On Inclined Curves According to Parallel Transport Frame in E4

TL;DR

This paper introduces a new class of inclined curves in four-dimensional space using the parallel transport frame, providing characterizations, examples, and visualizations with Mathematica.

Contribution

It defines inclined curves based on the parallel transport frame in E4 and offers new characterizations involving curvature functions, extending prior work on generalized helices.

Findings

New characterization of inclined curves in E4 using curvature integrals.

Equivalent conditions for generalized helices in E3 with Bishop frame.

Illustrative examples and visualizations created with Mathematica.

Abstract

In this paper, we introduce an inclined curves according to parallel transport frame. Also, we define a vector field called Darboux vector field of an inclined curve in and we give a new characterization such as: "\alpha: I \subset R \rightarrow E^4 is an inclined curve \Leftrightarrow k_1 \int k_1ds + k_2 \int \k_2 +k_3ds = 0" where k_1, k_2, K_3 are the principal curvature functions according to parallel transport frame of the curve and we give the similar characterizations such as "\alpha : I \subset R \rightarrow E^3 is a generalized helix \Leftrightarrow k_1 \int k_1ds + k_2 \int k_2ds = 0" where k_1, k_2 are the principal curvature functions according to Bishop frame of the curve \alpha. Moreover, we illustrate some examples and draw their figures with Mathematica Programme.