Bertrand Curves in three Dimensional Lie Groups
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}.
Keywords
Cite
@article{arxiv.1211.6424,
title = {Bertrand Curves in three Dimensional Lie Groups},
author = {O. Zeki Okuyucu and İsmail Gök and Yusuf Yaylı and Nejat Ekmekci},
journal= {arXiv preprint arXiv:1211.6424},
year = {2016}
}
Comments
11 pages. arXiv admin note: substantial text overlap with arXiv:1211.6141, arXiv:1203.1146