Bertrand Curves in three Dimensional Lie Groups

TL;DR

This paper characterizes Bertrand curves in three-dimensional Lie groups with bi-invariant metrics using harmonic curvature functions, extending classical curve theory to a Lie group setting.

Contribution

It introduces a new characterization of Bertrand curves in 3D Lie groups via harmonic curvature, generalizing known Euclidean results.

Findings

Bertrand curves satisfy a linear relation involving curvature and harmonic curvature.

Provides a necessary and sufficient condition for Bertrand curves in Lie groups.

Extends classical differential geometry of curves to Lie group contexts.

Abstract

In this paper, we give the defination of harmonic curvature function some special curves such as helix, slant curves, Mannheim curves and Bertrand curves. Then, we recall the characterizations of helices [8], slant curves (see [19]) and Mannheim curves (see [12]) in three dimensional Lie groups using their harmonic curvature function. Moreover, we define Bertrand curves in a three dimensional Lie group G with a bi-invariant metric and the main result in this paper is given as (Theorem 3.4): A curve ?{\alpha} with the Frenet apparatus {T,N,B,{\kappa},{\tau}} in G is a Bertrand curve if and only if {\lambda}{\kappa}+{\mu}{\kappa}H=1 where {\lambda},{\mu} ? are constants and H is the harmonic curvature function of the curve {\alpha}.