(k,m)-type Slant Helices According to Parallel Transport Frame in Euclidean 4-Space

Year 2021, , 261 - 269, 31.12.2021

Fatma Bulut , Feyzi Tartık

https://doi.org/10.47000/tjmcs.858489

Cited By: 1

Abstract

In this work, we describe a Frenet frame in 4-dimensional Euclidean space and call this frame as parallel transport frame (PTF). PTF is an alternative approach to defining a moving frame. This frame is obtained by rotating the tangent vector and the first binormal vector of a unit speed curve by an euler angle and then we give curvature functions according to PTF of the curve. Also, we introduce $(k,m)$-type slant helices according to PTF in Euclidean 4-Space. Additionally, we obtain the characterization of slant helices according to this frame in $\mathbb{E}^{4}$ and give an example of our main result.

Keywords

Slant helix, curvatures, Euclidean 4-space, parallel transport frame

References

• [1] Bektas¸, M., Yilmaz, M.Y., (k;m)-type Slant helices for partially null and pseudo null curves in Minkowski space $\mathbb{E}^{4}$, Applied Mathematics and Nonlinear Sciences, Mathematica, 5(1)(2020), 515–520.
• [2] Bishop, R.L., There is more than one way to frame a curve, Amer. Math. Monthly, 82(1975), 246–251.
• [3] Bulut, F., Bektas¸, M., Special helices on equiform differential geometry of spacelike curves in Minkowski space-time, Commun. Fac.Sci.Univ.Ank.Ser. A1 Math. Stat., 69(2)(2020), 51–62.
• [4] Gökçelik, F., Gök, İ., Ekmekci, F.N., Yayli, Y., Characterizations of inclined curves according to parallel transport frame in $\mathbb{E}^{4}$ and bishop frame in $\mathbb{E}^{3}$, Konuralp Journal of Mathematics, 7(1)(2019), 16–24.
• [5] Hacisalihoğlu, H.H., Diferensiyel Geometri, İnönü Üniversitesi Fen Edebiyat Fakultesi Yayınları (In Turkish), 1983.
• [6] http://en.wikipedia.org/wiki/Rotation matrix, Euler angles.
• [7] Izumiya, S., Takeuchi, N., New special curves and developable surfaces, Turk. J. Math., 28(2004), 153–164.
• [8] Soliman, M.A., Abdel-All, N.H., Hussien, R.A., Youssef, T., Evolution of space curves using type-3 Bishop frame, CJMS., 8(1)(2019), 58–73.
• [9] Yılmaz, M.Y., Bektas¸, M., Slant helices of (k,m)-type in $\mathbb{E}^{4}$, Acta Univ. Sapientiae, Mathematica, 10(2)(2018), 395–401.
• [10] Yılmaz, S., Turgut, M., A new version of Bishop frame and an application to spherical images, J. Math. Anal. Appl., 371(2010), 764–776.
• [11] Yılmaz, S., Turgut, M., On the characterizations of inclined curves in Minkowski space-time $\mathbb{E}_{1}^{4}$, International Mathematical Forum, 3(16)(2008), 783–792.