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

基于微分方程的射影子流形几何结构研究

基本信息

批准号：
09304010
负责人：
SAKAI Takeshi
金额：
$ 12.74万
依托单位：
Kobe University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (A)
财政年份：
1997
资助国家：
日本
起止时间：
1997 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09304010/
关键词：
projective submanifold hypergeometric system contact geometry projective differential geometry projective invariants affine differential geometry integrable system line congruence projective differential geometry affine differential geom

项目摘要

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.

本项目的主要成果如下：1.发表了关于曲线、残面和超曲面的射影微分几何，射影齐次曲面的分类，射影曲面的线汇与拉普拉斯变换之间的对应关系的研究等方面的注记; 2.从几个角度研究了超几何微分方程;比如，对Appell系统F D 22 D 2和F D 24 D 2定义的射影曲面的研究，确定与三次曲面的模空间有关的一致化方程，交理论的新公式，系统E（3，6）基本解的显式表示，A-超几何系统的组合研究，3.研究Painleve方程与Backlund变换之间的关系，用Hamilton结构刻画Painleve方程，4.通过引入微分不变量和Paileve方程描述三阶常微分方程的接触几何，与接触态有关的Schwarzian导数。5.研究各种几何结构;例如，齐性空间上的射影平坦结构，仿射对称空间上的Weyl-Yang-米尔斯联络，仿射对称空间上的para-kahler结构，黑森结构等. 6.利用DS-图和图上的调和分析研究三维流形. 7.特殊曲面的研究，如极小曲面，常平均曲率曲面，以及幂零流形和纽结理论的研究.沿着上述研究，新的代数计算计算机系统被开发出来，为我们的研究提供了数学上严格的计算机支持。