Research on geometries of Differential Systems and Parabolic geometries

我们建立了PD-流形的概念，它通过微分系的概念来刻画接触变换下一个未知函数的二阶偏微分方程组的几何。在此基础上，建立了二阶接触几何的两步约化定理，并利用这一约化定理构造了各类超定二阶偏微分方程组，从而将接触等价问题归结为抛物线几何(即与简单分次李代数相关的几何)。

We established the notion of PD-manifolds, which describes, through the concept of differentail systems, the geometry of systems of second order partial differential equations for one unknown function under contact transformations. Moerover we established two step Reduction Theorem, which is basic to Contact Geometry of Second Order.By utilizing this Reduction Theorem, we explicitly show how to construct various classes of overdetermined systems of second order partial differential equations, for which the contact equivalence problems are reduced to that of Parabolic Geometries(i.e., the geometries associated with the simple graded Lie algebras).