Research on geometries of Differential Systems and Parabolic geometries

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We established the notion of PD-manifolds, which describes, through the concept of differentail systems, the geometry of systems of second order partial differential equations for one unknown function under contact transformations. Moerover we established two step Reduction Theorem, which is basic to Contact Geometry of Second Order.By utilizing this Reduction Theorem, we explicitly show how to construct various classes of overdetermined systems of second order partial differential equations, for which the contact equivalence problems are reduced to that of Parabolic Geometries(i.e., the geometries associated with the simple graded Lie algebras).