Generalized Bishop frames of regular curves in $\mathbb{E}^{4}$

Abstract

We introduce and study generalized Bishop frames of regular curves, which are generalizations of the Frenet and Bishop frames for regular curves. There are four types B, C, D and F of generalized Bishop frames of regular curves in $\mathbb{E}^4$ up to the change of the order of vectors fixing the first one which is the tangent vector. We prove the following hierarchy among these four types of frames: Let $\gamma$ be a regular curve in $\mathbb{E}^4$. If $\gamma$ admits a frame of type F, which corresponds to the Frenet frame, then $\gamma$ admits a frame of type D. If $\gamma$ admits a frame of type D, then $\gamma$ admits a frame of type C. Since a frame of type B is a Bishop frame, by a result of Bishop, every regular curve in $\mathbb{E}^4$ admits such a frame. It follows that if $\gamma$ admits a frame of type D, then $\gamma$ clearly admits a frame of type B. We also construct examples of regular curves in $\mathbb{E}^4$ which show the hierarchy is strict.