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

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):

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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