IJPAM: Volume 15, No. 4 (2004)

ON A CHARACTERIZATION OF HELIX
FOR CURVES OF THE HEISENBERG GROUP

Vedat Asil$^1$, Mehmet Bektas$^2$
$^{1,2}$Department of Mathematics
Faculty of Art and Science
Firat University
23119 Elazig, TURKEY
$^1$e-mail: vasil@firat.edu.tr
$^2$e-mail: mbektas@firat.edu.tr


Abstract.T. Ikawa obtained in [5] 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 this result, i.e. $k_{\text{ }}$and $\tau $ are variable, but $\frac{k}{\tau }$ is constant.

Recently, N. Ekmekçi and K. Ilarslan obtained characterizations of timelike null helices in terms of principalnormal or binormal vector fields [4].

Furthermore, 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, we obtained characterizations of helices in terms of binormal vector field for a curve with respect to the Frenet frame of the three-dimensional Heisenberg group $H_{3}.$

Received: July 5, 2004

AMS Subject Classification: 53B30

Key Words and Phrases: Heisenberg group, general helix

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 15
Issue: 4