A natural Frenet frame for null curves on the lightlike cone in Minkowski space (mathbb{R} ^{4}_{2})

摘要

In this paper, we investigate the representation of curves on the lightlike cone $mathbb {Q}^{3}_{2}$ in Minkowski space $mathbb {R}^{4}_{2}$ by structure functions. In addition, with this representation, we classify all of the null curves on the lightlike cone $mathbb {Q}^{3}_{2}$ in four types, and we obtain a natural Frenet frame for these null curves. Furthermore, for this natural Frenet frame, we calculate curvature functions of a null curve, especially the curvature function $kappa _{2}=0$ , and we show that any null curve on the lightlike cone is a helix. Finally, we find all curves with constant curvature functions.