Generalized Bishop frames on curves on E^4

TL;DR

This paper introduces and analyzes four types of generalized Bishop frames for regular curves in four-dimensional Euclidean space, extending the classical Bishop and Frenet frames to higher dimensions and exploring their existence conditions.

Contribution

It classifies four types of generalized Bishop frames in extit{E}^4 and establishes conditions for their existence, extending the theory of frames for curves in higher dimensions.

Findings

Every regular curve admits a Bishop frame.

If a curve admits a Frenet frame, it admits all four types of generalized Bishop frames.

Curves with nowhere vanishing tangent derivative admit three types of frames, excluding the Frenet-related type F.

Abstract

We introduce and study generalized Bishop frames on regular curves, which are generalizations of the Frenet and Bishop frames for regular curves on higher dimensional spaces. There are four types of generalized Bishop frames on regular curves on $\mathbb{E}^{4}$ up to the change of the order of vectors fixing the first one which is the tangent vector. One of these four types of frames is a Bishop frame, and by a result of Bishop, every regular curve admits such a frame. We show that if a regular curve $\gamma$ on $\mathbb{E}^{4}$ admits a Frenet frame, then $\gamma$ admits all four types of generalized Bishop frames. We also show that if the derivative of the tangent vector of a regular curve is nowhere vanishing, then the curve admits all three types of generalized Bishop frames except a frame of type F, which is related to the Frenet frame.