On characterizations of general helices for ruled surfaces in the pseudo-Galilean space $G^{1}_{3}$-(Part-I)

Abstract

T. Ikawa obtained in [5] the following characteristic ordinary differential equation \[ \begin{array}{cc} \nabla _{X}\nabla _{X}\nabla _{X}X-K\nabla _{X}X=0,& K=k^{2}-\tau ^{2} \end{array} \] 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. Hacısalihoğlu generalized in [4] T. Ikawa’s this result, i.e., $k$ and $\tau$ are variable, but $\frac{k}{\tau}$ is constant. In [1] H. Balgetir, M. Bektaş 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, 4, 5] , we obtained characterizations of a curve with respect to the Frenet frame of ruled surfaces in the 3-dimensional pseudo-Galilean space $G_{3}^{1}$.