BERTRAND CURVES IN THREE DIMENSIONAL LIE GROUPS

Abstract

In this paper, we give the definition of harmonic curvature function some special curves such as helix, slant curves, Mannheim curves and Bertrand curves. Then, we recall the characterizations of helices [7], 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 7): A curve alpha : I subset of R -> G with the Frenet apparatus {T, N, B, kappa, tau} 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.