Natural and Conjugate Mates of Frenet Curves in Three-Dimensional Lie Group

TL;DR

This paper introduces the concepts of natural and conjugate mates of Frenet curves in three-dimensional Lie groups, exploring their relationships and special cases, with graphical illustrations in Euclidean space.

Contribution

It defines and analyzes natural and conjugate mates of Frenet curves in 3D Lie groups, extending classical curve theory to this setting.

Findings

Derived relationships between Frenet curves and their mates in Lie groups.

Characterized mates for special types of curves like helices and Salkowski curves.

Provided graphical visualizations in Euclidean space.

Abstract

In this study, we introduce the natural mate and conjugate mate of a Frenet curve in a three dimensional Lie group $ \mathbb{G} $ with bi-invariant metric. Also, we give some relationships between a Frenet curve and its natural mate or its conjugate mate in $ \mathbb{G} $. Especially, we obtain some results for the natural mate and the conjugate mate of a Frenet curve in $ \mathbb{G} $ when the Frenet curve is a general helix, a slant helix, a spherical curve, a rectifying curve, a Salkowski (constant curvature and non-constant torsion), anti-Salkowski (non-constant curvature and constant torsion), Bertrand curve. Finally, we give nice graphics with numeric solution in Euclidean 3-space as a commutative Lie group.