Characterizations of Inclined Curves According to Parallel Transport Frame in $E^{4}$ and Bishop Frame in $E^{3}$

Authors : Fatma ATEŞ, İsmail GOK, Faik Nejat EKMEKCİ, Yusuf YAYLI

Pages : 16-24

View : 18 | Download : 8

Publication Date : 2019-04-15

Article Type : Research Paper

Abstract :The aim of this paper is to introduce inclined curves according to parallel transport frame. This paper begins by defined a vector field D called Darboux vector field of an inclined curve in E 4 . It will then go on to an alternative characterization for the inclined curves “α : I ⊂ R −→ E 4 is an inclined curve ⇔ k1insert ignore into journalissuearticles values(s); Z k1insert ignore into journalissuearticles values(s);ds+k2insert ignore into journalissuearticles values(s); Z k2insert ignore into journalissuearticles values(s);ds+k3insert ignore into journalissuearticles values(s); Z k3insert ignore into journalissuearticles values(s);ds = 0” where k1insert ignore into journalissuearticles values(s);, k2insert ignore into journalissuearticles values(s);, k3insert ignore into journalissuearticles values(s); are the principal curvature functions according to parallel transport frame of the curve α and also, similar characterization for the generalized helices according to Bishop frame in E 3 is given by α : I ⊂ R −→ E 3 is a generalized helix ⇔ k1insert ignore into journalissuearticles values(s); Z k1insert ignore into journalissuearticles values(s);ds+k2insert ignore into journalissuearticles values(s); Z k2insert ignore into journalissuearticles values(s);ds = 0” where k1insert ignore into journalissuearticles values(s);, k2insert ignore into journalissuearticles values(s); are the principal curvature functions according to Bishop frame of the curve α. These curves have illustrated some examples and draw their figures with use of Mathematica programming language. Also, it is given an example for the inclined curve in E 4 and showed that the above condition is satisfied for this curve.
Keywords : Parallel transport frame, Inclined curve, Bishop frame, Generalized helix