Comments on Parallel Curves in 3-Dimensional Lie Group G

Year 2024, , 37 - 42, 01.10.2024

Ali Çakmak , Pelin Ezgi Epik

https://doi.org/10.46810/tdfd.1424728

Abstract

In this study, firstly, basic concepts in 3-dimensional Euclidean space and basic information about curves are given and some special curves are examined. Then, basic information about Lie algebra and Lie group basic concepts and curves are given and special curves such as helix, involute-evolute, Bertrand, Mannheim, Smarandache, are defined. Secondly, inspired by these special curves examined in the Lie group, the definitions of the parallel curve in the vector direction, the parallel curve in the direction of the vector B and the parallel curve in the direction of the linear combination of the vectors B and N of a curve according to the Frenet frame are given and characterized. Some theorems and results are obtained by finding the Frenet apparatus of these characterized curves. Finally, the findings are examined in more specific circumstances and new results are found.

Keywords

Lie group, Parallel curve, Some special curves

References

• O’ Neill B, Elementary Differential Geometry. Academic Press, New York. 1966.
• Willson FN. Theoretical and Practical Graphics. The Macmillan Company. 1898.
• Keskin Ö, Yüksel N, Karacan MK, İkiz H. Characterization of the Parallel Curve of the Adjoint Curve in E^3. General Mathematics Notes. 2016; 35:9-18.
• Aldossary MT, Gazwani MA. Montion of Parallel Curves and Surfaces in Euclidean 3-Space R3. Global J. Adv. Res. Class. Mod. Geo. 2020; 9:43-56.
• Bozkurt Z, Gök I, Okuyucu OZ, Ekmekçi FN. Characterizations of rectifying, normal and osculating curves in three-dimensional compact Lie groups. Life Science Journal. 2013; 10: 819-823.
• Çiftçi Ü. A generalization of Lancret’s theorem. Journal of Geometry and Physics. 2009; 59:1597-1603.
• Kızıltuğ S, Önder M. Associated Curves of Frenet Curves in Three-Dimensional Compact Lie Group. Miskolc Mathematical Notes 2015; 16:953-964.
• Okuyucu OZ, Gök I, Yaylı Y, Ekmekçi FN. Slant helices in three-dimensional Lie groups. Applied Mathematics and Computation. 2013, 221:672-683.
• Crouch P, Silva LF. The dynamic interpolation problem: On Riemannian manifolds, Lie groups, and symmetric spaces. J. Dyn. Control Syst. 1995; 1:177–202.
• Gök I, Okuyucu OZ, Yayli Y, Ekmekci N. Bertrand curves in three-dimensional Lie groups. Miskolc Mathematical Notes. 2016, 17:999–1010.
• Çakmak A. New Type Direction Curves in 3-Dimensional Compact Lie Group. Symmetry. 2019, 11:387.
• Hacısalihoğlu H. Introduction to High Differential Geometry. Fırat University, Faculty of Science Publications, Elâzığ. 2006.
• Okuyucu OZ. Special Curves on Some Manifolds. Ph.D. Thesis, Ankara University, Ankara, 2013.
• Kızılay E. Some Generalized Curves in Three Dimensional Compact Lie Groups, M.Sc. Thesis, Bilecik Seyh Edebali University, Bilecik, 2023.
• Değirmen C. On Curves in Three Dimensional Compact Lie Groups, M.Sc. Thesis, Bilecik Seyh Edebali University, Bilecik, 2017.