The Smarandache Curves on H

2016·0.63 MB·

by  Murat Savas

Gazi University Journal of Science
GU J Sci
29(1):69-77 (2016)

𝟐
The Smarandache Curves on 𝑯
𝟎

Murat SAVAS1 , Atakan Tugkan YAKUT2,, Tugba TAMIRCI1

1 Gazi University, Faculty of Sciences, Department of Mathematics, 06500 Teknikokullar,
Ankara, Turkey
2 Nigde University, Faculty of Arts and Sciences, Department of Mathematics, 51350,
Nigde, Turkey

Received: 09/07/2015 Accepted: 29/12/2015
ABSTRACT
In this study, we give special Smarandache curves according to the Sabban frame in hyperbolic space and new
Smarandache partners in de Sitter space. The existence of duality between Smarandache curves in hyperbolic and de
Sitter space is obtained. We also describe how we can depict picture of Smarandache partners in de Sitter space of a
curve in hyperbolic space. Finally, two examples are given to illustrate our main results.
Key words: Smarandache curves, de Sitter space, Sabban frame, Minkowski space.

1. INTRODUCTION
special curves, such as Bertrand, Mannheim, involute,
Regular curves have an important role in the theory of
evolute, and pedal curves. In the light of the literature, in
curves in differential geometry and relativity theory. In the
[11] authors introduced a special curve by Frenet-Serret
geometry of regular curves in Euclidean or Minkowskian
frame vector fields in Minkowski space-time. The new
spaces, it is well-known that one of the most important
special curve, which is named Smarandache curve
problem is the characterization and classification of these
according to the Sabban frame in the Euclidean unit sphere,
curves. In the theory of regular curves, there are some
is defined by Turgut and Yilmaz in Minkowski space-time
[11]. Smarandache curves in Euclidean or non-Euclidean
Corresponding author, e-mail: [email protected]
70 GU J Sci, 29(1):69-77 (2016)/ Murat SAVAS, Atakan Tugkan YAKUT, Tugba TAMIRCI

spaces have been recently of particular interest for be orthogonal if 〈𝑥,𝑦〉 =0. Next, we say that an arbitrary
𝐿
researchers. In Euclidean differential geometry, curve 𝛼=𝛼(𝑠) in 𝐸3 can locally be spacelike, timelike,
1
Smarandache curves of a curve are defined to be or null(lightlike) if all of its velocity vectors 𝛼′(𝑠) are,
combination of its position, tangent, and normal vectors. respectively, spacelike, timelike, or null for all 𝑠𝜖Ι. If
These curves have been also studied widely [1, 4, 6, 9, 11, ‖𝛼′(𝑠)‖ ≠0 for every 𝑠𝜖Ι, then α is a regular curve in
𝐿
12]. Smarandache curves play an important role in 𝐸3. A spacelike(timelike) regular curve 𝛼 is
1
Smarandache geometry. They are the objects of parameterized by a pseudo-arc length parameter 𝑠 , which
Smarandache geometry, i.e. a geometry which has at least is given by 𝛼:Ι⊂ℝ→𝐸3, and then the tangent vector
1
one Smarandachely denied axiom [2]. An axiom is said to 𝛼′(𝑠) along 𝛼 has unit length, that is
be Smarandachely denied if it behaves in at least two
different ways within the same space. Smarandache 〈𝛼′(𝑠),𝛼′(𝑠)〉𝐿 =1( 〈𝛼′(𝑠),𝛼′(𝑠)〉𝐿 =−1)
geometry has a significant role in the theory of relativity
for all 𝑠𝜖Ι. Let
and parallel universes. Ozturk U., et al. studied
Smarandache curves in hyperbolic space but they don’t 𝑥=(𝑥 ,𝑥 ,𝑥 ), 𝑦=(𝑦 ,𝑦 ,𝑦 ), 𝑧=(𝑧 ,𝑧 ,𝑧 )𝜖 𝐸3.
1 2 3 1 2 3 1 2 3 1
give dual Smarandache partners of these curves in de Sitter The Lorentzian pseudo-vector cross product is defined as
space [6]. We answer it for curves in hyperbolic space and follows:
show the Smarandache partners curve of these curves in de
𝑥∧𝑦=(−𝑥 𝑦 +𝑥 𝑦 , 𝑥 𝑦 −𝑥 𝑦 , 𝑥 𝑦 −𝑥 𝑦 ) (1)
Sitter space. We explain the Smarandache de Sitter duality 2 3 3 2 3 1 1 3 1 2 2 1
of curves in hyperbolic space. In this paper, we give the We remark that the following relations hold:
Smarandache partner curves in de Sitter space according to
the Sabban frame {𝛼,𝑡,𝜉} of a curve in hyperbolic space. (i) 〈𝑥∧𝑦,𝑧〉𝐿 =det (𝑥 𝑦 𝑧)
We obtain the geodesic curvatures and the expressions for
(ii) 𝑥∧(𝑦∧𝑧)=〈𝑥,𝑦〉 𝑧−〈𝑥,𝑧〉 𝑦
the Sabban frame’s vectors of special Smarandache curves 𝐿 𝐿
on de Sitter surface. In particular, we see that the timelike Let 𝛼:Ι⊂ℝ→𝐻2 be a regular unit speed curve lying
0
𝛼𝜉-Smarandache curve of a curve 𝛼 does not exist in de fully in 𝐻2. Then its position vector α is a timelike vector,
0
Sitter space. We give some examples of the Smarandache which implies that the tangent vector 𝑡=𝛼′ and normal
curves in hyperbolic space and its dual Smarandache vector 𝜉 are unit spacelike vector for all 𝑠𝜖Ι. We have the
curves in de Sitter space. Furthermore, we give some orthonormal Sabban frame {𝛼(𝑠),𝑡(𝑠),𝜉(𝑠)} along the
examples of special hyperbolic and de Sitter Smarandache curve 𝛼, where 𝜉(𝑠)=𝛼(𝑠)∧𝑡(𝑠) is the unit spacelike
curves, which are found in the study of Yakut et al. [12]. In vector. The corresponding Frenet formula of 𝛼, according
her Master thesis [9], Tamirci also studied the curves in de to the Sabban frame, is given by
Sitter and hyperbolic spaces using a similar framework.
𝛼′(𝑠)=𝑡(𝑠)
2. PRELIMINARIES {𝑡′(𝑠)=𝛼(𝑠)+𝜅𝑔(𝑠) 𝜉(𝑠) (2)
In this section, we use the basic notions and results in 𝜉′(𝑠)=−𝜅 (𝑠)𝑡(𝑠)
𝑔
Lorentzian geometry. For more detailed concepts, see

[7,8]. Let ℝ3 be the 3-dimensional vector space equipped
with the scalar product 〈 ,〉 which is defined by where 𝜅 (𝑠)=det (𝛼(𝑠),𝑡(𝑠),𝑡′(𝑠)) is the geodesic
𝑔
〈𝑥,𝑦〉 =−𝑥 𝑦 +𝑥 𝑦 +𝑥 𝑦 . curvature of 𝛼 on 𝐻02 and 𝑠 is the arc length parameter
𝐿 1 1 2 2 3 3 of α. In particular, the following relations hold:
The space 𝐸3=(ℝ3,〈 ,〉 ) is a pseudo-Euclidean space,
1 𝐿 𝜉=𝛼∧𝑡, −𝛼=𝑡∧𝜉, 𝑡=𝜉∧𝛼 (3)
or Minkowski 3-space. The unit pseudo-sphere (de Sitter
space) with index one 𝑆12 in 𝐸13 is given by Now we define a new curve 𝛽:Ι⊂ℝ→𝑆12 to be a
regular unit speed curve lying fully on 𝑆2 for all 𝑠𝜖Ι such
𝑆2 ={𝑥𝜖𝐸3|〈𝑥,𝑥〉 =1}. 1
1 1 𝐿 that its position vector 𝛽 is a unit spacelike vector
The unit pseudo-hyperbolic space according to the combination of the position, tangent, and
normal vectors of 𝛼. In this case 𝛽′=𝑡 may be a unit
𝐻2 ={𝑥𝜖𝐸3|〈𝑥,𝑥〉 =−1} 𝛽
0 1 𝐿 timelike or spacelike vector.
has two connected components 𝐻02,+ and 𝐻02,−. Each of Definition 2.1. A unit speed regular curve 𝛽(𝑠̅(𝑠)) lying
them can be taken as a model for the 2-dimensional fully in Minkowski 3-space, whose position vector is
hyperbolic space 𝐻02. In this paper, we take 𝐻02,+=𝐻02. associated with Sabban frame vectors on another regular
Recall that a nonzero vector 𝑥𝜖𝐸3 is spacelike if curve 𝛼(𝑠), is called a Smarandache curve[11].
1
〈𝑥,𝑥〉 >0, timelike if 〈𝑥,𝑥〉 <0, and null (lightlike) if
𝐿 𝐿 In the light of this definition, if a regular unit speed curve
〈𝑥,𝑥〉 =0. The norm (length) of a vector 𝑥𝜖𝐸3 is given
𝐿 1 𝛼:Ι⊂ℝ→𝐻2 is lying fully on 𝐻2 for all 𝑠𝜖Ι and its
by ‖𝑥‖ =√|〈𝑥,𝑥〉 | and two vectors 𝑥 and 𝑦 are said to 0 0
𝐿 𝐿 position vector 𝛼 is a unit timelike vector, then the
GU J Sci, 29(1):69-77 (2016)/ Murat SAVAS, Atakan Tugkan YAKUT, Tugba TAMIRCI 71

Smarandache curve 𝛽=𝛽(𝑠̅(𝑠)) of the curve 𝛼 is a where
regular unit speed curve lying fully in 𝑆2 or 𝐻2. In our
work we are interested in curves lying in1 𝑆2 an0d so we 𝑑𝑠̅=|𝑐1−𝑐2𝜅𝑔| (8)
1 𝑑𝑠 √2
have the following:
By substituting (8) into (7) we obtain a simple form of Eq.
a) The Smarandache curve 𝛽(𝑠̅(𝑠)) may be a spacelike 7 as follows,
curve on 𝑆2 or,
1
𝑡 =𝜀𝑡 (9)
𝛽
b) The Smarandache curve 𝛽(𝑠̅(𝑠)) may be a timelike
curve on 𝑆2 for all 𝑠𝜖Ι. where𝜀=1 if 𝑐 −𝑐 𝜅 >0for all 𝑠 and 𝜀=−1 if
1 1 2 𝑔
𝑐 −𝑐 𝜅 <0 for all 𝑠. It can be easily seen that the
Let {𝛼,𝑡,𝜉} and {𝛽,𝑡 ,𝜉 } be the moving Sabban frames 1 2 𝑔
𝛽 𝛽 tangent vector 𝑡 is a unit spacelike vector. Taking the
of 𝛼 and 𝛽, respectively. Then we have the following 𝛽
Lorentzian vector cross product of (4) with (9) we have
definitions and theorems of Smarandache curves
𝛽=𝛽(𝑠̅(𝑠)). 𝜉 =𝛽∧𝑡
𝛽 𝛽
3. CURVES ON 𝑯𝟐 AND ITS SPACELIKE 𝜀
𝟎 = (𝑐 𝛼+𝑐 𝜉) (10)
SMARANDACHE PARTNERS ON 𝑺𝟐 √2 2 1
𝟏
It is easily seen that 𝜉 is a unit timelike vector. On the
Let 𝛼 be a regular unit speed curve on 𝐻2. Then the 𝛽
0 other hand, by taking the derivative of the equation (9) with
Smarandache partner curve of α is either in de Sitter or in
respect to 𝑠, we find
hyperbolic space. β is called de Sitter dual of α in
hyperbolic space. In this section we obtain the spacelike 𝑑𝑡𝛽𝑑𝑠̅=𝜀(𝛼+𝜅 𝜉) (11)
Smarandache partners in de Sitter space of a curve in 𝑑𝑠̅ 𝑑𝑠 𝑔
hyperbolic space.
By substituting (8) into (11) we find
Definition 3.1. Let 𝛼=𝛼(𝑠) be a unit speed regular curve
lying fully on 𝐻2 with the moving Sabban frame {𝛼,𝑡,𝜉}. 𝑡′ = √2𝜀 (𝛼+𝜅 𝜉). (12)
The curve 𝛽:Ι⊂0ℝ→𝑆2 of α defined by 𝛽 |𝑐1−𝑐2𝜅𝑔| 𝑔
1
Consequently, from (4), (9), and (12), the geodesic
𝛽(𝑠̅(𝑠))=√12(𝑐1𝛼(𝑠)+𝑐2𝜉(𝑠)) (4) curvature 𝜅𝑔𝛽 of the curve 𝛽=𝛽(𝑠̅(𝑠)) is explicitly
is called the spacelike α𝜉-Smarandache curve of α and fully obtained by
lies on 𝑆12, where 𝑐1,𝑐2𝜖ℝ\{0} and −𝑐12+𝑐22=2. 𝜅𝛽 =det (𝛽,𝑡 ,𝑡′)= 𝑐1𝜅𝑔−𝑐2 (13)
Theorem 3.1. Let 𝛼:Ι⊂ℝ→𝐻2 be a unit speed regular 𝑔 𝛽 𝛽 |𝑐1−𝑐2𝜅𝑔|
0
curve lying fully on 𝐻2 with the Sabban frame {𝛼,𝑡,𝜉} Thus, the theorem is proved. In three theorems that follow,
0
and geodesic curvature 𝜅 . If 𝛽:Ι⊂ℝ→𝑆2 is the in a similar way as in Theorem 3.1 we obtain the Sabban
𝑔 1
α𝜉-Smarandache curve of α with the Sabban frame {𝛽,𝑡 ,𝜉 } and the geodesic curvature 𝜅𝛽of a
𝛽 𝛽 𝑔
frame {𝛽,𝑡𝛽,𝜉𝛽} then the relationships between the spacelike Smarandache curve. We omit the proofs of
Sabban frame of 𝛼 and its α𝜉-Smarandache curve are Theorems 3.2, 3.3, and 3.4, since they are analogous to the
given by proof of Theorem 3.1.
𝛽 𝑐1 0 𝑐2 𝛼 Definition 3.2. Let 𝛼=𝛼(𝑠) be a regular unit speed curve
√2 √2 lying fully on 𝐻2. Then the spacelike 𝛼𝑡-Smarandache
[𝑡𝛽]=[0 𝜀 0][𝑡] (5) curve 𝛽:Ι⊂ℝ→0𝑆2 of α defined by
𝜉𝛽 𝑐2𝜀 0 𝑐1𝜀 𝜉 1
√2 √2 𝛽(𝑠̅(𝑠))= 1 (𝑐 𝛼(𝑠)+𝑐 𝑡(𝑠)) (14)
where 𝜀=±1 and its geodesic curvature 𝜅𝛽 is given by √2 1 2
𝑔 where 𝑐 ,𝑐 𝜖ℝ\{0} and −𝑐2+𝑐2=2.
1 2 1 2
𝜅𝑔𝛽 =|𝑐𝑐11𝜅−𝑔𝑐2−𝜅𝑐𝑔2|. (6) Theorem 3.2. Let 𝛼:Ι⊂ℝ→𝐻02 be a regular unit speed
curve lying fully on 𝐻2 with the Sabban frame {𝛼,𝑡,𝜉}
0
Proof. Differentiating the equation (4) with respect to 𝑠 and geodesic curvature 𝜅 . If 𝛽:Ι⊂ℝ→𝑆2 is the
𝑔 1
and considering (2), we obtain
spacelike 𝛼𝑡-Smarandache curve of α, then its frame
𝑑𝛽𝑑𝑠̅ 1 {𝛽,𝑡 ,𝜉 } is given by
= (𝑐 −𝑐 𝜅 )𝑡. 𝛽 𝛽
𝑑𝑠̅𝑑𝑠 √2 1 2 𝑔
This can be rewritten as
𝑑𝑠̅ 1
𝑡 = (𝑐 −𝑐 𝜅 )𝑡 (7)
𝛽𝑑𝑠 √2 1 2 𝑔
72 GU J Sci, 29(1):69-77 (2016)/ Murat SAVAS, Atakan Tugkan YAKUT, Tugba TAMIRCI

𝑐1 𝑐2 0 Theorem 3.4. Let 𝛼:Ι⊂ℝ→𝐻2 be a regular unit speed
√2 √2 curve lying fully on 𝐻2 with th0e Sabban frame {𝛼,𝑡,𝜉}
𝛽 𝑐2 𝑐1 𝑐2𝜅𝑔 𝛼 0
and geodesic curvature 𝜅 . If 𝛽:Ι⊂ℝ→𝑆2 is the
[𝑡𝛽]= √𝑐22𝜅𝑔2−2 √𝑐22𝜅𝑔2−2 √𝑐22𝜅𝑔2−2 [𝑡] (15) 𝑔 1
𝜉𝛽 −𝑐22𝜅𝑔 −𝑐1𝑐2𝜅𝑔 −2 𝜉 𝛼gi𝑡v𝜉e-nS mbya randache curve of 𝛼, then its frame {𝛽,𝑡𝛽,𝜉𝛽} is
[√2(𝑐22𝜅𝑔2−2) √2(𝑐22𝜅𝑔2−2) √2(𝑐22𝜅𝑔2−2)]

The geodesic curvature 𝜅𝑔𝛽 of the curve β is given by 𝑐1 𝑐2 𝑐3
𝜅𝑔𝛽 =(𝑐22𝜅𝑔21−2)5⁄2(𝑐22𝜅𝑔𝜆1−𝑐1𝑐2𝜅𝑔𝜆2−2𝜆3) (16) [𝜉𝑡𝛽𝛽𝛽]= −𝑐22𝜅𝑔−𝑐3√√𝑐(2𝐴3−𝑐1+𝑐3𝜅𝑔) 𝑐2𝑐𝑐13−−√√𝑐𝑐𝐴331𝜅𝑐2𝑔𝜅𝑔 𝑐1(𝑐1−𝑐√√2𝑐3𝜅𝐴3𝜅𝑔𝑔)−𝑐22 ×[𝛼𝜉𝑡] (23)
where 𝑐22𝜅𝑔2>2 and [ √3𝐴 √3𝐴 √3𝐴 ]
where
𝜆 =−𝑐3𝜅 𝜅′ +𝑐 (𝑐2𝜅2−2)
1 2 𝑔 𝑔 1 2 𝑔
{𝜆 =−𝑐 𝑐2𝜅 𝜅′ +(𝑐 −𝑐 𝜅2)(𝑐2𝜅2−2) (17) 𝐴=(𝑐 −𝑐 𝜅 )2−𝑐2+𝑐2𝜅2 , (𝑐 −𝑐 𝜅 )2>𝑐2−
2 1 2 𝑔 𝑔 2 2 𝑔 2 𝑔 1 3 𝑔 2 2 𝑔 1 3 𝑔 2
𝜆3=−𝑐23𝜅𝑔2𝜅𝑔′ +(𝑐1𝜅𝑔+𝑐2𝜅𝑔′)(𝑐22𝜅𝑔2−2) 𝑐22𝜅𝑔2 and the Smarandache curve β is a spacelike curve.
Furthermore, the geodesic curvature 𝜅𝛽 of curve 𝛽 is
Definition 3.3. Let 𝛼:Ι⊂ℝ→𝐻2 be a regular unit speed 𝑔
0 given by
curve lying fully on 𝐻2. Then the spacelike
0
𝑡𝜉-Smarandache curve 𝛽:Ι⊂ℝ→𝑆12 of 𝛼 defined by 𝜅𝛽 =((𝑐2𝜅 +𝑐2𝜅 −𝑐 𝑐 )𝜆 +(−𝑐 𝑐 𝜅 +𝑐 𝑐 )𝜆 +
𝑔 2 𝑔 3 𝑔 1 3 1 1 2 𝑔 2 3 2
1
𝛽(𝑠̅(𝑠))=√2(𝑐1𝑡(𝑠)+𝑐2𝜉(𝑠)) (18) (𝑐12−𝑐1𝑐3𝜅𝑔−𝑐22)𝜆3)×(((𝑐1−𝑐3𝜅𝑔)2−𝑐22+
where 𝑐1,𝑐2𝜖ℝ\{0} and 𝑐12+𝑐22 =2. 𝑐2𝜅2)5⁄2)−1 (24)
Theorem 3.3. Let 𝛼:Ι⊂ℝ→𝐻2 be a regular unit speed 2 𝑔
0
curve lying fully on 𝐻2 with the Sabban frame {𝛼,𝑡,𝜉}
0 where
and geodesic curvature 𝜅 . If 𝛽:Ι⊂ℝ→𝑆2 is the
𝑔 1
spacelike 𝑡𝜉-Smarandache curve of 𝛼, then its frame 𝜆1=𝑐2(𝑐3𝜅𝑔′(𝑐1−𝑐3𝜅𝑔)−𝑐22𝜅𝑔𝜅𝑔′)
{𝛽,𝑡𝛽,𝜉𝛽} is given by +(𝑐1−𝑐3𝜅𝑔)((𝑐1−𝑐3𝜅𝑔)2−𝑐22+𝑐22𝜅𝑔2)
0 𝑐1 𝑐2 𝜆2=(𝑐1−𝑐3𝜅𝑔)(𝑐3𝜅𝑔′(𝑐1−𝑐3𝜅𝑔)−𝑐22𝜅𝑔𝜅𝑔′)
𝛽 𝑐1 −𝑐√22𝜅𝑔 𝑐√1𝜅2𝑔 𝛼 +(𝑐2−𝑐3𝜅𝑔′−𝑐2𝜅𝑔2)((𝑐1−𝑐3𝜅𝑔)2−𝑐22+𝑐22𝜅𝑔2) (25)
[𝑡𝛽]= √2𝜅𝑔2−𝑐12 √2𝜅𝑔2−𝑐12 √2𝜅𝑔2−𝑐12 [𝑡] (19) 𝜆3=𝑐2𝜅𝑔(𝑐3𝜅𝑔′(𝑐1−𝑐3𝜅𝑔)−𝑐22𝜅𝑔𝜅𝑔′)
𝜉𝛽 −2𝜅𝑔 𝑐1𝑐2 −𝑐12 𝜉 { +(𝜅𝑔(𝑐1−𝑐3𝜅𝑔)+𝑐2𝜅𝑔′)((𝑐1−𝑐3𝜅𝑔)2−𝑐22+𝑐22𝜅𝑔2)

[√2(2𝜅𝑔2−𝑐12) √2(2𝜅𝑔2−𝑐12) √2(2𝜅𝑔2−𝑐12)] Example 3.1. Let us consider a regular unit speed curve α
on 𝐻2 defined by
he geodesic curvature𝜅𝛽 of the curve β is given by 0
𝑔
𝜅𝛽 = 1 (2𝜅 𝜆 +𝑐 𝑐 𝜆 −𝑐2𝜆 ) (20) 𝛼(𝑠)=((𝑠−1)2+1, (𝑠−1)2, 𝑠−1).
𝑔 (2𝜅𝑔2−𝑐12)5⁄2 𝑔 1 1 2 2 1 3 2 2
Then the orthonormal Sabban frame {𝛼(𝑠),𝑡(𝑠),𝜉(𝑠)} of
where 𝑐2<2𝜅2 and
1 𝑔 α can be calculated as follows:
𝜆1=−2𝑐1𝜅𝑔𝜅𝑔′ −𝑐2𝜅𝑔(2𝜅𝑔2−𝑐12) 𝛼(𝑠)=((𝑠−1)2+1, (𝑠−1)2, 𝑠−1)
{𝜆2=2𝑐2𝜅𝑔2𝜅𝑔′ +(𝑐1−𝑐2𝜅𝑔′ −𝑐1𝜅𝑔2)(2𝜅𝑔2−𝑐12) (21) 2 2
𝜆 =−2𝑐 𝜅2𝜅′ +(−𝑐 𝜅2+𝑐 𝜅′)(2𝜅2−𝑐2) 𝑡(𝑠)=(𝑠−1, 𝑠−1, 1)
3 1 𝑔 𝑔 2 𝑔 1 𝑔 𝑔 1
𝜉(𝑠)=((𝑠−1)2, (𝑠−1)2−1, 𝑠−1)
{ 2 2
Definition 3.4. Let 𝛼:Ι⊂ℝ→𝐻2 be a regular unit speed
curve lying fully on 𝐻2. 0 Then the spacelike The geodesic curvature of 𝛼 is −1. In terms of the
0 definitions, we obtain the spacelike Smarandache curves
𝛼𝑡𝜉-Smarandache curve 𝛽:Ι⊂ℝ→𝑆2 of 𝛼 defined by
1 on 𝑆2 according to the Sabban frame on 𝐻2.
1 0
1
𝛽(𝑠̅(𝑠))= (𝑐 𝛼(𝑠)+𝑐 𝑡(𝑠)+𝑐 𝜉(𝑠)) (22)
√3 1 2 3 First, when we take 𝑐1=1 and 𝑐2=√3, then the
𝛼𝜉-Smarandache curve is spacelike and given by
where 𝑐 ,𝑐 ,𝑐 𝜖ℝ\{0} and −𝑐2+𝑐2+𝑐2=3.
1 2 3 1 2 3
GU J Sci, 29(1):69-77 (2016)/ Murat SAVAS, Atakan Tugkan YAKUT, Tugba TAMIRCI 73

1 1 1
𝛽 𝛼
1 1
1 1+√3 [𝑡𝛽]=[2 1 −2][𝑡]
𝛽(𝑠̅(𝑠))=√2(( 2 )(𝑠−1)2+1, 𝜉𝛽 3 1 1 𝜉
2 2
1+√3
( )(𝑠−1)2−√3, (𝑠−1)(1+√3)) and its geodesic curvature 𝜅𝛽 is −1.
2 𝑔
We give a curve 𝛼 in hyperbolic space and its
and the Sabban frame of the spacelike 𝛼𝜉-Smarandache
Smarandache partners in de Sitter space in Figure 1.
curve is given by
4. CURVES IN HYPERBOLIC SPACE AND DUAL
1 √3
𝛽 0 𝛼
√2 √2 TIMELIKE SMARANDACHE PARTNERS
[𝑡𝛽]= 0 1 0 [𝑡]
𝜉 √3 1 𝜉 In this section we obtain the timelike Smarandache partners
𝛽 0
[√2 √2] in de Sitter space of a curve in hyperbolic space.
and its geodesic curvature 𝜅𝛽 is −1. Theorem 4.1. Let 𝛼:Ι⊂ℝ→𝐻2 be a regular unit speed
𝑔 0
curve lying fully on 𝐻2 . Then the timelike
0
Second, when we take 𝑐1=1 and 𝑐2=√3, then the 𝛼𝜉-Smarandache curve 𝛽:Ι⊂ℝ→𝑆12 of 𝛼 does not
𝛼𝑡-Smarandache curve is a spacelike and given by exist.
1 (𝑠−1)2 Proof. Assume that 𝛼:Ι⊂ℝ→𝐻2 be a regular unit speed
𝛽(𝑠̅(𝑠))= ( +√3(𝑠−1)+1, 0
√2 2 curve lying fully on 𝐻2. Then the timelike
0
(𝑠−1)2
+√3(𝑠−1),(𝑠−1+√3)) 𝛼𝜉-Smarandache curve 𝛽:Ι⊂ℝ→𝑆12 of 𝛼 is defined by
2
1
𝛽(𝑠̅(𝑠))= (𝑐 𝛼(𝑠)+𝑐 𝜉(𝑠)) (26)
and the Sabban frame of the spacelike 𝛼𝑡-Smarandache √2 1 2
curve is given by
𝑐 ,𝑐 𝜖ℝ\{0}, −𝑐2+𝑐2 =2. Differentiating (26) with
1 2 1 2
1 √3 respect to 𝑠 and using (2), we obtain
𝛽 0 𝛼
√2 √2
𝑑𝛽𝑑𝑠̅ 1
[𝜉𝑡𝛽𝛽]= √33 √13 −−√√32 [𝜉𝑡] 𝛽′(𝑠)=𝑑𝑠̅𝑑𝑠=√2(𝑐1−𝑐2𝜅𝑔)𝑡
[√2 √2 ] 𝑑𝑠̅ 1
𝑡 = (𝑐 −𝑐 𝜅 )𝑡
and its geodesic curvature 𝜅𝑔𝛽 is −1. 𝛽𝑑𝑠 √2 1 2 𝑔
Third, when we take 𝑐 =1 and 𝑐 =1, then the where
1 2
𝑡𝜉-Smarandache curve is a spacelike curve and given by
𝛽(𝑠̅(𝑠))= 1 ((𝑠−1)2+𝑠−1, (𝑠−1)2+𝑠−2, 𝑠) 𝑑𝑑𝑠𝑠̅=√−(𝑐1−𝑐22𝜅𝑔)2
√2 2 2
which is a contradiction.
and the Sabban frame of the spacalike 𝑡𝜉-Smarandache
curve is given by In the corollaries which follow, in a similar way as in the
1 1 previous section, we obtain the Sabban frame {𝛽,𝑡𝛽,𝜉𝛽}
𝛽 0 𝛼
[𝑡𝛽]=[ 1 √12 −√21][𝑡] aSnmda ra ndthaec heg ceuordvees. iWc ec oumrvita ttuhree p ro𝜅o𝑔𝛽fs ooff T heao retimmse l4i.k2e,
𝜉𝛽 √2 1 − 1 𝜉 4.3, and 4.4.
√2 √2
and its geodesic curvature 𝜅𝛽 is −1. Definition 4.1. Let 𝛼:Ι⊂ℝ→𝐻02 be a regular unit speed
𝑔 curve lying fully on 𝐻2. Then, the timelike
0
Finally, when we take 𝑐1=√3, 𝑐2=√3 and 𝑐3=√3, 𝛼𝑡-Smarandache curve 𝛽:Ι⊂ℝ→𝑆12 of 𝛼 is defined by
1
then the 𝛼𝑡𝜉-Smarandache curve is a spacelike curve and 𝛽(𝑠̅(𝑠))= (𝑐 𝛼(𝑠)+𝑐 𝑡(𝑠))
given by √2 1 2
where 𝑐 ,𝑐 𝜖ℝ\{0}, and −𝑐2+𝑐2 =2.
𝛽(𝑠̅(𝑠))=((𝑠−1)2+𝑠,(𝑠−1)2+𝑠−2,2𝑠−1) 1 2 1 2

and the Sabban frame of the spacelike 𝛼𝑡𝜉-Smarandache
curve is given by
74 GU J Sci, 29(1):69-77 (2016)/ Murat SAVAS, Atakan Tugkan YAKUT, Tugba TAMIRCI

Corollary 4.1. Let 𝛼:Ι⊂ℝ→𝐻2 be a regular unit speed 𝜆 =𝑐3𝜅 𝜅′ +𝑐 (2−𝑐2𝜅2)
0 1 2 𝑔 𝑔 1 2 𝑔
curve lying fully on 𝐻2 with the Sabban frame {𝛼,𝑡,𝜉} {𝜆 =𝑐 𝑐2𝜅 𝜅′ +(𝑐 −𝑐 𝜅2)(2−𝑐2𝜅2)
0 2 1 2 𝑔 𝑔 2 2 𝑔 2 𝑔
and the geodesic curvature 𝜅𝑔. If 𝛽:Ι⊂ℝ→𝑆12 is the 𝜆 =𝑐3𝜅2𝜅′ +(𝑐 𝜅 +𝑐 𝜅′)(2−𝑐2𝜅2).
3 2 𝑔 𝑔 1 𝑔 2 𝑔 2 𝑔
timelike 𝛼𝑡-Smarandache curve of 𝛼, then its frame
{𝛽,𝑡 ,𝜉 } is given by
𝛽 𝛽
𝑐1 𝑐2 0 Definition 4.2. Let 𝛼:Ι⊂ℝ→𝐻02 be a regular unit
√2 √2
𝛽 𝑐2 𝑐1 𝑐2𝜅𝑔 𝛼 speed curve lying fully on 𝐻02. Then, the timelike

[𝑡𝛽]= √2−𝑐22𝜅𝑔2 √2−𝑐22𝜅𝑔2 √2−𝑐22𝜅𝑔2 [𝑡] 𝑡𝜉-Smarandache curve 𝛽:Ι⊂ℝ→𝑆12 of 𝛼 is defined by
𝜉𝛽 −𝑐22𝜅𝑔 −𝑐1𝑐2𝜅𝑔 −2 𝜉

1
[√2(2−𝑐22𝜅𝑔2) √2(2−𝑐22𝜅𝑔2) √2(2−𝑐22𝜅𝑔2)] 𝛽(𝑠̅(𝑠))=√2(𝑐1𝑡(𝑠)+𝑐2𝜉(𝑠)),
and the corresponding geodesic curvature 𝜅𝛽 is given by
𝑔 where 𝑐 ,𝑐 𝜖ℝ\{0}, and 𝑐2+𝑐2 =2.
1 2 1 2
𝜅𝛽 = 1 (𝑐2𝜅 𝜆 −𝑐 𝑐 𝜅 𝜆 −2𝜆 )
𝑔 (2−𝑐22𝜅𝑔2)5⁄2 2 𝑔 1 1 2 𝑔 2 3

where −𝑐2+𝑐2=2 with 𝑐 ,𝑐 𝜖ℝ\{0}, 𝑐2𝜅2 <2 and
1 2 1 2 2 𝑔

𝛼 is a unit speed curve o n 𝐻02 (black line)
𝛽 is a unit speed spacelike curve on 𝑆2 (red line)
1

𝛼𝜉-Smarandache curve
𝛼𝑡-Smarandache curve

𝑡𝜉-Smarandache curve 𝛼𝑡𝜉-Smarandache curve
Figure 1. Spacelike Smarandache partner curves of a curve 𝛼 on 𝐻2
0
GU J Sci, 29(1):69-77 (2016)/ Murat SAVAS, Atakan Tugkan YAKUT, Tugba TAMIRCI 75

𝛽
Corollary 4.2. Let 𝛼:Ι⊂ℝ→𝐻2 be a regular unit speed [𝑡𝛽]
0 𝜉
𝛽
curve lying fully on 𝐻2 with the Sabban frame {𝛼,𝑡,𝜉}
0 𝑐 𝑐 𝑐
1 2 3
and geodesic curvature 𝜅𝑔 . If 𝛽:Ι⊂ℝ→𝑆12 is the √𝑐3 𝑐 −√3𝑐 𝜅 𝑐√𝜅3 𝛼
2 1 3 𝑔 2 𝑔
timelike 𝑡𝜉-Smarandache curve of 𝛼, then its frame = √𝐴∗ √𝐴∗ √𝐴∗ [𝑡]
{𝛽,𝑡𝛽,𝜉𝛽} is given by −𝑐22𝜅𝑔−𝑐3(−𝑐1+𝑐3𝜅𝑔) 𝑐2𝑐3−𝑐1𝑐2𝜅𝑔 𝑐1(𝑐1−𝑐3𝜅𝑔)−𝑐22 𝜉
[ √3𝐴∗ √3𝐴∗ √3𝐴∗ ]

0 𝑐1 𝑐2 where
√2 √2
𝛽 𝑐1 −𝑐2𝜅𝑔 𝑐1𝜅𝑔 𝛼 𝐴∗=𝑐2−(𝑐 −𝑐 𝜅 )2−𝑐2𝜅2, (𝑐 −𝑐 𝜅 )2<𝑐2−𝑐2𝜅2
2 1 3 𝑔 2 𝑔 1 3 𝑔 2 2 𝑔
[𝜉𝑡𝛽𝛽]= √𝑐−122−𝜅2𝑔𝜅𝑔2 √𝑐𝑐121−𝑐22𝜅𝑔2 √𝑐1−2−𝑐122𝜅𝑔2 [𝜉𝑡] a nd the corresponding geodesic curvature 𝜅𝑔𝛽 is given by
[√2(𝑐12−2𝜅𝑔2) √2(𝑐12−2𝜅𝑔2) √2(𝑐12−2𝜅𝑔2)] (𝑐2𝜅 +𝑐2𝜅 −𝑐 𝑐 )𝜆 +(−𝑐 𝑐 𝜅 +𝑐 𝑐 )𝜆
𝜅𝛽=( 2 𝑔 3 𝑔 1 3 1 1 2 𝑔 2 3 2)
𝑔 +(𝑐12−𝑐1𝑐3𝜅𝑔−𝑐22)𝜆3
and the corresponding geodesic curvature 𝜅𝑔𝛽 is given by ×((𝑐22−(𝑐1−𝑐3𝜅𝑔)2−𝑐22𝜅𝑔2)5⁄2)−1

𝜅𝛽 = 1 (2𝜅 𝜆 +𝑐 𝑐 𝜆 +𝑐2𝜆 )
𝑔 (𝑐12−2𝜅𝑔2)5⁄2 𝑔 1 1 2 2 1 3 where 𝑐 ,𝑐 ,𝑐 𝜖ℝ\{0}, −𝑐2+𝑐2+𝑐2=3 and
1 2 3 1 2 3
where 𝑐1,𝑐2𝜖ℝ\{0}, 𝑐12+𝑐22 =2, 𝑐12 >2𝜅𝑔2 and 𝜆 =𝑐 (−𝑐 𝜅′(𝑐 −𝑐 𝜅 )+𝑐2𝜅 𝜅′)
1 2 3 𝑔 1 3 𝑔 2 𝑔 𝑔
+(𝑐 −𝑐 𝜅 )(𝑐2−(𝑐 −𝑐 𝜅 )2−𝑐2𝜅2)
𝜆 =2𝑐 𝜅 𝜅′ −𝑐 𝜅 (𝑐2−2𝜅2) 1 3 𝑔 2 1 3 𝑔 2 𝑔
1 1 𝑔 𝑔 2 𝑔 1 𝑔
{𝜆 =−2𝑐 𝜅2𝜅′ +(𝑐 −𝑐 𝜅′ −𝑐 𝜅2)(𝑐2−2𝜅2) 𝜆2=(𝑐1−𝑐3𝜅𝑔)(−𝑐3𝜅𝑔′(𝑐1−𝑐3𝜅𝑔)+𝑐22𝜅𝑔𝜅𝑔′)
2 2 𝑔 𝑔 1 2 𝑔 1 𝑔 1 𝑔
𝜆3=2𝑐1𝜅𝑔2𝜅𝑔′ +(𝑐1𝜅𝑔′−𝑐2𝜅𝑔2)(𝑐12−2𝜅𝑔2). +(𝑐2−𝑐3𝜅𝑔′−𝑐2𝜅𝑔2)(𝑐22−(𝑐1−𝑐3𝜅𝑔)2−𝑐22𝜅𝑔2)

𝜆3=𝑐2𝜅𝑔(−𝑐3𝜅𝑔′(𝑐1−𝑐3𝜅𝑔)+𝑐22𝜅𝑔𝜅𝑔′)

Definition 4.3. Let 𝛼:Ι⊂ℝ→𝐻02 be a regular unit speed { +(𝑐2𝜅𝑔′ +𝜅𝑔(𝑐1−𝑐3𝜅𝑔))(𝑐22−(𝑐1−𝑐3𝜅𝑔)2−𝑐22𝜅𝑔2)
curve lying fully on 𝐻2. Then 𝛼𝑡𝜉-Smarandache curve
0
𝛽:Ι⊂ℝ→𝑆12 of 𝛼 is defined by Example 4.1. Let us consider a regular unit speed curve 𝛼
on 𝐻2 defined by
0
1
𝛽(𝑠̅(𝑠))=√3(𝑐1𝛼(𝑠)+𝑐2𝑡(𝑠)+𝑐3𝜉(𝑠)) 𝛼(𝑠)=(cosh𝑠,sinh𝑠,0).
𝑐 ,𝑐 ,𝑐 𝜖ℝ\{0}, −𝑐2+𝑐2+𝑐2 =3. Then the orthonormal Sabban frame {𝛼,𝑡,𝜉} of the curve
1 2 3 1 2 3
𝛼 and the orthonormal Sabban frame {𝛽,𝑡 ,𝜉 } of the
𝛽 𝛽
Corollary 4.3. Let 𝛼:Ι⊂ℝ→𝐻02 be a regular unit speed curve 𝛽 and the geodesic curvature 𝜅𝛽 of a timelike
𝑔
curve lying fully on 𝐻2 with the Sabban frame {𝛼,𝑡,𝜉}
0 Smarandache curve can be calculated as in the previous
and geodesic curvature 𝜅 . If 𝛽:Ι⊂ℝ→𝑆2 is the
𝑔 1 example. The curve 𝛼 and its Smarandache partners are
timelike 𝛼𝑡𝜉-Smarandache curve of 𝛼, then its frame
given in Figure 2.
{𝛽,𝑡𝛽,𝜉𝛽} is given by
76 GU J Sci, 29(1):69-77 (2016)/ Murat SAVAS, Atakan Tugkan YAKUT, Tugba TAMIRCI

𝛼 is a unit speed curve on 𝐻2 (black line)
0
𝛽 is a unit speed timelike curve on 𝑆12 (red line)

𝛼𝑡-Smarandache curve
𝛼𝜉-Smarandache curve is undefined

𝑡𝜉-Smarandache curve 𝛼𝑡𝜉-Smarandache curve
Figure 2. Timelike Smarandache partner curves of a curve 𝛼 on 𝐻2
0
GU J Sci, 29(1):69-77 (2016)/ Murat SAVAS, Atakan Tugkan YAKUT, Tugba TAMIRCI 77

CONFLICT OF INTEREST

No conflict of interest was declared by the authors.

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