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.

[学科分类] 电力

[关键词] Constant curvature function;Helix;Lightlike cone;Natural Frenet frame;Null curve