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 α lying on a lightlike surface in Minkowski space E31 . We prove that α has two such frames and obtain generalized Darboux frame’s equations. We find the relations between the curvature functions kg , kn , τg of α with respect to its Darboux frame and the curvature functions ˜kg , ˜kn , ˜τg 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 ˜τg = 0 and give some examples.