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

1.利用Heisenberg框架丛重新证明了CR结构的Tanaka-Webster联络的存在性和唯一性，考虑到特定联络在各种几何结构中作为变换的Schwarzian导数起着重要作用.在CR结构的底层接触结构上，建立了Hamilton函数产生保持CR结构的接触变换的充分必要条件. su_2框架束被详细地检查，并且被证明在CR几何学中是有用的，例如，在分析Tanaka-Webster连接中.研究表明，CR结构的变形参数可以用复变函数来解释，并用复变函数来表示Levi形式和连接系数的精确形式.建立了高维接触变换的Schwarzian导数的概念，并将其用于求解一类偏微分方程组的等价问题.建立了CR几何中严格切触变换的Schwarzian导数的概念。结果得到了三维CR结构的基本方程，并在解空间上发现了一个自然的厄米特结构。最后解决了三维CR结构的等效问题。

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.