Geometry of Partial Differential Equation, Spin Structure and Twistor Theory

作为与几何方程组有关的几何结构的具体例子，我们利用扭量对应研究了下列对结构：a）射影和格拉斯曼B）射影接触和拉格朗日c）李接触和洛伦兹d）纯旋量和中性结构我们详细研究了这些结构，得到了它们的不变量并进行了分类.我们进一步将研究扩展到许多不同的方向。对于a），首席研究员和町田共同努力获得了作为Spencer上同调元素的完全不变量。对于B），首席研究员和吉川的共同工作是研究的起点。Ozawa和Head建立了一个偏微分方程组，给出了具体的接触变换，对于c）和d），我们得到了扭量图和不变量之间的关系。作为微分方程研究的一个新的扩展，我们得到了一个基本的共形结构系统，它在几何中可能有许多应用。

As concrete examples of geometric structures related to system of geometric equations, we investigated the following pair structures by using the twistor correspondence.a) projective and Grassmannian b) projective contact and Lagrangean c) Lie contact and Lorentzian d) pure spinor and neutral structuresWe studied in detail the structures, got invariants and classified them. Further we extended the investigations to many different directions.As for a), the head investigator and Machida worked together to obtain complete invariants as elements of Spencer cohomology.Concerning b), the joint work of the head and Yoshikawa was the starting point of the investigation. Ozawa and the head found a system of partial differential equations, which gives a concrete contact transformation.As for c) and d), we got the twistor diagrams and the relation of invariants. As a new extension of the study of differential equations, we got a fundamental system for the conformal structure, which may have many applications in geometry.