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International J.Math. Combin. Vol.3(2016), 1-16 


Spacelike Smarandache Curves of Timelike Curves in 


Anti de Sitter 3-Space 


Mahmut Mak and Hasan Altinbas 


(Ahi Evran University, The Faculty of Arts and Sciences, Department of Mathematics, Kırşehir, Turkey) 


E-mail: mmak@ahievran.edu.tr, hasan.altinbas@ahievran.edu.tr 


Abstract: In this paper, we investigate special spacelike Smarandache curves of timelike 
curves according to Sabban frame in Anti de Sitter 3-Space. Moreover, we give the rela- 
tionship between the base curve and its Smarandache curve associated with theirs Sabban 
Frames. However, we obtain some geometric results with respect to special cases of the 
base curve. Finally, we give some examples of such curves and draw theirs images under 


stereographic projections from Anti de Sitter 3-space to Minkowski 3-space. 


Key Words: Anti de Sitter space, Minkowski space, Semi Euclidean space, Smarandache 


curve. 


AMS(2010): 53A35, 53C25. 


§1. Introduction 


It is well known that there are three kinds of Lorentzian space. Minkowski space is a flat 
Lorentzian space and de Sitter space is a Lorentzian space with positive constant curvature. 
Lorentzian space with negative constant curvature is called Anti de Sitter space which is quite 
different from those of Minkowski space and de Sitter space according to causality. The Anti de 
Sitter space is a vacuum solution of the Einstein’s field equation with an attractive cosmological 
constant in the theory of relativity. The Anti de Sitter space is also important in the string 
theory and the brane world scenario. Due to this situation, it is a very significant space from 
the viewpoint of the astrophysics and geometry (Bousso and Randall, 2002; Maldacena, 1998; 
Witten, 1998). 

Smarandache geometry is a geometry which has at least one Smarandachely denied axiom. 
An axiom is said to be Smarandachely denied, if it behaves in at least two different ways 
within the same space (Ashbacher, 1997). Smarandache curves are the objects of Smarandache 
geometry. A regular curve in Minkowski space-time, whose position vector is composed by 
Frenet frame vectors on another regular curve, is called a Smarandache curve (Turgut and 
Yilmaz, 2008). Special Smarandache curves are studied in different ambient spaces by some 
authors (Bektaş and Yüce, 2013; Koc Ozturk et al., 2013; Taşköprü and Tosun, 2014; Turgut 
and Yimaz, 2008; Yakut et al., 2014). 


1Received December 03, 2015, Accepted August 2, 2016. 


2 Mahmut Mak and Hasan Altinbas 


This paper is organized as follows. In section 2, we give local diferential geometry of non- 
dejenerate regular curves in Anti de Sitter 3-space which is denoted by HÌ. We call that a curve 
is AdS curve in HÌ if the curve is immersed unit speed non-dejenerate curve in H}. In section 3, 
we consider that any spacelike AdS curve 8B whose position vector is composed by Frenet frame 
vectors on another timelike AdS curve a in H?. The AdS curve @ is called AdS Smarandache 
curve of a in H}. We define eleven different types of AdS Smarandache curve 3 of @ according 
to Sabban frame in H. Also, we give some relations between Sabban apparatus of a and 8 for 
all of possible cases. Moreover, we obtain some corollaries for the spacelike AdS Smarandache 
curve @ of AdS timelike curve œ which is a planar curve, horocycle or helix, respectively. In 
subsection 3.1, we define AdS stereographic projection, that is, the stereographic projection from 
H? to RÌ. Then, we give an example for base AdS curve and its AdS Smarandache curve, which 
are helices in HÌ. Finally, we draw the pictures of some AdS curves by using AdS stereographic 
projection in Minkowski 3-space. 


§2. Preliminary 


In this section, we give the basic theory of local differential geometry of non-degenerate curves 
in Anti de Sitter 3-space which is denoted by H}. For more detail and background about Anti 
de Sitter space, see (Chen et al., 2014; O’Neill, 1983).. 


Let R4 denote the four-dimensional semi Euclidean space with index two, that is, the real 
vector space R* endowed with the scalar product 





(x,y) = —T1Y1 — L2Y2 + T3Y3 + Lays 


for all æ = (21, £2, £3, £4), Y = (Y1, Y2, Y3, ya) E Rt. Let {e1, €2, €3, e4} be pseudo-orthonormal 
basis for R$. Then di; is Kronecker-delta function such that (e;,e;) = dij¢; for €1 = €2 = 
—1l, e3 = £4 = 1. 
A vector x € R3 is called spacelike, timelike and lightlike (null) if (2,2) > 0 (or æ = 0), 
(x,a@) <0 and (x,a) = 0, respectively. The norm of a vector x € R$ is defined by ||z|| = 
\(a,a)|. The signature of a vector æ is denoted by 


1, æ is spacelike 
sign(x) = 0, 2 is null 


—1, 2 is timelike 
The sets 


S3 = {ER} | (x,e)=1} 
Hy {x € R3 | (v,#) =—1} 


II 


are called de Sitter 3-space with index 2 (unit pseudosphere with dimension 3 and index 2 in 
R$) and Anti de Sitter 3-space (unit pseudohyperbolic space with dimension 3 and index 2 in 


Spacelike Smarandache Curves of Timelike Curves in Anti de Sitter 3-Space 3 


R4), respectively. 


The pseudo vector product of vectors æ! , æ? ,x° is given by 


—e€; —€Q €3 &4 
1 1 1 1 
x £ Wa 
1 2 3 4 
zl Az? AT? = 


zi © a3 i 
3 3 Bo aad 
zi T3 T3 Tj 


where {e1, e2, e3, e4} is the canonical basis of RẸ and x = (zį, x$, x$, x4), i = 1,2,3. Also, it 
is clear that 

(£, £! Aa? Ag’) = det(ax, x, x”, x°) 
for any x € R$. Therefore, x! A x? ^ x? is pseudo-orthogonal to any x’, i = 1, 2,3. 

We give the basic theory of non-degenerate curves in H}?. Let a : I > H? be regular curve 
(i.e., an immersed curve) for open subset J C R. The regular curve œ is said to be spacelike or 
timelike if & is a spacelike or timelike vector at any t € I where &(t) = da/dt. The such curves 
are called non-degenerate curve. Since a is a non-degenerate curve, it admits an arc length 
parametrization s = s(t). Thus, we can assume that a(s) is a unit speed curve. Then the unit 
tangent vector of æ is given by t(s) = a’(s). Since (a(s) ,a(s)) = —1, we have (a(s),t’(s)) = 
—6, where 6; = sign(t(s)). The vector t’(s) — 6,a(s) is pseudo-orthogonal to a(s) and t(s). 
In the case when (a@”(s),a’’(s)) # —1 and t(s) — ôia(s) # 0, the pirinciple normal vector 
and the binormal vector of æ is given by n(s) = eee. and b(s) = a(s) A t(s) A n(s), 
respectively. Also, geodesic curvature of œ are defined by Kg(s) = ||t’(s) — 6,a(s)||. Hence, we 
have pseudo-orthonormal frame field {a(s), t(s),(s), b(s)} of R$ along a. The frame is also 
called the Sabban frame of non-dejenerate curve a on HÌ such that 


t(s) An(s) A b(s) = 63a(s), n(s) A B(s) A a(s) = 01 63 t(s) 
b(s) A a(s) At(s) = —d263n(s), a(s)At(s) A n(s) = b(s). 


where sign(t(s)) = 61, sign(n(s)) = 62, sign(b(s)) = 63 and det(a,t,n,b) = —ds. 

Now, if the assumption is < @” (s), œ” (s) >4 —1, we can give two different Frenet-Serret 
formulas of @ according to the causal character. It means that if 6; = 1 (6, = —1), then @ is 
spacelike (timelike) curve in HÌ. In that case, the Frenet-Serret formulas are 


a’(s) 0 1 0 0 a(s) 
t(s) S 61 0 Kig(s) 0 t(s) (2) 
n'(s) 0 —61d2k,_(s) 0 —6163Tg(s) n(s) 
b'(s) 0 0 ô1ô2Tg (5) 0 b(s) 


— 91 det(a(s),a’(s),a’’(s),a’’’(s)) f 


where the geodesic torsion of œ is given by Tg(s) CaO 


Remark 2.1 The condition < a”(s),a”(s) >4 —1 is equivalent to kg(s) Æ 0. Moreover, we 


4 Mahmut Mak and Hasan Altinbas 


can show that K,(s) = 0 and t’(s) — 6,a(s) = 0 if and only if the non-degenerate curve @ is a 
geodesic in HÌ. 


We can give the following definitions by (Barros et al., 2001; Chen et al., 2014). 
Definition 2.2 Let a: I C R — HÌ is an immersed spacelike (timelike) curve according to the 
Sabban frame {a,t,n,b} with geodesic curvature kg and geodesic torsion Tg. Then, 

(1) If t =0 , a is called a planar curve in H3; 

(2) If kg =1 andt, =0, a is called a horocycle in H3}; 


(3) If Tg and kg are both non-zero constant, a is called a helix in HÌ. 


Remark 2.3 From now on, we call that æ is a spacelike (timelike) AdS curve if a : I C R— HÌ 


is an immersed spacelike (timelike) unit speed curve in H}. 


§3. Spacelike Smarandache Curves of Timelike Curves in HÌ 


In this section, we consider any timelike AdS curve œ = a(s) and define its spacelike AdS 
Smarandache curve B = G(s*) according to the Sabban frame {a(s), t(s), n(s),b(s)} of a in 
H? where s and s* is arc length parameter of a and 8, respectively. 


Definition 3.1 Let a = a(s) be a timelike AdS curve with Sabban frame p = {a,t,n, b} 
and geodesic curvature Kg and geodesic torsion Tg. Then the spacelike vjvj—Smarandache AdS 


curve B = 3(s*) of a is defined by 


a a Gals v;(s 
p(s S i(s) + bu;(s)), (3) 


where vi, vj E p fori # j and a,b E R such that 


EAA] F 


nb | a? +b? = -2 
(Undefined) 
Theorem 3.2 Leta = a(s) be a timelike AdS curve with Sabban frame y = {a,t,n,b} and 


geodesic curvature kg and geodesic torsion Tg. If B = B(s*) is spacelike vivj— Smarandache AdS 





curve with Sabban frame {8,tg,ng,bg} and geodesic curvature Kg, geodesic torsion Tg where 
vi, vj E p fori #j, then the Sabban apparatus of B can be constructed by the Sabban apparatus 


Spacelike Smarandache Curves of Timelike Curves in Anti de Sitter 3-Space 5 


of a such that 


a 
at b?kg(s)> -2>0 
an b?14(s)” — (bkg(s) +a)? > 0 


b?t4(s)” — a? >0 . (5) 





Proof We suppose that v;v; = at. Now, let B = G(s*) be spacelike at—Smarandache 
AdS curve of timelike AdS curve a = a(s). Then, 8 is defined by 


x = ae aa(s 8 
B(s*(s)) = Ja! (s) + bt(s)) (6) 


such that a? + b? = 2, a,b € R from the Definition 3.1. Differentiating both sides of (6) with 
respect to s, we get 











Po) = EE = T (aa"(s) +) 
and by using (2), 
ta(s*(s)) = = (atls) +b(—as) + m(s)n(s))). 
where 
ds* b2K4(s)” — 2 





with condition b?«(s)? — 2 > 0. 


From now on, unless otherwise stated, we won’t use the parameters ”s” and ”s*” in the 
? ? p 
following calculations for the sake of brevity). 


Hence, the tangent vector of spacelike at—Smarandache AdS curve @ is to be 


1 
ts = Ve (—ba + at + bkgn), (8) 


where o = b?K,? — 2. 


Differentiating both sides of (8) with respect to s, we have 


2 
ta’ = ue (Ara + Azt + A3n + A4b) (9) 


6 Mahmut Mak and Hasan Altınbaş 


by using again (2) and (7), where 











MA = Bkgk,’ — ao 
Ag = —ab kgk,! +b (rg? — 1) o (10) 
às = —2bkg' + akgo 
Mo = bDKgTgO . 
Now, we can compute 
1 
a i I ((2A1 — ao”) @ + (2g — bo?) t + 2A3n+2A4b) (11) 
and i 
lts’ — Bl| = a —o4 + 2 (ad, + bà2) 02 + 2 (—A1? — Ao? + As? + A4’). (12) 
From the equations (11) and (12), the principal normal vector of 6 is 
1 
ng = Th ((2A1 — ao?) a + (2A2 — bo?) t + 2Agn + 2A4b) (13) 
and the geodesic curvature of 8 is 
poe 
a=, (14) 
where 
u = —0* + 2 (aì + bra) 0? +2 (A1? — Ag? + A3? + Ag”). (15) 





Also, from the equations (6), (8) and (13), the binormal vector of G as pseudo vector 
product of B, tg and ng is given by 


1 
bg = van ((—b?KgA4) a + (abrgà4) t+ 2Aan + (=D? Kg A1 + abkigAz — 2A3) b) . (16) 


Finally, differentiating both sides of (9) with respect to s, we get 


t H —2 (2A10" — (Aq! = à2)o) a+ (2A20" = (Ai + do! + kgà3)o) t (17) 
B = 719 

c oe (2A30' — (KgA2 + Ag! — TgAg)o) n + (240 — (TgÀ3 + à4')o) b 
by using again (2) and (7). Hence, from the equations (6), (8), (9), (14) and (17), the geodesic 
torsion of 8 is 


pO 2 Kg (bA1 E ar2)(bTgA3 +aX4 + bAs’) = bkig (DAY = ano’) X4 


t= (18) 
OM \ 4275(A37 + Ag?) + abkg? Agàs — 2(A3’ A4 — A344”) 





under the condition a? + b? = 2. Thus, we obtain the Sabban aparatus of B for the choice 
viv; = at. 


It can be easily seen that the other types of vjv;—Smarandache curves 3 of œ by using 


Spacelike Smarandache Curves of Timelike Curves in Anti de Sitter 3-Space 7 











same method as the above. The proof is complete. 





Corollary 3.3 Let a = a(s) be a timelike AdS curve and B = B(s*) be spacelike vivj— Smarandache 


AdS curve of a, then the following table holds for the special cases of a under the conditions 


(4) and (5): 


C 





Definition 3.4 Let a = a(s) be a timelike AdS curve with Sabban frame p = {a,t,n, b} and 
geodesic curvature Kg and geodesic torsion Tg. Then the spacelike vivjuk— Smarandache AdS 


Curve B = B(s*) ofa is defined by 
B * = = Qvu;(S) + bv;(S) + CUR(S 
(s (s)) WEL i ( ) b 5 ( ) k( )), (19) 


where vi, vj, vk E Y fori # j Ak anda,b,c E R such that 





(20) 





Theorem 3.5 Let a = a(s) be a timelike AdS curve with Sabban frame y = {a,t,n,b} and 
geodesic curvature kg and geodesic torsion Tg. If B = B(s*) is spacelike vivjvg— Smarandache 
AdS curve with Sabban frame {B,tg,ng,bg} and geodesic curvature Kg, geodesic torsion Tg 


where vi, vj, uk E p fori A j Ak, then the Sabban apparatus of B can be constructed by the 


8 Mahmut Mak and Hasan Altınbaş 


Sabban apparatus of œ such that 


t 


a(s)? — 2ackg(s) + c2(r,(s)? —1) -—3>0 


(bkg(s) — crg(s))” — (2 +3) > 0 (21) 
b? + c*) rgs)? — (a+ bkg(s))” >0 
(ateg(s) — erg(s))” +8? (74(s)” — Kg(s)”) — a? > 0 


(- 2) 
( 





Proof We suppose that vivjuk = atb. Now, let B = B(s*) be spacelike atb—Smarandache 
AdS curve of timelike AdS curve a = a(s). Then, 8 is defined by 


si ed s) + cb(s 
p(s (8) = TAK (s) + bt(s) + cb(s)) (22) 


such that a? +b? —c? = 3, a,b,c € R from the Definition 3.4. Differentiating both sides of (22) 


with respect to s, we get 











Ix = dB ds* = 1 , , , 
Po) = FE = g (aa'(s) H) D) 
and by using (2), 
tols" (s)) FE = == (at(s) +b (-aa(s) + sgls)n(s)) + e(—r9(s)n(5))) 


where 
ds* (b rg(s)— cTals))? — (c? +3) 


i arr ar (23) 





with the condition (brg(s) — cT, (s))? — (2 +3) >0. 


(From now on, unless otherwise stated, we won’t use the parameters “s” and “s*” in the 


following calculations for the sake of brevity). 
Hence, the tangent vector of spacelike atb—Smarandache AdS curve 8 is to be 


tg = 





(—ba + at + (brg — cT) n), (24) 


al- 


where o = (bkg — CT)" — (c? +3). 
Differentiating both sides of (24) with respect to s, we have 


3 
tg’ = ue (Aia + At + Asn + 4b) (25) 


Spacelike Smarandache Curves of Timelike Curves in Anti de Sitter 3-Space 9 


by using again (2) and (23), where 











à = b (bkg — CTg) (bkg — CTg') — ao 
A2 = —a(bkg — CTg) (brg' — CT’) + (b (—1 + kg?) — CkgTg) 0 (26) 
às = — (34.7) (bkg' — cTg') + argo 
`M = Tg (bkg — CTg) 0 
Now, we can compute 
1 
tg’ — B= TA ((3A1 — ao?) a + (3A2 — bo”) t + 8A3n + (3A4 — co”) b) (27) 
and í 
lte’ — Bll = = 2 (a1 + bAg — c4) 02 +3 (A1? — Az? + A3? + 4°). (28) 
From the equations (27) and (28), the principal normal vector of 6 is 
1 
ng = Tai ((3A1 — ao’) a+ (32 — bo?) t + 3A3n + (34 — co”) b) (29) 
and the geodesic curvature of @ is 
a VE 
Kg = oo? (30) 
where 
u = —o* + 2 (ary + dAg — cà4) o? +3 (~A? — AQ” + Az? + Ag?) (31) 


Also, from the equations (22), (24) and (29), the binormal vector of B as pseudo vector 
product of 6, tg and ng is given by 





(c(bkg — CTg)A2 — (ac)A3 — b(bKg — cTg)A4) Q 
pa =o (c(bkg — CTg)A1 + (bc)A3 — a(bkg — CT) Aa) t 
VOE | — ((ac)A1 + (be)A2 — (Ê + 3)Aa) n 
— ((brg — cTg)(bA1 — aà2) + (c? + 3)A3) b 





Finally, differentiating both sides of (25) with respect to s, we get 


belt 3 | Cao- Qa! = Aa)o) a + (2d20! — (Ai + 2! + KgAs)o) t (33) 
6 =- 
o2 \ ob (2A30" — (Kgà2 + Az’ — Tgà4)a) n + (2Aga’ — (TgÀs + Aa’)o) b 


by using again (2) and (23). Hence, from the equations (22), (24), (25), (30) and (33), the 
geodesic torsion of 6 is 


c (aà — Xo (bkg — CTg)) (à2 — Ar’) — c(bA3 + Ài (bkg — CTg)) (ài + Kg A3 + 2") 
3 
Tg = a +4 (bkg — CTg) (b (A2 — à1”) +a (ài + Kg A3 + d2')) + c (aàı + bà2) (Kgà2 = Tg A4 + 3’) 


— (3 + Ê) Ag (KgA2 — Tgàa + AB’) + ((3 + 2) Az + (DAL — aà2) (bkg — CTg)) (TeAB + Aa’) 
(34) 
under the condition a? +b? — c? = 3. Thus, we obtain the Sabban aparatus of 3 for the choice 








10 Mahmut Mak and Hasan Altinbas 


vivjuk = atb. 


It can be easily seen that the other types of vivjvk—Smarandache curves 6 of œ by using 











same method as the above. The proof is complete. 





Corollary 3.6 Let a = a(s) be a timelike AdS curve and B = B(s*) be spacelike vivjvk— Smarandache 
AdS curve of a, then the following table holds for the special cases of a under the conditions 
(20) and (21): 


Definition 3.7 Let œ = a(s) be a timelike AdS curve with Sabban frame {a,t,n,b} and 
geodesic curvature kg and geodesic torsion Tg. Then the spacelike œtnb— Smarandache AdS 
curve B = B(s*) of a is defined by 





A(s*(s)) = loal) pa rene EE (35) 
where ao, bo, co, do E R such that 


ao? + b2 — co? — do? = 4. (36) 


Theorem 3.8 Let a = a(s) be a timelike AdS curve with Sabban frame {a,t,n,b} and 
geodesic curvature Kg and geodesic torsion Tg. If B = G(s*) is spacelike atnb—Smarandache 
AdS curve with Sabban frame {B,tg,ng,bg} and geodesic curvature Kg, geodesic torsion Tg, 
then the Sabban apparatus of B can be constructed by the Sabban apparatus of œ under the 


condition 
(borgs) — dotg(s))” — (ao + Cotig(s))” + co?T4(s)? — bo? > 0. (37) 


Proof Let B = B(s*) be spacelike atnb—Smarandache AdS curve of timelike AdS curve 
a=a(s). Then, 8 is defined by 


B(s*(s)) = Fj (aoa) + bot(s) + con(s) + dob(s)) (38) 


such that ap? + bo” — co? — do” = 4, ao, bo, co, do € R from the Definition 3.7. Differentiating 


Spacelike Smarandache Curves of Timelike Curves in Anti de Sitter 3-Space 11 


both sides of (38) with respect to s, we get 


_ dB dst 1 





BS) saa Wa (aoa (s) + bot (s) + con’ + dob’ (s)) 
and by using (2), 
ta(st(s)) = a (aot(s) + bo (—a(s) + g(s)n(s)) + co (Kg(s)t(s) + T9(s)b(s)) + do (—T9(s)n(s))) 
where 
ds* (borg (s) — dotg(s))” — (ao + cotig(s))” + co2T4(s)* — bo” 





ds 4 
with the condition (bok,g(s) — dotg(s))” — (ao + Cokg(s))? + cot _(s)* — by” > 0. 


66? 


From now on, unless otherwise stated, we won’t use the parameters “s” and “s*” in the 
? ? p 
following calculations for the sake of brevity). 


Hence, the tangent vector of spacelike atnb—Smarandache AdS curve @ is to be 


(—boa + (ao + cokg) t + (borg — doTg) Nn + coT,b) , (40) 


1 
ta = -z 
where o = (bokg — doTg)” — (ao + cokg)? + co? 7,7 — bo”. 
Differentiating both sides of (40) with respect to s, we have 
ta’ = = (Aa + Agt + Aan + Ab) (41) 


by using again (2) and (39) where 











ài = —bo (aoco + cõrg — bo (borg — doTg)) Kg’ + bo (CTs — do (borg — doTg)) Ty’ — (ao + CoKg) o 
— (—b6 (co + aokg) + bodo (ao — CoKg) Tg + Co (co? + do”) r2) Kg’ 
+ (bodokg (ao + cory) — (cå + dG) (ao + cosg) Ty) Te’ + (bo (Kg — 1) — dokgTg) o 
TE — (aoco (borg + doTg) + ror (dokgTg — bo (r2 — 1)) + bo (4 + d3) ) Kg’ 
+ (2aocodokg +e (do (1 + Ka) E bokgTg) + do (4 + do)) Tg 4 (aokg F co (Kg AD o 
TAE co (co (ao + Cokg) — bo (boKg — doTg)) TgKg' + 


Co 
Co (Ta (bodokg = (6 + do) Tg) + o) Tq + (bokg — doTg) To 
Now, we can compute 
1 
tg’ —-B= 32 ((4A1 — ago”) a+ (Ar2 = boo?) t+ (43 — coo?) n + (44 — doo?) b) (43) 


and 





1 
lts" — B\| = z —04 + 2 (aoà1 + boz — coàs — doà4) o2? + 4 (—A1? — Ag” + Az? + Ag”). 


12 Mahmut Mak and Hasan Altinbas 


From the equations (43) and (44), the principal normal vector of @ is 


1 
m= ((4A1 — aoa?) a + (4à2 — boo?) t + (4A3 — coo?) n + (444 — doo?) b) (45) 
and the geodesic curvature of @ is 
~ _ vB 
Kg = Be (46) 
where 
u = —0 + 2 (agdr + boà2 — coàs — doà4) o? + 4 (—A1? — Ag? + Ag? + 4?) (47) 


Also, from the equations (38),(40) and (45), the binormal vector of B as pseudo vector 
product of B, tg and ng is given by 


J; 
bg = Wie ((—bp tig A4 + co(—dokgA3 + aoa) = Ca(tA2 = Kg) _ do(doTg 2 + aoA3) 


+bo(coTgA3 + do(KgA2 + TyA4)) )@ + (bo(—do(KgA1 + A3) + (Co + Aokg)A4) 

+(cA1 — aocoAs + do(doA1 — aoà4))Tg)t + (aĝ A4 — bo(doA2 — boAa) 

—coA1(dokg — boTg) — Ao(dorA1 + Co(TeA2 — KgAa)))n + (Cog At = a Às 

—b6(KgA1 + Az) + bo(coAz + doTgà1) + ao(co(à1 — KgA3) + (borg — doTg)à2t))b) (48) 





Finally, differentiating both sides of (41) with respect to s, we get 


" —4 (2A10" = a’ ar d2) a) a+ (2A20" = (Ai + do! + KgA3)0) t 
a = (49) 


+ (2A30” — (KgA2 + à3' — TgA4)o) n+ (240 — (Tg à3 + à4')o) b 


by using again (2) and (49). Hence, from the equations (38), (40), (41), (46) and (49), the 
geodesic torsion of 6 is 


T = t ((boKgAa + (ao + corg) (doàs — CcoAa) + (cå + dG) Ty A2 

—bo(coTgA3 + do(KgA2 + TgA4)))(à2 — 4) 
+(bo(—do(KgA1 + A3) + (co + dokg) Aa) + (Coa — agcodAs 
+dotg(doA1 — aoA4)))(A1 + Kg A3 + A2) + (do((@o + Cokg)Ar 
+boA2) — (ao(ao + cong) + b3)A4 — CoTg(boA1 — @oA2z))(KgA2 — A4Ty + A5) 
(—cokgA1 + aiA3 + b3(KgA1 + A3) — bolcoàz + doA1 Tg) + @o(Co(—A1 + Kg As) 
+A2(—bokg + doTg)))(A3Tg + A4)) (50) 























under the condition (36). The proof is complete. 





Corollary 3.9 Let a = a(s) be a timelike AdS curve and B = B(s*) be spacelike atnb— Smarandache 
AdS curve of a, then the following table holds for the special cases of œ under the conditions 
(36) and (87): 


Spacelike Smarandache Curves of Timelike Curves in Anti de Sitter 3-Space 13 


a a is planar curve | œ is horocycle | a is helix 





Consequently, we can give the following corollaries by Corollary 3.3, Corollary 3.6, Corol- 
lary 3.9. 


Corollary 3.10 Let a be a timelike horocycle in HÌ. Then, there exist no spacelike Smaran- 
dache AdS curve of a in HÌ. 


Corollary 3.11 Let a be a timelike AdS curve and B be any spacelike Smarandache AdS curve 
of a. Then, a is helix if and only if B is helix. 


§4. Examples and AdS Stereographic Projection 


Let RÌ denote Minkowski 3-space (three-dimensional semi Euclidean space with index one), 
that is, the real vector space R? endowed with the scalar product 


(@,Y), = -TIY + T2 Y2 + T3 Y3 
for all & = (T7, T2, £3), Y = (Yi, Y2, Y3) € RÌ. The set 
Si = {ze R}|(z, Zz), =1} 


is called de Sitter plane (unit pseudosphere with dimension 2 and index 1 in RẸ). Then, the 
stereographic projection ® from H3 to R? and its inverse is given by 








® : HANS RAS? d(x) = (| B “M 
At <i oY 1) (a) Lae l+ Tay 


and 
= = ye 14+ (%,z) 277 273 273 
®-': RAS? > HC, 8t (@) = | —, — _, ——_ , — 
LSD pts (2) 1—(#,2%),’1—(%,@),’1—(#,@),’1- (2,2), 
according to set T = {x € H? | zı = —1}, respectively. It is easily seen that ® is conformal 
map. 


Hence, the stereographic projection © of HÌ is called AdS stereographic projection. Now, 


we can give the following important proposition about projection regions of any AdS curve. 


Proposition 4.1 Let ® be AdS stereographic projection. Then the following statements are 
satisfied for all x € HÌ: 


(a) xı > -14 (®(a),8(x)), <1; 


* 


(b) zı < -1 & (®(x),®(a)), >1. 


* 


14 Mahmut Mak and Hasan Altinbas 


Now, we give an example for timelike AdS curve as helix and some spacelike Smarandache 
AdS curves of the base curve. Besides, we draw pictures of these curves by using Mathematica. 


Example 4.2 Let AdS curve a be 
a(s) = ( vBeosn( vs, 21/4 cosh(V5s) + y 1 + V2sinh(V5s), 
V2sinh(V2s), y 1 + V2cosh(V5s) + 2/4 sinh vs) : 


Then the tangent vector of œ is given by 


t(s) = (z sinh(V2s), 4/5 (1 + v2) cosh V5s + 2'/4V/5 sinh(V5s), 


2 cosh(vV2s), 21/45 cosh(V5s) + 4/5 (1 + v2) sont V5) f 





and since 


(t(s), t(s)) =—1, 


a is timelike AdS curve. By direct calculations, we get easily the following rest of Sabban 


frame’s elements of a: 


n(s) = (envas, 23/4 cosh(V5s) + 4/2 (1 + v2) sinh(V5s), 
sinh(V2s), 4/2 (1 + v2) cosh(V5s) + 23/4 sont V5) 


b(s) = (v3 sinh(V2s), 21/1 + V2 cosh(V5s) + 25/4 sinh(V5s), 
V5 cosh(V2s), 25/4 cosh(V5s) + 2V 1 + VBsini(V5s)) : 
and the geodesic curvatures of œ are obtained by 
Kg = 3v2, Tg = — v10. 


Thus, @ is a helix in HÌ. Now, we can define some spacelike Smarandache AdS curves of @ as 


the following: 


an[3(s*(s)) = 5 ( 3a(s) za n(s)) 
anbb(*(5)) = 4 (VGax(s) - V2n(s) + b(s)) 


atnbp(s*(s)) = $ (als) — $4(s) + $n(s) + $0(s)) 


Spacelike Smarandache Curves of Timelike Curves in Anti de Sitter 3-Space 15 


and theirs geodesic curvatures are obtained by 


ankg = 1.9647, anTg = —0.0619 
anbkg = 1.9773, anbTg = —0.0126 
atnbkg = 2.0067, atnbT = —0.0044 


in numeric form, respectively. Hence, the above spacelike Smarandache AdS curves of @ are 
also helix in HÊ, seeing Figure 1. 


curve a in-Smarandache curve 8 























(b) 


atnb -Smarandache curve F 


anb-Smarandache curve 























Figure 1 


16 


Mahmut Mak and Hasan Altinbas 


where, (a) is the timelike AdS helix a, (b) the spacelike an-Smarandache AdS helix of a, (c) 
the spacelike anb-Smarandache AdS helix of aœ and (d) the spacelike atnb-Smarandache AdS 


helix of a. 


§5. 


Conflict of Interests 


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


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1 
2 











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