Research on Geometric Structures, Schwarzian Derivatives of Maps and Partial Differential Equations

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1. The existence and the uniqueness of the so-called Tanaka-Webster connection of a CR structure was reproved by using a Heisenberg frame bundle, in consideration that specific connections are expected to play an important role as the Schwarzian derivatives of transformations in various geometric structures.2. On the underlying contact structure of a CR structure, a necessary and sufficient condition of a Hamiltonian function to generate a contact transformation that preserves the CR structure is established.3. su_2 frame bundle was examined in detail, and shown to be useful in CR geometry, for example, in analyzing the Tanaka-Webster connections.4. It is shown that the deformation parameter of CR structure is explained by complex valued functions, which was used to exhibit the Levi form and the connection coefficients in precise form.5. The notion of Schwarzian derivative for higher dimensional contact transformations was established, which was used to solve an equivalence problem of a certain system of partial differential equations.6. The notion of Schwarzian derivative in CR geometry for strictly contact transformations was established. As the result, the fundamental equation of 3-dimensional CR structure was obtained, and a natural Hermitian structure on the solution space was detected. Finally the equivalence problem of 3 dimensional CR structure was solved.