De Sitter空间是爱因斯坦场方程的真空解之一，也是半黎曼几何中的伪球模型。本项目主要研究de Sitter空间中含奇点的子流形，借助拉格朗日/勒让德奇点理论，将这些子流形视作焦散面和波前等拓扑意义下的研究对象。本项目主要包括以下几个方面：.(1)研究3维de Sitter空间中曲线的渐屈线和焦曲面，寻找曲线新的几何不变量，通过曲线与模型子流形的切触关系，研究渐屈线和焦曲面的奇点与几何不变量的内在联系。.(2)利用勒让德对偶定理研究de Sitter空间中勒让德对偶子流形的几何拓扑性质，拟给出单参数族伪球之间的勒让德对偶定理。.(3)类光子流形是退化子流形，利用奇点理论和退化子流形的微分几何，研究de Sitter空间中类光子流形的奇点和几何性质。.希望通过本项目的研究，促进奇点理论应用研究的发展，并为退化子流形研究提供新的方法和思路。

De Sitter space is not only one of the vacuum solutions of the Einstein field equation, but also a pseudo-spherical model in semi-Riemannian geometry. This project will focus on the submanifolds with singular points on de Sitter space. By Lagrangian/Legendrian singularity theory, the submanifolds are viewed as caustics and wave fronts in the sense of topology. This project mainly includes the following aspects:.(1) Studying geometric properties of evolutes and focal surfaces for curves on three dimensional de Sitter space, looking for new geometric invariants of curves, giving the relations between the geometric invariants and the singularities of evolutes and focal surfaces by the contact between curves and model submanifolds..(2) By Legendrian dual theorem, studying geometry and topological properties of Legendrian dual submanifolds on de Sitter space, giving one-parameter pseudo-spherical Legendrian dual theorem. .(3) Lightlike manifolds are degenerate submanifolds, we study the geometric properties of lightlike submanifolds on de Sitter space by singularity theory and the differential geometry of degenerate submanifolds..By this project, we hope to find new methods or thoughts for studying degenerate submanifolds from singularity theory, and promote the development of the application of singularity theory.