The twistor correspondence between different geometric structures and its application

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An aspect of the twistor theory is to see a relation between different geometric structures defined by a double fibration.1. (1) A pure spinor structure has the model space of the orthonormal frame bundle of the unit sphere. Considering the null plane bundle, we can define the twistor space, which has a neutral conformal structure. 1. (2) A Lagrangian structure has the model space of the Lagrangian Grassmann manifold. Considering the null plane bundle, we can define the twistor space, which has a projective contact structure.2. (1) We consider twistor integral representations of solutions of the sub-Laplacian defined on a space equipped with a contact structure. The twistor space is all the null spaces associated with a Heisenberg group, 2. (2) Restricting Goursat equations to finite type, we consider local equivalence problems in terms of curvatures of normal Cartan connections.3. (1) We consider flag manifolds of second kind by various double fiberings defined from a vector space equipped with an inner product or a symplectic form. We generalize the system of hypergeometric equations associated with geometric structures. 3. (2) We consider generalized instantons of gauge fields (half-flat partial connections) on non-holonomic systems (non-integrable distributions).