On generalized Darboux frame of a spacelike curve lying on a lightlike surface in Minkowski space $\mathbb{E}^{3}_{1}$

Abstract

In this paper we introduce generalized Darboux frame of a spacelike curve $\alpha$ lying on a lightlike surface in Minkowski space $\mathbb{E}_{1}^{3}$. We prove that $\alpha$ has two such frames and obtain generalized Darboux frame's equations. We find the relations between the curvature functions $k_g$, $k_n$, $\tau_g$ of $\alpha$ with respect to its Darboux frame and the curvature functions $\tilde{k}_{g}$, $\tilde{k}_{n}$, $\tilde{\tau}_{g}$ with respect to generalized Darboux frames. We show that such frames exist along a spacelike straight line lying on a ruled surface which is not entirely lightlike, but contains some lightlike points. We define lightlike ruled surfaces on which the tangent and the binormal indicatrix of a null Cartan curve are the principal curvature lines having $\tilde{\tau}_{g}=0$ and give some examples.