IJPAM: Volume 12, No. 4 (2004)

A CHARACTERIZATION FOR CURVES
OF THE HEISENBERG GROUP

Mehmet Bektas$^1$, Essin Turhan$^2$
$^{1,2}$Department of Mathematics
Firat University
23119 Elazig, TURKEY
$^1$e-mail: mbektas@firat.edu.tr
$^2$e-mail: eturhan@firat.edu.tr


Abstract.T. Ikawa obtained in [4] the following characteristic ordinary differential equation: \begin{equation*}
\nabla _{X}\nabla _{X}\nabla _{X}X-K\nabla _{X}X=0,\qquad K=k^{2}-\tau ^{2}\,,
\end{equation*}for the circular helix which corresponds to the case that the curvatures $k$ and $\tau $ of a time-like curve $\alpha $ on the Lorentzian manifold $M$ are constant.

N. Ekmekçi and H. H. Hacisalihoglu generalized in [3] T. Ikawa's result, i.e. $k_{\text{ }}$ and $\tau $ are variable, but $\frac{k}{\tau }$ is constant.

In [1] H.Balgetir, M.Bektas and M. Ergüt obtained a geometric characterization of Null Frenet curve with constant ratio of curvature and torsion (called null general helix).

In this paper, making use of method in [1], [3], [4], we obtained characterizations of a curve with respect to the Frenet frame of the three-dimensional Heisenberg group $H_{3}.$

Received: February 20, 2004

AMS Subject Classification: 53B30

Key Words and Phrases: Heisenberg group, helix

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 12
Issue: 4