Smarandache Curves on S 2. Key Words: Smarandache Curves, Sabban Frame, Geodesic Curvature. Contents. 1 Introduction Preliminaries 52

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1 Bol. Soc. Paran. Mat. (s.) v. (04): c SPM ISSN on line ISSN in press SPM: doi:0.569/bspm.vi.94 Smarandache Curves on S Kemal Taşköprü and Murat Tosun abstract: In this paper, we introduce special Smarandache curves according to Sabban frame on S and we give some characterization of Smarandache curves. Besides, we illustrate examples of our results. Key Words: Smarandache Curves, Sabban Frame, Geodesic Curvature. Contents Introduction 5 Preliminaries 5 Smarandache Curves According to Sabban Frame on S 5. γt-smarandache Curves td-smarandache Curves γtd-smarandache Curves Example Introduction The differential geometry of curves is usual starting point of students in field of differential geometry which is the field concerned with studying curves, surfaces, etc. with the use the concept of derivatives in calculus. Thus, implicit in the discussion, we assume that the defining functions are sufficiently differentiable, i.e., they have no concerns of cusps, etc. Curves are usually studied as subsets of an ambient space with a notion of equivalence. For example, one may study curves in the plane, the usual three dimensional space, curves on a sphere, etc. There are many important consequences and properties of curves in differential geometry. In the light of the existing studies, authors introduced new curves. Special Smarandache curves are one of them. A regular curve in Minkowski spacetime, whose position vector is composed by Frenet frame vectors on another regular curve, is called a Smarandache curve [8]. Special Smarandache curves have been studied by some authors ([],[],[],[4],[8]). In this paper, we study special Smarandache curves such asγt, td, γtd - Smarandache curves according to Sabban frame in Euclidean unit sphere S. We hope these results will be helpful to mathematicians who are specialized on mathematical modeling. 000 Mathematics Subject Classification: 5A04, 5C40 5 Typeset by B S P M style. c Soc. Paran. de Mat.

2 5 Kemal Taşköprü and Murat Tosun. Preliminaries The Euclidean -space E provided with the standard flat metric given by, = dx +dx +dx (x,x,x ) is a rectangular coordinate system of E. Recall that, the norm of an arbitrary vector X E is given by X = X,X. The curve α is called a unit speed curve if velocity vector α of α satisfies α =. For vectors v,w E, it is said to be orthogonal if and only if v,w = 0. The sphere of radius r = and with center in the origin in the space E is defined by S = {P = (P,P,P ) P,P = }. Denote by {T,N,B} the moving Frenet frame along the curve α in E. For an arbitrary curve α E, with first and second curvature, κ and τ respectively, the Frenet formulae is given by [5] T = κn N = κt +τb B = τn. Now, we give a new frame different from Frenet frame. Let γ be a unit speed spherical curve. We denote s as the arc-length parameter of γ. Let us denote t(s) = γ (s), and we call t(s) a unit tangent vector of γ. We now set a vector d(s) = γ(s) t(s) along γ. This frame is called the Sabban frame of γ on S (Sphere of unit radius). Then we have the following spherical Frenet formulae of γ: γ = t t = γ +κ g d d = κ g t is called the geodesic curvature of κ g on S and κ g = t,d [6].. Smarandache Curves According to Sabban Frame on S In this section, we investigate Smarandache curves according to the Sabban frame ons.letγ = γ(s) andβ = β(s ) be a unit speed regular spherical curves on S, and {γ,t,d} and {γ β,,d β } be the Sabban frame of these curves, respectively... γt-smarandache Curves Definition.. Let S be a unit sphere in E and suppose that the unit speed regular curve γ = γ(s) lying fully on S. In this case, γt - Smarandache curve can be defined by β(s ) = (γ +t). (.)

3 Smarandache Curves on S 5 Now we can compute Sabban invariants of γt - Smarandache curves. Differentiating the equation. with respect to s, we have and β (s ) = dβ ds = (γ +t ) ds = (t+κ g d γ) Thus, the tangent vector of curve β is to be = ds = +κg. (.) +κg ( γ +t+κ gd). (.) Differentiating the equation. with respect to s, we get ds ds = (+κ g ) λ = κ g κ g κ g λ = κ g κ g κ g κ g 4 λ = κ g +κ g +κ g. Substituting the equation. into equation.4, we reach = (λ γ +λ t+λ d) (.4) (+κ g ) (λ γ +λ t+λ d). (.5) Considering the equations. and., it easily seen that d β = β = 4+κg (κ gγ +( κ g )t+d). (.6) From the equation.5 and.6, the geodesic curvature of β(s ) is κ g β =,d β = (+κ g ) (λ κ g +λ ( κ g )+λ ).

4 54 Kemal Taşköprü and Murat Tosun.. td-smarandache Curves Definition.. Let S be a unit sphere in E and suppose that the unit speed regular curve γ = γ(s) lying fully on S. In this case, td - Smarandache curve can be defined by β(s ) = (t+d). (.7) Now we can compute Sabban invariants of td - Smarandache curves. Differentiating the equation.7 with respect to s, we have and β (s ) = dβ ds = (t +d ) ds = ( γ +κ g d κ g t) ds = +κg. (.8) In that case, the tangent vector of curve β is as follows = +κg ( γ κ gt+κ g d). (.9) Differentiating the equation.9 with respect to s, it is obtained that ds ds = (+κ g ) (λ γ +λ t+λ d) (.0) λ = κ g κ g +κ g +κ g λ = κ g κ g κ g 4 λ = κ g +κ g κ g 4. Substituting the equation.8 into equation.0, we get t β = (+κ g ) (λ γ +λ t+λ d). (.) Using the equations.7 and.9, we easily find d β = β = So, the geodesic curvature of β(s ) is as follows +4κg (κ gγ t+(+κ g )d). (.)

5 Smarandache Curves on S 55 κ g β =,d β = (+κ g ).. γtd-smarandache Curves (λ κ g λ +λ (+κ g )). Definition.. Let S be a unit sphere in E and suppose that the unit speed regular curve γ = γ(s) lying fully on S. Denote the Sabban frame of γ(s), {γ,t,d}. In this case, γtd - Smarandache curve can be defined by β(s ) = (γ +t+d). (.) Lastly, let us calculate Sabban invariants of γtd - Smarandache curves. Differentiating the equation. with respect to s, we have and β (s ) = dβ ds = (γ +t +d ) ds = (t γ +κ g d κ g t) ds = ( κg +κ g ). (.4) Thus, the tangent vector of curve β is = ( κg +κ g ) ( γ +( κ g)t+κ g d). (.5) Differentiating the equation.5 with respect to s, it is obtained that ds ds = ( κ g +κ g ) (λ γ +λ t+λ d) (.6) λ = κ g +κ g κ g +4κ g 4κ g +κ g λ = κ g κ g κ g 4κ g +κ g +κ g κ g 4 λ = κ g κ g +κ g 4κ g +κ g +4κ g κ g 4. Substituting the equation.4 into equation.6, we reach t β = 4( κ g +κ g ) (λ γ +λ t+λ d). (.7) Using the equations. and.5, we have d β = β = 6 κg +κ g ((κ g )γ +( κ g )t+( κ g )d). (.8)

6 56 Kemal Taşköprü and Murat Tosun From the equation.7 and.8, the geodesic curvature of β(s ) is κ g β =,d β = 4 ( κ g+κ g ) (λ (κ g )+λ ( κ g )+λ ( κ g ))..4. Example Let us consider the unit speed spherical curve: γ(s) = {cos(s)tanh(s),sin(s)tanh(s),sech(s)}. It is rendered in Figure. Figure : γ = γ(s) Considering the Sabban frame and the Sabban formulae, we have geodesic curvature of γ: κ g = sech(t). In terms of the definitions, we obtain Smarandache curves according to Sabban frame on S, see Figures -4.

7 Smarandache Curves on S 57 Figure : γt - Smarandache Curve Figure : td - Smarandache Curve

8 58 Kemal Taşköprü and Murat Tosun Figure 4: γtd - Smarandache Curve References. Ali A. T., Special Smarandache Curves in the Euclidean Space, International Journal of Mathematical Combinatorics, Vol., 0-6, 00.. Bayrak N., Bektaş Ö. and Yüce S., Special Smarandache Curves in E, arxiv: v [math.ho] 5 Apr 0.. Bektaş Ö. and Yüce S., Special Smarandache Curves According to Darboux Frame in Euclidean - Space, arxiv: v [math. DG], 0 Mar Çetin M., Tunçer Y. and Karacan M. K., Smarandache Curves According to Bishop Frame in Euclidean - Space, arxiv: 06. 0v [math. DG], 6 Jun Do Carmo, M. P., Differential Geometry of Curves and Surfaces, Prentice Hall, Englewood Cliffs, NJ, Koenderink J., Solid Shape, MIT Press, Cambridge, MA, O Neill B., Elemantary Differential Geometry, Academic Press Inc. New York, Turgut M. and Yılmaz S., Smarandache Curves in Minkowski Space-time, International Journal of Mathematical Combinatorics, Vol., 5-55, 008.

9 Smarandache Curves on S 59 Kemal Taşköprü Sakarya University, Department of Mathematics, Faculty of Art and Science, 5487 Sakarya, Turkey ktaskopru@gmail.com and Murat Tosun Sakarya University, Department of Mathematics, Faculty of Art and Science, 5487 Sakarya, Turkey tosun@sakarya.edu.tr