Characterizations of a Bertrand Curve According to Darboux Vector

Year 2022, Volume: 10 Issue: 1, 112 - 117, 15.04.2022

Süleyman Şenyurt , Osman Çakır

Abstract

In this paper, we first take a Bertrand curve pair and then we use Darboux vector instead of mean curvature vector to give characterizations of Bertrand partner curve by means of the Bertrand curve. By making use of the relations between the Frenet frames of the Bertrand curve pair we give the differential equations and sufficient conditions of harmonicity(biharmonic or 1-type harmonic) of the Bertrand partner curve in terms of the Darboux vector of the Bertrand curve. After driving the conclusions we write an example to demonstrate how our assumptions come true

Keywords

Bertrand curve pair, differential equation,, biharmonic curve,, Laplace operator

Project Number

yok

References

• [1] B. Y. Chen and S. Ishikawa, Biharmonic Surface in Pseudo-Euclidean Spaces, Mem. Fac. Sci. Kyushu Univ. Ser. A, Vol:45, No.2 (1991) , p. 323–347.
• [2] O. Cakır and S. Senyurt, Harmonicity and Differential Equation of Involute of a Curve in E^3, Thermal Science, Vol: 23, No.6 (2019), p. 2119–2125.
• [3] K. Arslan, H. Kocayigit and M. Onder, Characterizations of Space Curves with 1-type Darboux Instantaneou Rotation Vector, Commun. Korean Math. ¨Soc., Vol: 31, No.2 (2016), p. 379–388.
• [4] S. Senyurt and O. C¸ akır, Characterizations of Curves According to Frenet Frame in Euclidean Space, Turk. J. Math. Comput. Sci., Vol: 11, No.1 (2019),p. 48–52.
• [5] S. Senyurt and O. Cakır, Diferential Equations for a Space Curve According to the Unit Darboux Vector, Turk. J. Math. Comput. Sci., Vol: 9, No.1 (2018), p. 91–97.
• [6] Sabuncuoglu A., Diferensiyel Geometri, Nobel Akademik Yayincilik, Ankara, 2014.
• [7] S. Senyurt, A. Calıskan and U. Celik, N∗C∗–Smarandache Curve of Bertrand Curves Pair According to Frenet Frame, International J.Math. Combin.,Vol:1, (2016), p. 1–7.
• [8] O. Cakır and S. Senyurt, On Harmonicity and Differential Equations of a Bertrand Curve in E^3, arXiv: 2103.0301