c3κg)t+c2κgη).
Taking the Lorentzian inner product in (54) we have
(55)〈tβ,tβ〉L(ds¯ds)2=13(c22+c22κg2-(c1+c3κg)2).
For (c1+c3κg)2>c22+c22κg2, tβ is a unit timelike vector. It follows that
(56)ds¯ds=(c1+c3κg)2-c22-c22κg23.
Therefore, the unit timelike tangent vector of the curve β is given by
(57)tβ=1(c1+c3κg)2-c22-c22κg2(c2α+(c1+c3κg)t+c2κgη).
On the other hand, taking the cross-product of (49) with (57) it can be easily seen that
(58)ηβ=β∧tβ=((-c22κg+c3(c1+c3κg))α+(c2c3-c1c2κg)t+(c1(c1+c3κg)-c22)η)×(3(c1+c3κg)2-3c22-3c22κg2)-1.
This means that the ηβ is a unit spacelike vector. In order to obtain the tangent vector of β let us differentiate (57) with respect to s. We find
(59)dtβds¯ds¯ds=1((c1+c3κg)2-c22-c22κg2)3/2(λ1α+λ2t+λ3η),
where
(60)λ1=c2(-c3κg′(c1+c3κg)+c22κgκg′)+(c1+c3κg)((c1+c3κg)2-c22-c22κg2),λ2=(c1+c3κg)(-c3κg′(c1+c3κg)+c22κgκg′)+(c2+c3κg′+c2κg2)((c1+c3κg)2-c22-c22κg2),λ3=c2κg(-c3κg′(c1+c3κg)+c22κgκg′)+(κg(c1+c3κg)+c2κg′)×((c1+c3κg)2-c22-c22κg2),
and by combining (56) and (59) we get
(61)tβ′=3((c1+c3κg)2-c22-c22κg2)2(λ1α+λ2t+λ3η).
Finally, the geodesic curvatureκgβ of the curve β=β(s¯(s)) is given by
(62)κgβ=det⁡(β,tβ,tβ′)=((c22κg-c32κg-c1c3)λ1+(-c1c2κg+c2c3)λ2+(c12+c1c3κg-c22)λ3)×(((c1+c3κg)2-c22-c22κg2)5/2)-1.
The proof of ε=-1 case is similar.Corollary 13.
Letα:I⊂R→S12 be a regular unit speed timelike curve lying fully on S12 with the Sabban frame {α,t,η} and geodesic curvature κg. If β:I⊂R→S12 is the αtη-spacelike Smarandache curve of α, then, for A=c22+c22κg2-(c1+c3κg)2, its frame {β,tβ,ηβ} is given by(63)[βtβηβ]=[c13c23c33c2Ac1+c3κgAc2κgA-c22κg+c3(c1+c3κg)3A-c1c2κg+c2c33Ac1(c1+c3κg)-c223A][αtη].Furthermore, the geodesic curvature κgβ of curve β is given by
(64)κgβ=((c22κg-c32κg-c1c3)λ1+(-c1c2κg+c2c3)λ2+(c12+c1c3κg-c22)λ3)×((c22+c22κg2-(c1+c3κg)2)5/2)-1,
where λ1,λ2, and λ3 can be calculated as in Theorem 12.Example 14.
Let us consider a unit speed timelike curveα on S12 defined by
(65)α(s)=(2sinh⁡⁡s,cosh⁡⁡s,sinh⁡⁡s).
Then the orthonormal Sabban frame {α(s),t(s),η(s)} of α can be calculated as follows:
(66)α(s)=(2sinh⁡⁡s,cosh⁡⁡s,sinh⁡⁡s),t(s)=(2cosh⁡⁡s,sinh⁡⁡s,cosh⁡⁡s),η(s)=(-1,0,-2).
The geodesic curvature of α is 0. In terms of the definitions, we obtain Smarandache curves according to Sabban frame on S12.
Firstly, when we takec1=1 and c2=1, then the timelike αη-Smarandache curve is given by
(67)β(s¯(s))=12(2sinh⁡⁡s-1,cosh⁡⁡s,sinh⁡⁡s-2)
and the Sabban frame of the αη-Smarandache curve is given by
(68)[βtβηβ]=[1201201012012][αtη],
and its geodesic curvature κgβ is -1. Here the hyperbolic αη-Smarandache curve is undefined.
Secondly, when we takec1=3 and c2=7, then the timelike αt-Smarandache curve is given by
(69)β(s¯(s))=12(32sinh⁡⁡s+14cosh⁡⁡s,3cosh⁡⁡s+7sinh⁡⁡s,3sinh⁡⁡s+7cosh⁡⁡s),
and if we take c1=7 and c2=3, then the hyperbolic αt-Smarandache curve is given by
(70)β(s¯(s))=12(14sinh⁡⁡s+32cosh⁡⁡s,7cosh⁡⁡s+3sinh⁡⁡s,7sinh⁡⁡s+3cosh⁡⁡s).
Thirdly, when we take c1=2 and c2=2, then the spacelike tη-Smarandache curve is given by
(71)β(s¯(s))=(2(cosh⁡⁡s-1),sinh⁡⁡s,cosh⁡⁡s-2)
and if we take c1=2 and c2=2, then the hyperbolic tη-Smarandache curve is given by
(72)β(s¯(s))=(2cosh⁡⁡s-1,2sinh⁡⁡s,2(cosh⁡⁡s-1)).
Finally, when we take c1=2, c2=2, and c3=1, then the αtη-Smarandache curve is a timelike curve and given by
(73)β(s¯(s))=13(22sinh⁡⁡s+2cosh⁡⁡s-1,2cosh⁡⁡s+2sinh⁡⁡s,2sinh⁡⁡s+2cosh⁡⁡s-2),
and if we take c1=2, c2=22, and c3=3, then the αtη-Smarandache curve is a hyperbolic curve and given by
(74)β(s¯(s))=13(2sinh⁡⁡s+4cosh⁡⁡s-3,2cosh⁡⁡s+22sinh⁡⁡s,2sinh⁡⁡s+22cosh⁡⁡s-6).
On the other hand, in the last case, if we take c1=1, c2=2, and c3=2 (i.e., c1<c2<c3), then the αtη-Smarandache curve is spacelike and given by
(75)β(s¯(s))=13(2sinh⁡⁡s+2cosh⁡⁡s-2,cosh⁡⁡s+2sinh⁡⁡s,sinh⁡⁡s+2cosh⁡⁡s-22).The Sabban frames and geodesic curvatures ofαt, tη, and αtη-Smarandache curves can be easily obtained by using a similar way to the above. Also we give the curve α and its Smarandache partners in Figure 1.Figure 1
Smarandache curves of a timelike curveα.



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*Source: 193586-2014-08-21.xml*