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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. 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