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

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.

arXiv comment16 pages