奇点理论在微分拓扑和微分几何学中的应用研究

The singularity theory was born from intersection and integration of many branches of mathematics, it is a powerful tool for studying the singular phenomena in the nature. It can be applied to various fields, such as differential topology, differential geometry, differential equation, and theoretical physics. The purpose of this project is to conduct researches on classifying singularities of maps and properties of geometric and topology of degenerate submanifolds, and to complete classic theories of differential topology and differential geometry. .Mainly concerned the following:.1. Classification of singularities and finite determinacy of smooth maps; .2. Differential geometry of null submanifolds; .3. The properties of geometric and the topological of singular submanifolds;.4. The relationship between singularities of the maps and geometric invariants of the submanifolds.

奇点理论是众多数学分支交叉与融合而诞生的，它是研究自然界中奇异现象的有力工具，在微分拓扑、微分几何、微分方程以及理论物理等领域都有重要的应用。本项目拟在奇点理论的视角下研究映射的奇点分类问题以及退化子流形的几何和拓扑性质，以充实经典微分拓扑和微分几何学的研究成果。着重研究：.1..光滑映射的奇点分类和有限决定性；.2..类光子流形的局部微分几何学；.3..奇异子流形的几何和拓扑性质；.4..映射的奇点与子流形的几何不变量之间的关系。

本项目中，我们主要利用奇点理论研究了光滑映射的奇点分类和光滑映射的奇点与子流形的几何不变量之间的联系。. 项目执行期间获得了如下主要结果：.1）建立了可含有奇点的(n, m)-尖型曲线的局部微分几何。作为应用我们给出了光学系统中的焦线和波面的对偶关系。.2）利用勒让德奇点理论给出了子流形的几何、拓扑性质以及映射的奇点与这类子流形的几何不变量之间的联系。.3）建立了光球子空间的子流形的局部微分几何。给出了在伪球之间的Legendre对偶框架下的光球子流形的对偶子流形的奇点分类。.4）给出了高余维光滑映射芽的有限决定性的充要条件。.5）给出了3维混合型曲线的局部微分几何，并给出了类光曲线的构造方法。