Characterization of proper curves and proper helix lying on $S_{1}^2(r)$

Abstract

In this paper, we analyse the proper curve $\gamma(s)$ lying on the pseudo-sphere. We develop an orthogonal frame $\lbrace V_{1}, V_{2}, V_{3} \rbrace$ along the proper curve, lying on pseudosphere. Next, we find the condition for $\gamma(s)$ to become $V_{k} -$ slant helix in Minkowski space. Moreover, we find another curve $\beta(\bar{s})$ lying on pseudosphere or hyperbolic plane heaving $V_{2} = \bar{V_{2}}$ for which $\lbrace \bar{V_{1}},\bar{V_{2}},\bar{V_{3}} \rbrace$, an orthogonal frame along $\beta(\bar{s})$. Finally, we find the condition for curve $\gamma(s)$ to lie in a plane.