Study of geometric structures on projective submanifolds in view of differential equations

该项目的结果被总结为遵循：1.关于曲线、划痕和超表面的预测差异几何学的出版说明，关于预测均匀表面的分类，以及一项关于线性并发和拉普拉斯变换的预测表面之间的正确研究。2.从几点观点出发的高精度测量差异方程的研究。说，一项由Appell系统定义的投影表面研究，F-D22-D2和F-D24-D2，确定与立方体表面模块空间的统一方程，一种新的交集理论形式，系统E（3，6）的基本解决方案的解释性表示，A-高血压计量系统的组合研究，以及D-模块的算法研究，3.对疼痛和后向转换之间的关系保持的调查，以及汉密尔顿结构术语中疼痛平衡的特征，和Paileve equation.4.通过介绍差分变量和与Contactomorphisms.5.各种几何结构的研究;通过Paileve equation.4.通过介绍差分变量和形成的与施瓦茨衍生物相关的几何学研究。关于同源空间、Weyl-Yang-Mills连接para-kahler结构的投影平面结构，以及类似的.6.使用DS-diagram和Harmonic分析对图形的3尺寸流形的研究. 7.研究特殊表面，如微型表面，具有恒定平均曲线的表面，以及另一个，研究镍电位流形和结理论。一个新的用于代数计算的计算机系统已经开发，我们的研究提供了一个数学上合理的计算机支持。

The result of the project is summarized as follows :1.Publishing of notes on the projective differential geometry of curves, scurfaces, and hypersurfaces, classification of projectively homogeneous surfaces, and a study of correspondences between line congruence and Laplace transforms of projective surfaces.2.Study of hypergeometric differential equations from several point of views ; say, a study of projective surfaces defined by Appell's systems FィイD22ィエD2 and FィイD24ィエD2, determination of the uniformizing equation associated with the moduli space of cubic surfaces, a new formulation of intersection theory, explicit presentation of fundamental solutions of the system E(3,6), a combinatorial study of A-hypergeometric systems, and algorithmic study of D-modules.3.Investigation of the relation holding between Painlevequation and Backlund transformation, characterization of Painleve equation in terms of Hamiltonian structure, and description of timelike Bonnet surfaces by Paileve equation.4.Contact geometric study of 3rd order ordinary differential equations by introducing differential invariants and formulation of Schwarzian derivatives related with contactomorphisms.5.Study of various geometric structures ; e.g., projective flat structures on homogeneous spaces, Weyl-Yang-Mills connection para-kahler structures on affine symmetric spaces, Hesse structures, and so on.6.Study of 3-dimensional manifolds using DS-diagram and harmonic analysis on graphs.7.Study of special surfaces such as minimal surfaces, surfaces with constant mean curvature and, further, a study of nilpotent manifolds and knot theory.Along with the studies above, new computer systems for algebraic computation was developed, by which mathematically rigorous computer support was available for our study.