Mihriban ALYAMAÇ KÜLAHCI, Mehmet BEKTAŞ, AHMET BİLİCİ

On Classifications of Normal and Osculating Curves in 3-dimensional Sasakian Space

This study provides the de…nition of rectifying, normal and osculating curves in 3-dimensional Sasakian space with their characterizations. Furthermore, the di¤erential equations obtained from these characterizations are solved and their figures are presented in the text. Sasakian manifolds were introduced in 1960 by the Japenese geometer Shigeo Sasaki [19]. There was not much activity in this field after the mid-1970s, until the advent of String theory. Since then Sasakian manifolds have gained importance in physics and geometry. For physicists and geometers, the study of Sasakian space has its own interest, so it has been extensively studied area of scientific researchs [1, 3, 4, 7, 9, 20]. In the 3-dimensional Sasakian space, to each regular curve γ, it is possible to associate three mutually ortogonal unit vector fields. The vectors V1, V2, V3 are called the tangent, the principal normal and the binormal vector field, respectively. The planes spanned by the vector fields, {V1, V2} , {V1, V3} and {V2, V3} are defined as the osculating plane, the rectifying plane and the normal plane, respectively. In the Euclidean space E3, the notion of rectifying curves was introduced by Chen in [10]. In addition, he showed in [11] that there exist a simple relationship between the rectifying curves and centrodes, which play some important roles in mechanics, kinematics as well as differentialgeometry.

On Classifications of Normal and Osculating Curves in 3-dimensional Sasakian Space

Keywords:

Rectifying curve, normal curve, osculating curve,

PDF

___

1) Alegre P, Blair D E, Carriazo A, Generalized Sasakian Space Forms, Israel Journal of Mathematics 2004; 141: 157-183. 2) AliAT,OnderMA,SomeCharacterizationsofSpacelikeRectifyingCurvesintheMinkowski Space-time, Global Journal of Science Frontier Research Mathematics & Decision Sciences 2012; 12(1). 3) Baikoussis C, Blair D E, On Legendre Curves In Contact 3-Manifolds, Geometria Dedicata 1994;49: 135-142. 4) Belkhelfa M, H¬r¬ca I E, Rosca R, Verstraelen L, On Legendre curves in Riemannian and Lorentzian Sasaki spaces, Soochow J Math 2002; 28(11): 81-91. 5) Blair D E, Contact Manifolds in Riemannian Geometry, Lecture Notes in Math. Vol. 509, Berlin: Springer, 1976. 6) Blair D E, Riemannian Geometry of Contact and Symplectic Manifolds, Boston: Birkhauser, 2002. 7) Camc¬ Ç, Yayl¬ Y, Hac¬saliho¼ glu H H, On the characterization of spherical curves in 3dimensional Sasakian spaces, J Math Anal Appl 2008;342: 1151-1159.118) Camc¬ Ç, Kula L, · Ilarslan K, Characterizations of the position vector of a surface curve in Euclidean 3-space, An ¸ St Ovidius Constanta 2011;19(3): 59-70. 9) Cappelletti-Montano, B., Nicola, A., Yudin, I., Curvature Properties of Quasi-Sasakian Manifolds, International Journal of Geometric Methods in Modern Physics, 2013; Vol.10, No.8, 1360008. 10) Chen B Y, When does the position vector of a space curve always lie in its rectifying plane? Amer Math Monthly 2003; 110: 147-152. 11) Chen B Y, Dillen F, Rectifying curves as centrodes and extremal curves, Bull Inst Math Academia Sinica 2005; 33(2): 77-90. 12) Grbovic M, Nesovic E, Some relations between rectifying and normal curves in Minkowski 3-space, Math Commun 2012; 17: 655-664. 13) GüngörMA,TosunM,Somecharacterizationsofquaternionicrectifyingcurves, Di¤erential Geometry-Dynamical Systems 2011; 13: 89-100. 14) · Ilarslan K, Spacelike normal curves in Minkowski space, Turk J Math 2005; 29: 53-63. 15) · Ilarslan K, Nesovic E, Timelike and null normal curves in Minkowski space E3 1, Indian J Pure Appl Math 2004; 35: 881-888. 16) · Ilarslan K, Nesovic E, On Rectifying Curves as Centrodes and Extremal Curves in the Minkowski 3-space, Novi Sad J Math 2007; 37(1): 53-64. 17) · Ilarslan K, Nesovic E, Petrovic-Torgasev M, Some characterizations of rectifying curves in Minkowski 3-space, Novi Sad J Math 2003; 33(2): 23-32. 18) Öztekin H, Ö¼ grenmi¸ s A O, Normal and Rectifying Curves In Pseudo-Galilean Space G1 3 and Their Characterizations, J Math Comput Sci 2012; 2(1): 91-100. 19) Sasaki S, On di¤erentiable manifolds with certain structures which are closely related to almost contact structure I. Tohoku Math J 1960; 2: 459-476. 20) Sharma, R., Ghosh, A., Sasakian 3-Manifold Asa RicciSoliton Represents in the Heisenberg Group, International Journal of Geometric Methods in Modern Physics, 2011; Vol.08, No.01, 149154. 21) Y¬lmaz, M. Y., Külahc¬, M., The Classi…cations of Quaternionic Osculating Curves In Q4, Georgian Mathematical Journal, 2017;1-6.