Contact Geometry of Second Order

研究结果总结如下。首席研究员研究了有限型线性微分方程系统（完整系统）的等价问题与其基本解的投影嵌入图像（投影解）之间的对应关系。作为Seashi理论在前一个问题中的应用，我们证明了超几何系统E（n，k）的射影解的象不位于Grassmannian流形Gr（k-1，n-1）（Plucker嵌入的象）中，除了E（3，6）。Izumiya对拟线性一阶偏微分方程解曲面的通有奇性进行了分类，并对空间维为1的Hamilton-Jacobi方程粘性解的通有分歧进行了分类. Kiyohara定义了Liouville流形和Kahler-Liouville流形的概念，这两类黎曼流形的测地线流是可积的，并详细研究了它们的结构。石川证明了Thom的横截性定理，并在球面上发现了一个新的所谓“Cl-度量”族。Mather型，并给出了各向同性映射的辛稳定性和拉格朗日稳定性的Mather型或Arnold型的刻画.Kawazumi发展了计算上同调群的新工具超椭圆映射类群的有限域，综述了用复解析Gelfond-Fuchs上同调研究Riemann曲面模空间的上同调群。

The summary of Research Results is as follows. The head investigator studied the correspondence between the equivalence problem of the systems of linear differential equations of finite type (Holonomic system) and that of the projective embedding images of their fundamental solutions (projective solution). As the application of Seashi's theory for the former problem, we showed that the image of the projective solution of the Hyper geometric system E(n, k) does not lie in the Grassmannian manifold Gr(k-1 , n-1) (the image of Plucker embedding) except for E(3,6). Furthermore we discussed the generalization of E.Cartan's paper on 5 variables.Izumiya classified the generic singularities of solution surfaces forquasi-linear first order partial differential equations and also classified the generic bifurcation of viscosity solutions for Hamilton-Jacobi equations of space dimension 1.Kiyohara defined the notion of Liouville manifolds and Kahler-Liouville manifolds, which are two classes of riemannian manifolds whose geodesic flows are integrable and studied their structures in detail. He carried out a part of classification and found out a new family of so-called "Cl-metrics" on the Sphere.Ishikawa showed the transversality theorem of Thom-Mather type for the space of isotropic mappings of corank 1 into symplectic manifolds and gave the characterization of Mather type or Arnold type for the Symplectic stability and Lagrange stability of the isotropic mappings.Kawazumi developed new tools to calculate the cohomology groups over finite fields of the hyperelliptic mapping class groups and gave an overview on the study of the cohomology groups of the moduli spaces of Riemann surfaces by the complex analytic Gelfond-Fuchs cohomology.