Study of geometric structures on projective submanifolds in view of differential equations

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The result of the project is summarized as follows :1.Publishing of notes on the projective differential geometry of curves, scurfaces, and hypersurfaces, classification of projectively homogeneous surfaces, and a study of correspondences between line congruence and Laplace transforms of projective surfaces.2.Study of hypergeometric differential equations from several point of views ; say, a study of projective surfaces defined by Appell's systems FィイD22ィエD2 and FィイD24ィエD2, determination of the uniformizing equation associated with the moduli space of cubic surfaces, a new formulation of intersection theory, explicit presentation of fundamental solutions of the system E(3,6), a combinatorial study of A-hypergeometric systems, and algorithmic study of D-modules.3.Investigation of the relation holding between Painlevequation and Backlund transformation, characterization of Painleve equation in terms of Hamiltonian structure, and description of timelike Bonnet surfaces by Paileve equation.4.Contact geometric study of 3rd order ordinary differential equations by introducing differential invariants and formulation of Schwarzian derivatives related with contactomorphisms.5.Study of various geometric structures ; e.g., projective flat structures on homogeneous spaces, Weyl-Yang-Mills connection para-kahler structures on affine symmetric spaces, Hesse structures, and so on.6.Study of 3-dimensional manifolds using DS-diagram and harmonic analysis on graphs.7.Study of special surfaces such as minimal surfaces, surfaces with constant mean curvature and, further, a study of nilpotent manifolds and knot theory.Along with the studies above, new computer systems for algebraic computation was developed, by which mathematically rigorous computer support was available for our study.