Global view of curves lying on $H_{0}^2(-r) \subset E_{1}^3$

Abstract

In this paper, we study the geometry of the proper curve and proper helix of order 2 lying on the hyperbolic plane $H_{0}^2(-r)$, globally from Minkowski space $E_{1}^3$. We develop the Frenet frame (orthogonal frame) along the proper curve of order 2 using connection $\tilde{\nabla}$ on $E_ {1}^3$ and connection $\nabla$ on $H_ {0} ^ 2(-r)$. The Frenet frame for the proper curve and proper helix of order 2 depends on the curvature of the proper curve and proper helix of order 2 in the hyperbolic plane $ H_ {0} ^ 2(-r)$. Finally, we find the condition for a proper curve of order 2 with non constant curvature to become a $V_{k} -$slant helix in $E_{1}^3$.