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

Year 2022, Volume: 51 Issue: 5, 1288 - 1303, 01.10.2022

Buddhadev Pal , Santosh Kumar

https://doi.org/10.15672/hujms.960966

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.

Keywords

Minkowski space, pseudo sphere, proper helix of order 2, proper curve of order 2, $V_{k}$-slant helix

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