Research on Contact transformations and Geometric structures of Schwarzian Derivatives

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We established the notion of Schwarzian derivative for contact transformations, that is a generalization of the classical Schwarzian derivative of one complex variable functions. Explicitly, the result is the following:(1) to obtain the system of partial differential equations to construct contact transformations, which should be called the fundamental equation for the contact geometry,(2) to obtain a necessary and sufficient condition for the fundamental equations to be integrable,(3) to verify that the coefficients of the fundamental equation satisfy certain properties as the Schwarzian derivative of the contact transformations.The method that is developed in the above research was applied to the conformal geometry, and is succeedingly planed to be applied to CR and other geometric structures.