Parallel Transport Frame in 4-dimensional Euclidean Space

TL;DR

This paper introduces the parallel transport frame in 4D Euclidean space, generalizing known 3D relations, and uses it to characterize special curves with simplified conditions.

Contribution

It extends the concept of the parallel transport frame to 4D Euclidean space and derives new characterizations of curves using this frame.

Findings

Relation between parallel transport and Frenet frames in 4D

Conditions for spherical curves in 4D

Characterizations of curves in normal, rectifying, and osculating planes

Abstract

In this work, we give parallel transport frame of a curve and we introduce the relations between the frame and Frenet frame of the curve in 4-dimensional Euclidean space. The relation which is well known in Euclidean 3-space is generalized for the first time in 4-dimensional Euclidean space. Then we obtain the condition for spherical curves using the parallel transport frame of them. The condition in terms of \kappa and {\tau} is so complicated but in terms of k_{1} and k_{2} is simple. So, parallel transport frame is important to make easy some complicated characterizations. Moreover, we characterize curves whose position vectors lie in their normal, rectifying and osculating planes in 4-dimensional Euclidean space E^{4}.