Characterizations of Curves According to Frenet Frame in Euclidean 3-Space

Abstract

In this paper, we investigate the conditions of being an harmonic curve and research differential equations characterizing any differentiable curve in Euclidean 3-space. By means of the Laplacian image of the mean curvature vector field of a curve, it is stated which type of harmonic the curve is. Then we write the theorems related to the characterization of the curves and proved these theorems. When the differentiable curve, used throughout this paper, is specifically  replaced to the unit speed curve then it is seen that the results coincide with the study [4].   In addition we elucidate the characterizations of helix as an example.

Keywords

• regular curve,connection,mean curvature,biharmonic,differential equation

References

1. Arslan, K., Kocayigit, H., Onder, M., Characterizations of Space Curves with 1-type Darboux Instantaneous Rotation Vector, Commun. Korean Math. Soc., 31(2)(2016), 379--388.
2. Chen, B. Y., Ishikawa, S., Biharmonic Surface in Pseudo-Euclidean Spaces , Mem. Fac. Sci. Kyushu Univ., A45(1991), 323--347.
3. Hacısalihoğlu, H. H. Diferensiyel Geometri $3^{rd}$ ed., Ankara, 1988.
4. Kocayiğit, H., Hacısalihoğlu, H. H., 1-type Curves and Biharmonic Curves in Euclidean 3-Space, Int. Elect. Journ. of Geo., 4(1)(2011), 97--101.
5. Kocayiğit, H., Hacısalihoğlu, Hilmi H., Biharmonic Curves in Contact Geometry, Commun. Fac. Sci. Univ. Ank. Series A1, 2061(2)(2012), 35-43.
6. Kocayiğit, H., Onder, M., Hacısalihoğlu H. H., Harmonic 1-type Curves and Weak Biharmonic Curves in Lorentzian 3-Space, Ana. Stiin. A. Uni. Al. I. Cuza. D. I. (S.N.) Mat., f(1)(2014), 109--124.
7. Sabuncuoğlu, A., Diferensiyel Geometri $5^{th}$ ed., Ankara, 2014.
8. Şenyurt, S., Çakır O., Diferential Equations for a Space Curve According to the Unit Darboux Vector, Turk. J. Math. Comput. Sci., 9(2018), 91--97.