Theory of Differential Forms and Twistor Structures

在本研究中，我们用两年的时间研究了由首席研究员构造的扭量图。该图给出了各种几何结构之间的关系。利用微分形式的对合系统理论，阐明了该图与粒子论、场论的关系。我们的主要研究目的是将扭量理论应用于经典力学和量子力学。特别是，我们详细研究了以下几何结构：由三阶常微分方程、射影结构、Grassmann结构、Lie接触结构等定义的结构，我们证明了在许多情况下，一个结构的一个方程可以转化为另一个几何结构的一个简单方程，作为具体结果，首席研究员A.Y.Yoshikawa女士研究了三阶常微分方程接触同态下的等价问题，证明了完全不变量由两条几何曲率给出，并确定了曲率的具体形式。利用扭量理论将这一结果与相对论几何联系起来，为射影接触几何奠定了基础，并进一步研究了与二阶微分方程组几何有关的Grassmann结构，阐明了半平坦性与零丛扭量理论的关系。

In this reserch during two years, we have studied the twistor diagram constructed by the head investigator. The diagram gives relations between various geometric structures. We use the theory of involutive system of differential forms and clarified the relations of the diagram with the theory of particles and the field theory. Main purpose of our studies is application of the twistor theory to the classical and quantum mechanics. Especially, we investigated the following geometric strustures in detail : The structure defined by third order ordinary differential equations, projective structures, Grassmann structures, Lie contact structures, etc. We showed that in many cases one equation of a structure is transformed to a simple equation of other geometric structure.As concrete results, the head investigator with Mrs.A.Y.Yoshikawa studied the equivalence problem under contact diffeomorphism of third order ordinary differential equations, proved that the complete invariants are given by two geometric curvatures and decided the concrete form of the curvatures. By the twistor theory, this result is connected to the geometry of the relativity and gives the fundation of projective contact geometry.Further the head investigator studied the Grassmann structures related to the geometry of the system of second order differential equations and made clear the relation between the half flatness and the twistor theory of the null bundles.