The hyperbolic worldsheets and the hyperbolic worldline generated by null Cartan curves are defined and their geometric properties are investigated. As applications of singularity theory, the singularities of the hyperbolic worldsheets and the hyperbolic worldline are classified by using the approach of the unfolding theory in singularity theory. It is shown that under appropriate conditions, the hyperbolic worldsheet is diffeomorphic to cuspidal edge or swallowtail type of singularity and the hyperbolic worldline is diffeomorphic to cusp. An important geometric invariant which has a close relation with the singularities of the hyperbolic worldsheets and worldlines is found such that the singularities of the hyperbolic worldsheets and worldlines can be characterized by the invariant. Meanwhile, the contact of the spacelike normal curve of a null Cartan curve with hyperbolic quadric or world hypersheet is discussed in detail. In addition, the dual relationships between the spacelike normal curve of a null Cartan curve and the hyperbolic worldsheet are described. Moreover, it is demonstrated that the spacelike normal curve of a null Cartan curve and the hyperbolic worldsheet are [Formula: see text]-dual each other.

中文翻译：

---

de Sitter 3 空间中零嘉当曲线的双曲世界表和世界线

定义了由零嘉当曲线生成的双曲世界表和双曲世界线，并研究了它们的几何特性。作为奇点理论的应用，采用奇点理论中展开理论的方法对双曲世界片和双曲世界线的奇点进行分类。结果表明，在适当的条件下，双曲世界线微同胚于尖点边缘或燕尾型奇点，双曲世界线微同胚于尖点。一个重要的几何不变量与双曲世界表和世界线的奇异性密切相关，使得双曲世界表和世界线的奇异性可以用该不变量来表征。同时，详细讨论了零嘉当曲线的类空间正态曲线与双曲二次曲线或世界超表的接触。此外，还描述了零嘉当曲线的空间正态曲线与双曲世界表之间的对偶关系。此外，证明了零嘉当曲线的类空间法线曲线和双曲世界表是[公式：见正文]-彼此对偶的。