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

Abstract

Abstract In this paper, we investigate the representation of curves on the lightlike cone Q 2 3 $\mathbb {Q}^{3}_{2}$ in Minkowski space R 2 4 $\mathbb {R}^{4}_{2}$ by structure functions. In addition, with this representation, we classify all of the null curves on the lightlike cone Q 2 3 $\mathbb {Q}^{3}_{2}$ 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 κ 2 = 0 $\kappa _{2}=0$ , and we show that any null curve on the lightlike cone is a helix. Finally, we find all curves with constant curvature functions.

Keywords

• Constant curvature function
• Helix
• Lightlike cone
• Natural Frenet frame
• Null curve