Theory of Differential Forms and Twistor Structures

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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.