1. Bertrand curves in three dimensional Lie groups
- Author
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Nejat Ekmekci, Ismail Gk, Yusuf Yayli, and O. Zeki Okuyucu
- Subjects
Mathematics - Differential Geometry ,Numerical Analysis ,Control and Optimization ,Algebra and Number Theory ,Frenet–Serret formulas ,Mathematics::History and Overview ,010102 general mathematics ,Mathematical analysis ,Curvature function ,Lie group ,Harmonic (mathematics) ,02 engineering and technology ,01 natural sciences ,Differential Geometry (math.DG) ,Metric (mathematics) ,Helix ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Discrete Mathematics and Combinatorics ,020201 artificial intelligence & image processing ,Mathematics::Differential Geometry ,0101 mathematics ,Analysis ,Mathematics - 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}., Comment: 11 pages. arXiv admin note: substantial text overlap with arXiv:1211.6141, arXiv:1203.1146
- Published
- 2017