Characterizations of Bertrand curve pairs via new Frenet formulas

Yıl 2023, Cilt: 41 Sayı: 6, 1115 - 1120, 29.12.2023

Süleyman Şenyurt Osman Çakır

Öz

In this article, we first introduce new Frenet formulas by making use of the properties of connected curves. Then applying these formulas we show that some decisive properties of Bertrand partner curve can be given in terms of a Bertrand curve. More precisely, we offer dif-ferential equations of the Bertrand partner curve with respect to both Levi-Civita and normal Levi-Civita connections in terms of the Bertrand curve. We also give harmonicity conditions of the partner curve of Bertrand curve pair by the same method. We obtain some new results and finally we give an example to support our allegations.

Anahtar Kelimeler

Bertrand Partner Curve, Levi-Civita Connection, Curvature Vector, Biharmonic Curve, Laplace Operator

Kaynakça

• REFERENCES
• [1] Babaarslan M, Yayli Y. On helices and Bertrand curves in Euclidean 3-space. Maths and Comput Applications 2013;18:1–11. [CrossRef]
• [2] Ekmekci N, Ilarslan K. On Bertrand curves and their characterization. Diff Geom Dynam Systems 2001;17. [CrossRef]
• [3] Okuyucu OZ, Gok I, Yayli Y, Ekmekci N, et al. Bertrand curves in three dimensional Lie groups. Miskolc Maths Notes 2017;17:999–1010. [CrossRef]
• [4] Chen BY, Ishikawa S. Biharmonic Surface in Pseudo-Euclidean Spaces. Mem Fac Sci Kyushu Univ Ser A 1991;45:323–347. [CrossRef]
• [5] Çakır O, Senyurt S. Harmonicity and Differential Equation of Involute of a Curve in E^3. Thermal Science 2019;23:2119–2125.
• [6] Kocayigit H, Hacisalihoglu HH. 1-Type curves and biharmonic curves in Euclidean 3-space. Int Elect Journ of Geo 2011;4:97–101.
• [7] Senyurt S, Çakır O. Characterizations of Curves According to Frenet Frame in Euclidean Space. Turk J Math Comput Sci 2019;11:48–52.
• [8] Yokus A, Baskonus HM, Sulaiman TA, Bulut H. Numerical simulation and solutions of the two-component second order KdV evolutionary system. Numerical Methods for Partial Differential Equations 2018;34:211–227. [CrossRef]
• [9] Sulaiman TA, Bulut H, Yokus A, Baskonus HM. On the exact and numerical solutions to the coupled Boussinesq equation arising in ocean engineering. Indian Journal of Physics 2019;93:647–656. [CrossRef]
• [10] Din A, Li Y. Controlling heroin addiction via age-structured modeling. Adv Differ Equ 2020;521. [ 11] Din A, Li Y, Shah MA, et al. The Complex Dynamics of Hepatitis B Infected Individuals with Optimal Control. Journal of systems science and complexity 2021. [CrossRef] [12] Din A, Li Y, Tahir K, Hassan T, Asaf K, Wajahat AK, et al. Mathematical analysis of dengue stochastic epidemic model. Results in Physics 2020. [CrossRef]
• [13] Senyurt S, Çakır O. Diferential Equations for a Space Curve According to the Unit Darboux Vector. Turk J Math Comput Sci 2018;9:91–97. [CrossRef]
• [14] Sabuncuoglu A. Diferensiyel Geometri. Nobel Akademik Yayincilik, Ankara, 2014.
• [15] Çakır O, Senyurt S. Calculation of the differential equations and harmonicity of the involute curve according to unit Darboux vector with a new method. Turkish Journal of Science 2020;5:62–77.