Slant helices in Euclidean 4-space $E^4$

TL;DR

This paper characterizes slant helices in four-dimensional Euclidean space by analyzing their curvatures and the constant angle condition between the principal normal vector and a fixed direction.

Contribution

It provides new characterizations of slant helices in $E^4$ based on their curvature functions and geometric properties.

Findings

Characterizations of slant helices via curvature functions

Conditions for the principal normal vector to make a constant angle with a fixed direction

New geometric insights into curves in four-dimensional space

Abstract

We consider a unit speed curve $\alpha$ in Euclidean four-dimensional space $E^4$ and denote the Frenet frame by $\{T,N,B_1,B_2\}$. We say that $\alpha$ is a slant helix if its principal normal vector $N$ makes a constant angle with a fixed direction $U$. In this work we give different characterizations of such curves in terms of their curvatures.