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

Year 2019, , 120 - 127, 30.04.2019
https://doi.org/10.36753/mathenot.521075

Abstract

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. 

References

  • 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.

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

Year 2019, , 120 - 127, 30.04.2019
https://doi.org/10.36753/mathenot.521075

Abstract

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. 

References

  • 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.
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Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Mihriban Alyamac Kulahci 0000-0002-8621-5779

Mehmet Bektaş 0000-0002-8621-5779

Ahmet Bilici 0000-0002-8621-5779

Publication Date April 30, 2019
Submission Date February 1, 2019
Acceptance Date March 28, 2019
Published in Issue Year 2019

Cite

APA Alyamac Kulahci, M., Bektaş, M., & Bilici, A. (2019). On Classifications of Normal and Osculating Curves in 3-dimensional Sasakian Space. Mathematical Sciences and Applications E-Notes, 7(1), 120-127. https://doi.org/10.36753/mathenot.521075
AMA Alyamac Kulahci M, Bektaş M, Bilici A. On Classifications of Normal and Osculating Curves in 3-dimensional Sasakian Space. Math. Sci. Appl. E-Notes. April 2019;7(1):120-127. doi:10.36753/mathenot.521075
Chicago Alyamac Kulahci, Mihriban, Mehmet Bektaş, and Ahmet Bilici. “On Classifications of Normal and Osculating Curves in 3-Dimensional Sasakian Space”. Mathematical Sciences and Applications E-Notes 7, no. 1 (April 2019): 120-27. https://doi.org/10.36753/mathenot.521075.
EndNote Alyamac Kulahci M, Bektaş M, Bilici A (April 1, 2019) On Classifications of Normal and Osculating Curves in 3-dimensional Sasakian Space. Mathematical Sciences and Applications E-Notes 7 1 120–127.
IEEE M. Alyamac Kulahci, M. Bektaş, and A. Bilici, “On Classifications of Normal and Osculating Curves in 3-dimensional Sasakian Space”, Math. Sci. Appl. E-Notes, vol. 7, no. 1, pp. 120–127, 2019, doi: 10.36753/mathenot.521075.
ISNAD Alyamac Kulahci, Mihriban et al. “On Classifications of Normal and Osculating Curves in 3-Dimensional Sasakian Space”. Mathematical Sciences and Applications E-Notes 7/1 (April 2019), 120-127. https://doi.org/10.36753/mathenot.521075.
JAMA Alyamac Kulahci M, Bektaş M, Bilici A. On Classifications of Normal and Osculating Curves in 3-dimensional Sasakian Space. Math. Sci. Appl. E-Notes. 2019;7:120–127.
MLA Alyamac Kulahci, Mihriban et al. “On Classifications of Normal and Osculating Curves in 3-Dimensional Sasakian Space”. Mathematical Sciences and Applications E-Notes, vol. 7, no. 1, 2019, pp. 120-7, doi:10.36753/mathenot.521075.
Vancouver Alyamac Kulahci M, Bektaş M, Bilici A. On Classifications of Normal and Osculating Curves in 3-dimensional Sasakian Space. Math. Sci. Appl. E-Notes. 2019;7(1):120-7.

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