Geometry of Partial Differential Equation, Spin Structure and Twistor Theory

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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.