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.

中文翻译：

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Minkowski空间\（\ mathbb {R} ^ {4} _ {2} \）中光锥上的空曲线的自然Frenet框架

在本文中，我们通过结构函数研究了Minkowski空间$ \ mathbb {R} ^ {4} _ {2} $中的光锥$ \ mathbb {Q} ^ {3} _ {2} $上曲线的表示形式。另外，通过这种表示，我们将光锥$ \ mathbb {Q} ^ {3} _ {2} $上的所有空曲线分为四种类型，并为这些空曲线获得自然的Frenet帧。此外，对于此自然Frenet帧，我们计算零曲线的曲率函数，尤其是曲率函数$ \ kappa _ {2} = 0 $，并且我们证明，光锥上的任何零曲线都是螺旋。最后，我们找到所有具有恒定曲率函数的曲线。