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Gazi University Journal of Science aG 
GUJ Sci G\U sys 
29(1):69-77 (2016) © 


The Smarandache Curves on H$ 


Murat SAVAS! , Atakan Tugkan YAKUT” *, Tugba TAMIRCI' 


! Gazi University, Faculty of Sciences, Department of Mathematics, 06500 Teknikokullar, 
Ankara, Turkey 
? 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, 
evolute, and pedal curves. In the light of the literature, in 
[11] authors introduced a special curve by Frenet-Serret 
frame vector fields in Minkowski space-time. The new 
special curve, which is named Smarandache curve 
according to the Sabban frame in the Euclidean unit sphere, 
is defined by Turgut and Yilmaz in Minkowski space-time 
[11]. Smarandache curves in Euclidean or non-Euclidean 


Regular curves have an important role in the theory of 
curves in differential geometry and relativity theory. In the 
geometry of regular curves in Euclidean or Minkowskian 
spaces, it is well-known that one of the most important 
problem is the characterization and classification of these 
curves. In the theory of regular curves, there are some 


“Corresponding author, e-mail: sevaty @ nigde.edu.tr 


70 GU J Sci, 29(1):69-77 (2016)/ Murat SAVAS, Atakan Tugkan YAKUT, Tugba TAMIRCI 


spaces have been recently of particular interest for 
researchers. In Euclidean differential geometry, 
Smarandache curves of a curve are defined to be 
combination of its position, tangent, and normal vectors. 
These curves have been also studied widely [1, 4, 6, 9, 11, 
12]. Smarandache curves play an important role in 
Smarandache geometry. They are the objects of 
Smarandache geometry, i.e. a geometry which has at least 
one Smarandachely denied axiom [2]. An axiom is said to 
be Smarandachely denied if it behaves in at least two 
different ways within the same space. Smarandache 
geometry has a significant role in the theory of relativity 
and parallel universes. Ozturk U., et al. studied 
Smarandache curves in hyperbolic space but they don’t 
give dual Smarandache partners of these curves in de Sitter 
space [6]. We answer it for curves in hyperbolic space and 
show the Smarandache partners curve of these curves in de 
Sitter space. We explain the Smarandache de Sitter duality 
of curves in hyperbolic space. In this paper, we give the 
Smarandache partner curves in de Sitter space according to 
the Sabban frame {a, t, &} of a curve in hyperbolic space. 
We obtain the geodesic curvatures and the expressions for 
the Sabban frame’s vectors of special Smarandache curves 
on de Sitter surface. In particular, we see that the timelike 
aé-Smarandache curve of a curve æ does not exist in de 
Sitter space. We give some examples of the Smarandache 
curves in hyperbolic space and its dual Smarandache 
curves in de Sitter space. Furthermore, we give some 
examples of special hyperbolic and de Sitter Smarandache 
curves, which are found in the study of Yakut et al. [12]. In 
her Master thesis [9], Tamirci also studied the curves in de 
Sitter and hyperbolic spaces using a similar framework. 


2. PRELIMINARIES 


In this section, we use the basic notions and results in 
Lorentzian geometry. For more detailed concepts, see 
[7,8]. Let R? be the 3-dimensional vector space equipped 
with the scalar product (,) which is defined by 


(x, Y), = —X1Y1 + X2Y2 + X33. 


The space Ef = (R?,(,),) is a pseudo-Euclidean space, 
or Minkowski 3-space. The unit pseudo-sphere (de Sitter 
space) with index one S? in E? is given by 


S? = {xeE}|(x,x), = 1}. 
The unit pseudo-hyperbolic space 
Hg = {xeE? (x, x), = —1} 


has two connected components Hj, and Hg_. Each of 
them can be taken as a model for the 2-dimensional 
hyperbolic space Hg. In this paper, we take Hê, = HŽ. 
Recall that a nonzero vector xeE} is spacelike if 
(x,x), > 0, timelike if (x,x), < 0, and null (lightlike) if 
(x, x), = 0. The norm (length) of a vector xeE? is given 


by |lxll, = 4 |(x,x),| and two vectors x and y are said to 


be orthogonal if (x, y}, = 0. Next, we say that an arbitrary 
curve a@ = a(s) in E? can locally be spacelike, timelike, 
or null(lightlike) if all of its velocity vectors a@’(s) are, 
respectively, spacelike, timelike, or null for all sel. If 
lla'(s)||, # 0 for every sel, then a is a regular curve in 
E. A spacelike(timelike) regular curve œ is 
parameterized by a pseudo-arc length parameter s, which 
is given by a:1 c R > Ef, and then the tangent vector 
a'(s) along a has unit length, that is 


(a’(s),a'(s)), = 1((a"(s), a'(s)), = —1) 


for all sel. Let 


x= (x4, X2, X3), y = (y1, Y2, Y3), Z= (21, Z2, Z3)E E 
The Lorentzian pseudo-vector cross product is defined as 
follows: 


x AY = (—xX2¥3 + X3Y2, X3V1 — X1Y3, X1Y2 — X2y1 ) (1) 
We remark that the following relations hold: 

(i) (x Ay, Z}, = det(x y z) 

(ii) x A Q AZ) = (x, y}LZ — (xX, Z}LY 


Let a:1 c R > HẸ be a regular unit speed curve lying 
fully in Hé. Then its position vector a is a timelike vector, 
which implies that the tangent vector t = a’ and normal 
vector € are unit spacelike vector for all sel. We have the 
orthonormal Sabban frame {a(s),t(s),&(s)} along the 
curve a, where (s) = a(s) At(s) is the unit spacelike 
vector. The corresponding Frenet formula of æ, according 
to the Sabban frame, is given by 


a'(s) = t(s) 
t'(s) = a(s) + kg(S) E(s) (2) 
č’ (s) = —K,(s)t(s) 


where Kg(s) = det(a(s),t(s),t'(s)) is the geodesic 
curvature of a on Hé and s is the arc length parameter 
of a. In particular, the following relations hold: 


€=aAt, -a@=taAt, t=ENa (3) 


Now we define a new curve B:1C R> sS? to be a 
regular unit speed curve lying fully on S? for all sel such 
that its position vector f is a unit spacelike vector 
according to the combination of the position, tangent, and 
normal vectors of a. In this case p’ = tg may be a unit 
timelike or spacelike vector. 


Definition 2.1. A unit speed regular curve B(s(s)) lying 
fully in Minkowski 3-space, whose position vector is 
associated with Sabban frame vectors on another regular 
curve a(s), is called a Smarandache curve[11]. 


In the light of this definition, if a regular unit speed curve 
a:1cC R > HÊ is lying fully on HÊ for all sel and its 
position vector œ 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 Bf = B(5(s)) of the curve æ is a 
regular unit speed curve lying fully in S or Hé. In our 
work we are interested in curves lying in SŽ and so we 
have the following: 


a) The Smarandache curve f(S(s)) may be a spacelike 
curve on S? or, 


b) The Smarandache curve f(S5(s)) may be a timelike 
curve on S? forall sel. 


Let {a,t,é} and {f, tp, Ee} be the moving Sabban frames 
of æ and fp, respectively. Then we have the following 
definitions and theorems of Smarandache curves 


P = B(s(s)). 


3. CURVES ON HZ AND ITS SPACELIKE 
SMARANDACHE PARTNERS ON S? 


Let æ be a regular unit speed curve on HÊ. Then the 
Smarandache partner curve of a is either in de Sitter or in 
hyperbolic space. B is called de Sitter dual of a in 
hyperbolic space. In this section we obtain the spacelike 
Smarandache partners in de Sitter space of a curve in 
hyperbolic space. 


Definition 3.1. Let a = a(s) be a unit speed regular curve 
lying fully on Hé with the moving Sabban frame {a, t, č}. 
The curve 8:1 c R > S? of a defined by 


B(5(s)) = 5 (cas) + 2€(s)) (4) 


is called the spacelike ag-Smarandache curve of a and fully 
lies on SŽ, where c1, c2E€ER\{0} and —c? + c2 = 2. 


Theorem 3.1. Let a:I1 c R > Hé be a unit speed regular 
curve lying fully on Hë with the Sabban frame {a, t, ë} 
and geodesic curvature kg. If p:Ic R > SŽ is the 
ač-Smarandache curve of a with the Sabban 
frame fp, tg, $ B} then the relationships between the 
Sabban frame of æ and its ač-Smarandache curve are 
given by 


b] |e ° zle 
fel=!0 = olļļt (5) 
T fF o SEILS 


V2 v2 
where € = +1 and its geodesic curvature KË is given by 
C1Kg—C 
5 = EER, (6) 
g |c1-c2Kgl 


Proof. Differentiating the equation (4) with respect to s 
and considering (2), we obtain 

dßdaš __ 1 

dē ds V2 (opaka 

This can be rewritten as 


d5 1 
= N (Cy — C2Kg)t (7) 


where 

ds _ |¢1-Cakg| 

be 8 
ds V2 (8) 


By substituting (8) into (7) we obtain a simple form of Eq. 
7 as follows, 


where € = 1 if cy — C2Kg >Ofor all s and € = —1 if 
Cy — C2Kg < 0 for all s. It can be easily seen that the 
tangent vector tg is a unit spacelike vector. Taking the 
Lorentzian vector cross product of (4) with (9) we have 


Sg =P Atg 
= Z (cza + 16) (10) 


It is easily seen that ¢g is a unit timelike vector. On the 
other hand, by taking the derivative of the equation (9) with 
respect to s, we find 


dtg ds 


rn E(a + K€) (11) 
By substituting (8) into (11) we find 

T V2e 

B 7 eee + Kg6). (12) 


Consequently, from (4), (9), and (12), the geodesic 


curvature KË of the curve p = B(S(s)) is explicitly 


obtained by 
C1 Kg x C2 


|c1-—cokg| 


ER = 
Kg = det(G, tg, tg) = (13) 
Thus, the theorem is proved. In three theorems that follow, 
in a similar way as in Theorem 3.1 we obtain the Sabban 
frame fp, tg, $ g} and the geodesic curvature KË of a 
spacelike Smarandache curve. We omit the proofs of 
Theorems 3.2, 3.3, and 3.4, since they are analogous to the 
proof of Theorem 3.1. 


Definition 3.2. Let a = a(s) be a regular unit speed curve 
lying fully on H#. Then the spacelike at-Smarandache 
curve 6:1 c R > S? of a defined by 


B(5(s)) = Z (uals) + c2t(s)) (14) 
where c4, c2E€ER\{0} and —c? + c2 = 2. 


Theorem 3.2. Let a:I1 c R > Hé be a regular unit speed 
curve lying fully on HÊ with the Sabban frame {a, t, č} 
and geodesic curvature Kg. If p:Ic R> S? is the 
spacelike at-Smarandache curve of a, then its frame 


fp, tg, Ep} is given by 


12 GU J Sci, 29(1):69-77 (2016)/ Murat SAVAS, Atakan Tugkan YAKUT, Tugba TAMIRCI 


Cy C2 











2 2 : 
p 2 a 2u a 
€5K,—2 cłkg—2 [c3x2—2 fl (15) 
—=cÎKg —C1C2Kg -2 3 
2(c3Kk3=2) [2(c2K2-2) 2(cZK5—2) 
B 


The geodesic curvature Kk 


g 
KP = T — C1C2KgA2 — 243) (16) 


of the curve £ is given by 


á (c3KG-2 


where c3KZ > 2 and 


Ay = —CZKgKg + C1 (ch Kg — 2) 
A2 = —c1Cfkgkg + (C2 — Co.KG) (CZ KZ — 2) (17) 
A3 = —cZKGKG + (C1kg + c2kg)(cfKg — 2) 


Definition 3.3. Let a: I c R > Hé be a regular unit speed 
curve lying fully on H. Then the spacelike 
té-Smarandache curve p:I c R > S? of æ defined by 


B(5(s)) = 5 (cts) + cẸ(s)) (18) 


where c4, c2ER\{0} and c? + c? = 2. 


Theorem 3.3. Let a:I1 c R > Hé be a regular unit speed 
curve lying fully on HÊ with the Sabban frame {a, t, č} 
and geodesic curvature Kg. If p:Ic R> S? is the 
spacelike tě -Smarandache curve of a, then its frame 


fp, tg, čeg} is given by 








0 EL £2 
v2 v2 
B ci —C2Kg C1Kg a 
tg | = [2xz—c? [2x2 —c? [2x3 -c2 fel (19) 
B -2kg C1C2 ze ¢ 
{2Q2xg-c2)J2(axg-c?) —|2(2x§-c?) 
he geodesic curvature KË of the curve B is given by 
Bra 1 
Kg = ——y; (2g Ay + CCÀ» H c?A3) (20) 
(2x9-c1) 


where cf < 2k and 


Ay = —2c1KgKg — C2Kg (2K — cf) 
Ag = 2czkĉkg + (c1 — cokg — ck) (2k? — c?) (21) 
Ag = —2c,K2Kg + (—czk2 + cikg )(2k2 — c?) 


Definition 3.4. Let a: I c R > Hé be a regular unit speed 
curve lying fully on Hê. Then the spacelike 
atč-Smarandache curve B:1 c R > S? of a defined by 


B(5(s)) = Z (cals) + cat(s) + c3Ẹ(s)) 22) 


where c4, C2, czER\{0} and —c? + c2 +c? = 3. 


Theorem 3.4. Let a:I1 c R > Hé be a regular unit speed 
curve lying fully on Hé with the Sabban frame {a, t, €} 
and geodesic curvature ky. If B:1C R> Sf is the 
até-Smarandache curve of a, then its frame fp, tp, é B} iS 
given by 





SE oo 3 
B V3 V3 V3 a 
t C2 C1—C3Kg C2Kg 
B\ = VA VA VA xit (23) 
Šg —chKg—C3(—cy+¢3Kg) C2C3—C1C2kg c1ı(c1-c3kg)-cf Š 
V34 V3A V3A 
where 
ies 2 2 259 2 2 
= (cy — C3Kg) — c3 +C3Kĝ , (cy — C3Kg) > C3 — 
c? KG and the Smarandache curve B is a spacelike curve. 


Furthermore, the geodesic curvature Ke 
given by 
KË = (k + cĝkg — C1C3)Ay + (—c1C2Kg + €2€3)A2 + 


(c? — C1C3Kg — c2)A3) x (a — czk) — c2 + 
Si -1 
c3K§) (24) 


where 


of curve p is 


A, = €2(c3Ky(c, — c3kg) — c2kgkg ) 

+(c, — c3kg) (Cc — czk) — c2 + ck) 
Ag = (c1 — C3kg )(c3kg (c1 — C3kg) — cĉkgkg) 

+(c3 — C3kg— czk?) (Cc — czk) — c2 + ck) 
Ag = €2Kg(C3kg(C, — C3Kkg) — CZK gKz ) 


+(kg(c1 — C3Kg) + C2K; ) (Cc — ek — c2 + ae) 


(25) 


Example 3.1. Let us consider a regular unit speed curve a 
on Hé defined by 


L432 432 
a(s) = (=+ 1, G = ; s—1). 





Then the orthonormal Sabban frame {a(s), t(s), &(s)} of 
a can be calculated as follows: 


if Gat)" CS ae 
a(s) = ( =k 1, eS 1) 
t(s) =(s—1, s—1, 1) 


(s-1)? (s-1)? 
(s) = (S p -1, s—1) 














The geodesic curvature of œ is —1. In terms of the 
definitions, we obtain the spacelike Smarandache curves 
on S$? according to the Sabban frame on Hg. 


First, when we take c} =1 and c, =vV3, then the 
aé-Smarandache curve is spacelike and given by 


GU J Sci, 29(1):69-77 (2016)/ Murat SAVAS, Atakan Tugkan YAKUT, Tugba TAMIRCI 73 


p(s(s)) = a(€ S)is-pi+1 


(S8)o-m-Va evan) 








and the Sabban frame of the spacelike aé-Smarandache 
curve is given by 


1 v3 
L] |z ° zle 
tp =/0 0 t 
Šg V3 o 4JL 
V2 V2 
and its geodesic curvature KË is —1. 


Second, when we take c} =1 and c, = V3, then the 
at-Smarandache curve is a spacelike and given by 





B(S5(s)) = (£ = + ¥3(s—1) +1, 


(s —1)? 
2 





H36- 9, (6-1 +8) 


and the Sabban frame of the spacelike a@t-Smarandache 
curve is given by 


a ae, i 
f Z Z a 
fe@)/=|V¥3 1 -v3 | 
SB 3 V3 

a 8 
and its geodesic curvature KË is —1. 


Third, when we take c;=1 and cz =1, then the 
t€-Smarandache curve is a spacelike curve and given by 


B(5(s)) = (e> +s-—1, eo +s-—2, s) 


and the Sabban frame of the spacalike tě-Smarandache 


curve is given by 

1 

f 0 z Iya 

tp =) 1 —1 | 
AIL 

še] |v2 - 

B 


and its geodesic curvature Kg 





al- = Al 


is —1. 
Finally, when we take c, = V3, cy = V3 and c3 = V3, 


then the até-Smarandache curve is a spacelike curve and 
given by 


B(5(s)) = ((s — 1)? + s, (s — 1)? + s — 2,2s — 1) 


and the Sabban frame of the spacelike até-Smarandache 
curve is given by 


1 1 1 
e Ei he 
ol- A 
3 KS 
SB 2 1 2 
and its geodesic curvature KË is —1. 


We give a curve œ in hyperbolic space and its 
Smarandache partners in de Sitter space in Figure 1. 


4. CURVES IN HYPERBOLIC SPACE AND DUAL 
TIMELIKE SMARANDACHE PARTNERS 


In this section we obtain the timelike Smarandache partners 
in de Sitter space of a curve in hyperbolic space. 


Theorem 4.1. Let a:1 c R > Hé be a regular unit speed 
curve lying fully on H . Then the timelike 
ač -Smarandache curve B:1C R > S? of æ does not 
exist. 


Proof. Assume that a: I c R > HÊ be a regular unit speed 
curve lying fully on Hé. Then the timelike 


aé-Smarandache curve B:1 c R > S? of æ is defined by 
= 1 
B(5(s)) = = (c1@(s) F c2Ẹ(s)) (26) 


C1 2ER\{0}, —c? + cå = 2. Differentiating (26) with 
respect to s and using (2), we obtain 

i dfdS 1 
p'(s) = dds p~ — C2Kg)t 


ds _ 1 
P ds — T C2Kg) 


where 


dS _ (cy—-C2Kg)* 
ds 2 


which is a contradiction. 
In the corollaries which follow, in a similar way as in the 
previous section, we obtain the Sabban frame {£ , tg, $ B} 


and the geodesic curvature Kh of a timelike 
Smarandache curve. We omit the proofs of Theorems 4.2, 
4.3, and 4.4. 


Definition 4.1. Let a:1 c R > HŽ be a regular unit speed 
curve lying fully on Hé. Then, the timelike 
at-Smarandache curve B:1 c R > S? of a is defined by 


B(5(s)) = S(c.a(s) + et(s)) 


where C,,C,€R\{0}, and —c? + c? = 2. 


74 GU J Sci, 29(1):69-77 (2016)/ Murat SAVAS, Atakan Tugkan YAKUT, Tugba TAMIRCI 


Corollary 4.1. Let a:1 c R > Hé be a regular unit speed 


curve lying fully on Hé with the Sabban frame {a, t, č} 
and the geodesic curvature Kg. If p:I c R> S? is the 
timelike at-Smarandache curve of a, then its frame 


fp, tg, če} is given by 


oN 


bez) —c3KG) fo(o—c33) —c3KG) Ta c5KG) 
and the corresponding geodesic curvature KË is given by 


P 1 2 — = 
Kg = 2g (E Kgdy — C1C2KgÀ2 2A3) 


C1 C2 
V2 
C4 _ Cang 
par a [2—c2 x2 ms C3 KG si; | 


where —c? + c? = 2 with c,,c,éR\{0}, ciki < 2 and 


Ay = cžkgky + c(2 — cġkĝ) 
Az = cC kgkg + (c2 — cak) (2 — cfr?) 


Ag = cfkfkg + (CyKg + Cokg)(2 — cf Ké). 


Definition 4.2. Let a:1 c R > Hé be a regular unit 
speed curve lying fully on Hé. Then, the timelike 


t€-Smarandache curve B:1 c R > S? of æ is defined by 


B(5(s)) = 5 (aat(s) + c2§(s)), 


where c1, C,€R\{0}, and c? + c# = 2. 


a is a unit speed curve on Hé (black line) 
B is a unit speed spacelike curve on S? (red line) 


t€-Smarandache curve 





at&-Smarandache curve 


Figure 1. Spacelike Smarandache partner curves of acurve a on Hé 


GU J Sci, 29(1):69-77 (2016)/ Murat SAVAS, Atakan Tugkan YAKUT, Tugba TAMIRCI 75 


Corollary 4.2. Let a:1 c R > HÊ be a regular unit speed 
curve lying fully on Hé with the Sabban frame {a, t, €} 
and geodesic curvature kg. If p:Ic R> S? is the 


timelike tě -Smarandache curve of a, then its frame 


fp, tg, če} is given by 


fi T2 
V2 V2 
Cu [c2-23 pe t | 

2kg — — C&C a ae 

[2(c?-2x2) |2(c?-2x2) |2(c?-2x2) 

and the corresponding geodesic curvature Kh is given by 
P 
Kg = ——- (2k yA + C1 Cz Az + Cy 2 A3) 


(c2-22) /2 
where c1, c2ER\{0}, cf + cf = 2, cf > 2K and 
Ay = 2c1kgk} — Czkg(c? — 2x2) 


Ag = —2cgK2K, + (c1 — cokg — ck) (c? — 2x2) 


Ag = 2cyK2Kg + (cukg—czk2)(c? — 2K). 


Definition 4.3. Let a: I c R > Hé be a regular unit speed 
curve lying fully on HÊ. Then at&-Smarandache curve 


B:1C R > S? of a is defined by 


B(5(s)) = 3 (cra (s) + czt (s) + c3Ẹ(s)) 
C1, Cp, CzER\{0}, —c? + c¥ + cZ = 3. 


Corollary 4.3. Let a:1 c R > HÊ be a regular unit speed 
curve lying fully on Hé with the Sabban frame {a, t, ë} 
and geodesic curvature kg. If p:Ic R> S? is the 


timelike aætě -Smarandache curve of a, then its frame 


fp, tg, Ee} is given by 


4 








Cy C2 C3 
V3 V3 V3 
Co Cy — C3Kg C2Kg a 
= VA VA" va | 
—c?kg — c3(—c1 + C3Kg) C263 — C1C2Kg (Cc, — C3kg) — c2 
3A* v3 3 A* 


where 

Ane 7 2,2 2 2 Be. 
= cf—-(c, — 3k) — 3g, (c1 — C3kg) < c3- cêk? 

and the corresponding geodesic curvature Ke is given by 


Bae. ~ + c3Kg — C1C3)A, +(- C1C2Kg + om 
“9 = +(c? — c1c3kg — c2 )A; 


5/, -1 
x (a-a = czk) — c2) ) 


where c4, C2, czER\{0}, —c? + c2 + cê = 3 and 


A, = cz(—c3Kg(c1 = C3Kq) T C3 Kg kg) 
+(c1 — C3Kg) (c3 —(c,- caka) — A 
Ag = (c1 — c3kg )(—c3k4 (c1 — c3kg) + cłkgkg) 
+(c — C3Kg—C2K2) (c3 —(c,- C3Kg) — 3x3) 
A3 = C2Kg(—cgkg (cy = C3Kg) K C3 Kg kg) 


+ (cx; F Kkg(c1 = C3Kg)) (c3 E (c E ezko) = 3x3) 


Example 4.1. Let us consider a regular unit speed curve q 
on Hé defined by 
a(s) = (coshs, sinhs,0). 


Then the orthonormal Sabban frame {a,t,€} of the curve 


a and the orthonormal Sabban frame fp, tg, $ B} of the 


curve f and the geodesic curvature KP 


a of a timelike 


Smarandache curve can be calculated as in the previous 
example. The curve œ and its Smarandache partners are 


given in Figure 2. 


76 





GU J Sci, 29(1):69-77 (2016)/ Murat SAVAS, Atakan Tugkan YAKUT, Tugba TAMIRCI 


a is a unit speed curve on Hé (black line) 
B is a unit speed timelike curve on S? (red line) 


at-Smarandache curve 


t¢-Smarandache curve até-Smarandache curve 


Figure 2. Timelike Smarandache partner curves of acurve a on Hé 


GU J Sci, 29(1):69-77 (2016)/ Murat SAVAS, Atakan Tugkan YAKUT, Tugba TAMIRCI 


CONFLICT OF INTEREST 


No conflict of interest was declared by the authors. 


REFERENCES 


[1] Ali, A. T., “Special Smarandache Curves in the 
Euclidean Space”, International Journal of 
Mathematical Combinatorics, Vol.2, 30-36, (2010). 


[2] Ashbacher, Cc. Smarandache geometries, 
Smarandache Notions Journal, Vol.8(13), 212-215, 
(1997). 


[3] Asil, V., Korpinar, T. and Bas, S., “Inextensible 
flows of timelike curves with Sabban frame in S?”, 


Siauliai Math. Semin., Vol.7(15), 5-12, (2012). 


[4] Cetin, M., Tuncer, Y. and Karacan, M. K., 
“Smarandache Curves According to Bishop Frame in 
Euclidean 3-Space”, Gen. Math. Notes, Vol.20(2), 
50-66, (2014). 


[5] Izumiya, S., Pei, D. H., Sano, T. and Toru, E., 
“Evolutes of hyperbolic plane curves”, Acta Math. 
Sinica (English Series), Vol.20(3), 543-550, (2004). 


[6] Koc Ozturk, E. B., Ozturk, U., Ilarslan, K. and 
Nesovic, E., “On Pseudohyperbolical Smarandache 
Curves in Minkowski 3-Space”, Int. J. of Math. and 
Math. Sci., 7, (2013). 


[7] O’Neill, B., Semi-Riemannian Geometry with 
Applications to Relativity, Academic Press, San 
Diego, London, (1983). 


[8] Sato, T., “Pseudo-spherical evolutes of curves on a 
spacelike surface in three dimensional 
Lorentz-Minkowski space”, J. Geom. Vol.103(2), 
319-331, (2012). 


[9] Tamirci, T., “Curves on surface in three dimensional 
Lorentz-Minkowski space”, Master Thesis, Niğde 
University Graduate Scholl Of Natural and Applied 
Sciences, Niğde, (2014). 


[10] Taskopru, K. and Tosun, M., “Smarandache Curves 
on S2”, Bol. Soc. Paran. Mat. (3s.) Vol. 32(1), 


51-59, (2014). 


[11] Turgut M. and Yilmaz S., “Smarandache Curves in 
Minkowski Space-time”, International J. Math. 
Combin., Vol.3, 51-55, (2008). 


[12] Yakut, A., Savas, M. and Tamirci T., “The 


Smarandache Curves on S? and Its Duality on Hé”, 


Journal of Applied Mathematics, 12, (2014). 


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