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

Mihriban ALYAMAC KULAHCİ; Mehmet BEKTAŞ; Ahmet BİLİCİ

doi:https://doi.org/10.36753/mathenot.521075

Article Details

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

Authors:

Mihriban ALYAMAC KULAHCİ | Mehmet BEKTAŞ | Ahmet BİLİCİ

Publisher:

Murat TOSUN

Main Topics:

Mathematics

Journal:

Mathematical Sciences and Applications E-Notes 2019 Vol.7 Issue 1, pp.120-127

Document Type:

Article

Publication Date:

4/2019

Document Language:

English

Document ID (Org No.):

2000541943

Frequency:

Quarterly

eISSN:

2147-6268

DOI:

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. .

 


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. 

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Article Details

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

Authors: Mihriban ALYAMAC KULAHCİ | Mehmet BEKTAŞ | Ahmet BİLİCİ

Publisher: Murat TOSUN

Main Topics: Mathematics

Journal: Mathematical Sciences and Applications E-Notes 2019 Vol.7 Issue 1, pp.120-127

Document Type: Article

Publication Date: 4/2019

Document Language: English

Document ID (Org No.): 2000541943

Frequency: Quarterly

eISSN: 2147-6268

DOI: https://doi.org/10.36753/mathenot.521075

Issues

Authors:

Mihriban ALYAMAC KULAHCİ | Mehmet BEKTAŞ | Ahmet BİLİCİ

Publisher:

Murat TOSUN

Main Topics:

Mathematics

Journal:

Mathematical Sciences and Applications E-Notes 2019 Vol.7 Issue 1, pp.120-127

Document Type:

Article

Publication Date:

4/2019

Document Language:

English

Document ID (Org No.):

2000541943

Frequency:

Quarterly

eISSN:

2147-6268

DOI:

https://doi.org/10.36753/mathenot.521075