Characterizations of Inclined Curves According to Parallel Transport Frame in E 4 and Bishop Frame in E 3

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 ⇔ k1(s) Z k1(s)ds+k2(s) Z k2(s)ds+k3(s) Z k3(s)ds = 0” where k1(s), k2(s), k3(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 E3 is given by α : I ⊂ R −→ E 3 is a generalized helix ⇔ k1(s) Z k1(s)ds+k2(s) Z k2(s)ds = 0” where k1(s), k2(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.