Research of Systems of Partial Differential Equations from the view point of Contact Geometry

本项目的目的是研究作为几何对象的偏微分方程系统，即，作为Jet空间的子流形，从微分几何和奇点理论的角度出发，以微分方程组的切触等价问题为中心，在项目的最后一年，我们对以下六个项目进行了研究：（1）二阶偏微分方程组对一个未知函数的切触等价问题.特别是二阶G2型偏微分方程的研究。嘉当(2)冲击波解的形成和Monge-Ampere方程解的奇性。(3)由偏微分方程符号导出的分次李代数的研究及高阶有限型偏微分方程组的接触等价性。(4)有限型线性偏微分方程的等价问题在射影子流形理论和高斯-施瓦茨理论中的应用。(5)外微分系统中体裁概念的网络几何表征。(6)关于测地线流完全可积的黎曼流形的研究，主要研究者总结了“二阶超定系统的G2-几何”中（1）的内容。Izumiya在《Sugaku Exposition》杂志上总结了（2）的内容。（3）的内容由首席研究员和Yatsui在“与对称空间相关的有限型高阶微分方程的几何学”中总结。关于（4）的内容，Sato和Ozawa在切触同态的情况下构造了“Schwarzian导数”。Kiyohara在“On Kahler-Liouville manifold”中总结了（6）的内容。

The purpose of this project is to study systems of partial differential equations as geometric objects, i.e., as submanifolds of Jet spaces, from the view points of differential geometry and singularities theory, the central theme of which is the contact equivalence problems of systems of differential equations.For the last year of the project, we did our research to summarize the following our original 6 projects :(1) Contact equivalence problem of systems of partial differential equations of second order for one unknown function. Especially the research of G2type partial differential equations of second order d'apres E. Cartan.(2) Formation of shock wave solutions and singularities of solutions of Monge-Ampere equations.(3) The research of graded Lie algebras induced from symbols of partial differential equations and the contact equivalence of systems of higher order partial differential equations of finite type.(4) Application of the equivalence problem for linear partial differential equations of finite type to projective submanifolds theory and Gauss-Schwarz theory.(5) Characterization of the notion of genre in exterior differential systems in terms of Web geometry.(6) The research of riemannian monifolds whose geodesic flows are completely integrable.The head investigator summarized the content of (1) in "G2-geometry of overdetermined systems of second order". Izumiya summarized the content of (2) in the journal "Sugaku Exposition". The contents of (3) was summarized by the head investigator and Yatsui in "Geometry of higher order differential equations of finite type associated with Symmetric spaces". As for the content of (4), Sato and Ozawa contributed to construct "Schwarzian derivative" in case of contact diffeomorphisms. The content of (6) was summarized by Kiyohara in "On Kahler-Liouville manifolds".