Bertrand and Mannheim curves of framed curves in the 4-dimensional Euclidean space

TL;DR

This paper explores Bertrand and Mannheim curves within the context of framed curves in 4D Euclidean space, providing conditions for their existence and highlighting differences from classical regular curves.

Contribution

It introduces definitions and characterizations of Bertrand and Mannheim curves for framed curves in 4D Euclidean space, extending classical concepts to include singular points.

Findings

Bertrand curves of regular curves do not exist under certain conditions.

Bertrand curves can exist as framed curves even when they do not for regular curves.

Necessary and sufficient conditions for Bertrand and Mannheim framed curves are established.

Abstract

A Bertrand curve in the 4-dimensional Euclidean space is a space curve whose first normal line is the same as the first normal line of another curve. On the other hand, a Mannheim curve in the 4-dimensional Euclidean space is a space curve whose first normal line is the same as the second or third normal line of another curve. By definitions, another curve is a parallel curve with respect to the direction of the first normal vector. As smooth curves with singular points, we consider framed curves in the Euclidean space. Then we define and investigate Bertrand and Mannheim curves of framed curves. We give necessary and sufficient conditions of Bertrand and Mannheim curves of both regular and framed curves. It is well-known that the Bertrand curves of regular curves do not exist under a condition. However, even if regular curves, Bertrand curves exist as framed curves.