The Smarandache Curves on H 0

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1 Gazi University Journal of Science GU J Sci 9():69-77 (6) The Smarandache Curves on H Murat SAVAS, Atakan Tugkan YAKUT,, Tugba TAMIRCI Gazi University, Faculty of Sciences, Department of Mathematics, 65 Teknikokullar, Ankara, Turkey Nigde University, Faculty of Arts and Sciences, Department of Mathematics, 55, Nigde, Turkey ABSTRACT Received: 9/7/5 Accepted: 9//5 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.. INTRODUCTION 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 characterization and classification of these curves. In the theory of regular curves, there are some special curves, such as Bertrand, Mannheim, involute, evolute, and pedal curves. In the light of the literature, in ] 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 ]. Smarandache curves in Euclidean or non-euclidean Corresponding author, sevaty@nigde.edu.tr

2 7 GU J Sci, 9():69-77 (6)/ 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, 4, 6, 9,, ]. 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 ]. 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 {, 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 -Smarandache curve of a curve 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. ]. In her Master thesis 9], Tamirci also studied the curves in de Sitter and hyperbolic spaces using a similar framework.. 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 -dimensional vector space equipped with the scalar product, which is defined by x, y L = x y + x y + x y. The space E = (R,, L ) is a pseudo-euclidean space, or Minkowski -space. The unit pseudo-sphere (de Sitter space) with index one S in E is given by S = {xεe x, x L = }. The unit pseudo-hyperbolic space H = {xεe x, x L = } has two connected components H,+ and H,. Each of them can be taken as a model for the -dimensional hyperbolic space H. In this paper, we take H,+ = H. Recall that a nonzero vector xεe is spacelike if x, x L >, timelike if x, x L <, and null (lightlike) if x, x L =. The norm (length) of a vector xεe is given by x L = x, x L and two vectors x and y are said to be orthogonal if x, y L =. Next, we say that an arbitrary curve = (s) in E can locally be spacelike, timelike, or null(lightlike) if all of its velocity vectors (s) are, respectively, spacelike, timelike, or null for all sει. If (s) L for every sει, then is a regular curve in E. A spacelike(timelike) regular curve is parameterized by a pseudo-arc length parameter s, which is given by : Ι R E, and then the tangent vector (s) along has unit length, that is (s), (s) L = ( (s), (s) L = ) for all sει. Let x = (x, x, x ), y = (y, y, y ), z = (z, z, z )ε E. The Lorentzian pseudo-vector cross product is defined as follows: x y = ( x y + x y, x y x y, x y x y ) () We remark that the following relations hold: (i) x y, z L = det (x y z) (ii) x (y z) = x, y L z x, z L y Let : Ι R H curve lying fully in H. Then its position vector is a timelike vector, which implies that the tangent vector t = and normal vector are unit spacelike vector for all sει. We have the orthonormal Sabban frame {(s), t(s), (s)} along the curve, (s) = (s) t(s) unit spacelike vector. The corresponding Frenet formula of, according to the Sabban frame, is given by (s) = t(s) { t (s) = (s) + κ g (s) (s) () (s) = κ g (s)t(s) κ g (s) = det ((s), t(s), t (s)) geodesic curvature of on H and s arc length parameter of. In particular, the following relations hold: = t, = t, t = () Now we define a new curve : Ι R S to be a regular unit speed curve lying fully on S for all sει such that its position vector is a unit spacelike vector according to the combination of the position, tangent, and normal vectors of. In this case = t may be a unit timelike or spacelike vector. Definition.. A unit speed regular curve (s (s)) lying fully in Minkowski -space, whose position vector is associated with Sabban frame vectors on another regular curve (s), is called a Smarandache curve]. In the light of this definition, if a regular unit speed curve : Ι R H is lying fully on H for all sει and its position vector is a unit timelike vector, then the

3 GU J Sci, 9():69-77 (6)/ Murat SAVAS, Atakan Tugkan YAKUT, Tugba TAMIRCI 7 Smarandache curve = (s (s)) of the curve is a regular unit speed curve lying fully in S or H. In our work we are interested in curves lying in S and so we have the following: a) The Smarandache curve (s (s)) may be a spacelike curve on S or, b) The Smarandache curve (s (s)) may be a timelike curve on S for all sει. Let {, t, } and {, t, } be the moving Sabban frames of and, respectively. Then we have the following definitions and theorems of Smarandache curves = (s (s)).. CURVES ON H AND ITS SPACELIKE SMARANDACHE PARTNERS ON S Let curve on H. Then the Smarandache partner curve of is either in de Sitter or in hyperbolic space. is called de Sitter dual of in hyperbolic space. In this section we obtain the spacelike Smarandache partners in de Sitter space of a curve in hyperbolic space. Definition.. Let = (s) be a unit speed regular curve lying fully on H with the moving Sabban frame {, t, }. The curve : Ι R S of defined by (s (s)) = ((s) + c (s)) (4) is called the spacelike -Smarandache curve of and fully lies on S,, c εr\{} and + c =. Theorem.. Let : Ι R H be a unit speed regular and geodesic curvature κ g. If : Ι R S -Smarandache curve of with the Sabban frame {, t, } then the relationships between the Sabban frame of and its -Smarandache curve are given by ] = ε ] t ] (5) c ε t c ε ε = ± and its geodesic curvature κ g is given by κ g = κ g c c κ g. (6) Proof. Differentiating the equation (4) with respect to s and considering (), we obtain d ds = ( c κ g )t. This can be rewritten as t ds = ( c κ g )t (7) = c κ g ds By substituting (8) into (7) we obtain a simple form of Eq. 7 as follows, (8) t = εt (9) ε = if c κ g > for all s and ε = if c κ g < for all s. It can be easily seen that the tangent vector t is a unit spacelike vector. Taking the Lorentzian vector cross product of (4) with (9) we have = t = ε (c + ) () It is easily seen that is a unit timelike vector. On the other hand, by taking the derivative of the equation (9) with respect to s, we find dt = ε( + κ ds g) () By substituting (8) into () we find t = ε c κ g ( + κ g). () Consequently, from (4), (9), and (), the geodesic curvature κ g of the curve = (s (s)) is explicitly obtained by κ g = det (, t, t ) = κ g c c κ g () Thus, the theorem is proved. In three theorems that follow, in a similar way as in Theorem. we obtain the Sabban frame {, t, } and the geodesic curvature κ g of a spacelike Smarandache curve. We omit the proofs of Theorems.,., and.4, since they are analogous to the proof of Theorem.. Definition.. Let = (s) curve lying fully on H. Then the spacelike t-smarandache curve : Ι R S of defined by (s (s)) = ((s) + c t(s)) (4), c εr\{} and + c =. Theorem.. Let : Ι R H and geodesic curvature κ g. If : Ι R S spacelike t -Smarandache curve of, then its frame {, t, } is given by

4 7 GU J Sci, 9():69-77 (6)/ Murat SAVAS, Atakan Tugkan YAKUT, Tugba TAMIRCI t ] = c c κ g c κ g (c κ g ) c c κ g c κ g (c κ g ) The geodesic curvature κ g κ g = (c κ g ) 5 c κ g > and c κ g c κ g (c κ g ) ] t ] (5) of the curve is given by (c κ g λ c κ g λ λ ) (6) λ = c κ g κ g + (c κ g ) { λ = c κ g κ g + (c c κ g )(c κ g ) λ = c κ g κ g + ( κ g + c κ g )(c κ g ) (7) Definition.. Let : Ι R H. Then the spacelike t-smarandache curve : Ι R S of defined by (s (s)) = (t(s) + c (s)) (8), c εr\{} and + c =. Theorem.. Let : Ι R H and geodesic curvature κ g. If : Ι R S spacelike t -Smarandache curve of, then its frame {, t, } is given by t ] = κ g κ g (κ g ) he geodesic curvature κ g κ g = (κ g ) 5 < κ g and c κ g κ g c (κ g ) c κ g κ g (κ g ) ] t ] (9) of the curve is given by (κ gλ + c λ λ ) () λ = κ g κ g c κ g (κ g ) { λ = c κ g κ g + ( c κ g κ g )(κ g c ) λ = κ g κ g + ( c κ g + κ g )(κ g ) () Definition.4. Let : Ι R H. Then the spacelike t-smarandache curve : Ι R S of defined by (s (s)) = ((s) + c t(s) + c (s)) (), c, c εr\{} and + c + c =. Theorem.4. Let : Ι R H and geodesic curvature κ g. If : Ι R S t-smarandache curve of, then its frame {, t, } is given by t ] = c A c κ g c ( +c κ g ) A c c κ g A c c c κ g A c c κ g A ( c κ g ) c A ] t ] () A = ( c κ g ) c + c κ g, ( c κ g ) > c c κ g and the Smarandache curve is a spacelike curve. Furthermore, the geodesic curvature κ g of curve is given by κ g = ((c κ g + c κ g c )λ + ( c κ g + c c )λ + ( c κ g c )λ ) ((( c κ g ) c + 5 c κ g ) ) { λ = c (c κ g ( c κ g ) c κ g κ g ) +( c κ g ) (( c κ g ) c + c κ g ) λ = ( c κ g )(c κ g ( c κ g ) c κ g κ g ) +(c c κ g c κ g ) (( c κ g ) c + c κ g ) λ = c κ g (c κ g ( c κ g ) c κ g κ g ) +(κ g ( c κ g ) + c κ g ) (( c κ g ) c + c κ g ) (4) (5) Example.. Let us consider a regular unit speed curve on H defined by (s) = ( (s ) +, (s ), s ). Then the orthonormal Sabban frame {(s), t(s), (s)} of can be calculated as follows: { (s ) (s) = ( (s ) +, t(s) = (s, s, ) (s) = ( (s ), (s ), s ), s ) The geodesic curvature of is. In terms of the definitions, we obtain the spacelike Smarandache curves on S according to the Sabban frame on H. First, when we take = and c =, then the -Smarandache curve is spacelike and given by

5 GU J Sci, 9():69-77 (6)/ Murat SAVAS, Atakan Tugkan YAKUT, Tugba TAMIRCI 7 (s (s)) = + (( ) (s ) +, ( + ) (s ), (s )( + )) and the Sabban frame of the spacelike -Smarandache curve is given by t ] = ] t ] and its geodesic curvature κ g is. Second, when we take = and c =, then the t-smarandache curve is a spacelike and given by (s (s)) = ) ((s + (s ) +, (s ) + (s ), (s + )) and the Sabban frame of the spacelike t-smarandache curve is given by t ] = t ] ] and its geodesic curvature κ g is. Third, when we take = and c =, then the t-smarandache curve is a spacelike curve and given by (s (s)) = ((s ) + s, (s ) + s, s) and the Sabban frame of the spacalike t-smarandache curve is given by t ] = ] t ] and its geodesic curvature κ g is. Finally, when we take =, c = and c =, then the t-smarandache curve is a spacelike curve and given by (s (s)) = ((s ) + s, (s ) + s,s ) and the Sabban frame of the spacelike t-smarandache curve is given by t ] = ] t ] and its geodesic curvature κ g is. We give a curve in hyperbolic space and its Smarandache partners in de Sitter space in Figure. 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.. Let : Ι R H. Then the timelike -Smarandache curve : Ι R S of does not exist. Proof. Assume that : Ι R H. Then the timelike -Smarandache curve : Ι R S of is defined by (s (s)) = ((s) + c (s)) (6), c εr\{}, + c =. Differentiating (6) with respect to s and using (), we obtain (s) = d ds = ( c κ g )t t ds = ( c κ g )t = ( c κ g ) ds which is a contradiction. In the corollaries which follow, in a similar way as in the previous section, we obtain the Sabban frame {, t, } and the geodesic curvature κ g of a timelike Smarandache curve. We omit the proofs of Theorems 4., 4., and 4.4. Definition 4.. Let : Ι R H. Then, the timelike t-smarandache curve : Ι R S of is defined by (s (s)) = ((s) + c t(s)), c εr\{}, and + c =.

6 74 GU J Sci, 9():69-77 (6)/ Murat SAVAS, Atakan Tugkan YAKUT, Tugba TAMIRCI Corollary 4.. Let : Ι R H and the geodesic curvature κ g. If : Ι R S timelike t -Smarandache curve of, then its frame {, t, } is given by t ] = c c κ g c κg ( c κ g ) c c κ g c κ g ( c κ g ) c κ g c κ g ( c κ g ) ] t ] and the corresponding geodesic curvature κ g is given by κ g = ( c κ g ) 5 (c κ g λ c κ g λ λ ) λ = c κ g κ g + ( c κ g ) { λ = c κ g κ g + (c c κ g )( c κ g ) λ = c κ g κ g + ( κ g + c κ g )( c κ g ). Definition 4.. Let : Ι R H be a regular unit speed curve lying fully on H. Then, the timelike t-smarandache curve : Ι R S of is defined by (s (s)) = (c t(s) + c (s)),, c εr\{}, and c + c =. + c = with, c εr\{}, c κ g < and is a unit speed curve on H (black line) is a unit speed spacelike curve on S (red line) -Smarandache curve t-smarandache curve t-smarandache curve t-smarandache curve Figure. Spacelike Smarandache partner curves of a curve on H

7 GU J Sci, 9():69-77 (6)/ Murat SAVAS, Atakan Tugkan YAKUT, Tugba TAMIRCI 75 Corollary 4.. Let : Ι R H and geodesic curvature κ g. If : Ι R S timelike t -Smarandache curve of, then its frame {, t, } is given by t ] = c A c κ g c ( + c κ g ) A c c κ g c c κ g A A c c c κ g ( c κ g ) c A A ] t ] t ] = κ g κ g ( κ g ) c κ g κ g c ( κ g ) c κ g κ g ( κ g ) ] t ] and the corresponding geodesic curvature κ g is given by κ g = ( κ g ) 5 (κ gλ + c λ + λ ), c εr\{}, + c =, > κ g and λ = κ g κ g c κ g ( κ g ) { λ = c κ g κ g + ( c κ g κ g )(c κ g ) λ = κ g κ g + ( κ g c κ g )( κ g ). Definition 4.. Let : Ι R H. Then t-smarandache curve : Ι R S of is defined by (s (s)) = ((s) + c t(s) + c (s)), c, c εr\{}, + c + c =. Corollary 4.. Let : Ι R H and geodesic curvature κ g. If : Ι R S timelike t -Smarandache curve of, then its frame {, t, } is given by A = c ( c κ g ) c κ g, ( c κ g ) < c c κ g and the corresponding geodesic curvature κ g is given by κ g = ( (c κ g + c κ g c )λ + ( c κ g + c c )λ +( c κ g c )λ ) ((c ( c κ g ) 5 c κ g ) ), c, c εr\{}, + c + c = and { λ = c ( c κ g ( c κ g ) + c κ g κ g ) +( c κ g ) (c ( c κ g ) c κ g ) λ = ( c κ g )( c κ g ( c κ g ) + c κ g κ g ) +(c c κ g c κ g ) (c ( c κ g ) c κ g ) λ = c κ g ( c κ g ( c κ g ) + c κ g κ g ) + (c κ g + κ g ( c κ g )) (c ( c κ g ) c κ g ) Example 4.. Let us consider a regular unit speed curve on H defined by (s) = (cosh s, sinh s, ). Then the orthonormal Sabban frame {, t, } of the curve and the orthonormal Sabban frame {, t, } of the curve and the geodesic curvature κ g of a timelike Smarandache curve can be calculated as in the previous example. The curve and its Smarandache partners are given in Figure.

8 76 GU J Sci, 9():69-77 (6)/ Murat SAVAS, Atakan Tugkan YAKUT, Tugba TAMIRCI is a unit speed curve on H (black line) is a unit speed timelike curve on S (red line) -Smarandache curve is undefined t-smarandache curve t-smarandache curve t-smarandache curve Figure. Timelike Smarandache partner curves of a curve on H

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