The twistor correspondence between different geometric structures and its application

扭转理论的一个方面是看到由双重纤维定义的不同几何结构之间的关系。(1)纯旋量结构具有单位球的正交框架束的模型空间。考虑零平面束，我们可以定义具有中性共形结构的扭转空间。1. (2)拉格朗日结构具有拉格朗日格拉斯曼流形的模型空间。考虑零平面束，我们可以定义具有射影接触结构的扭转空间。(1)考虑在具有接触结构的空间上定义的次拉普拉斯算子解的扭或积分表示。扭转空间是所有与海森堡群相关的零空间2。(2)将Goursat方程限制为有限型，考虑了法向Cartan连接曲率下的局部等价问题。(1)在具有内积或辛形式的向量空间中，用各种双纤维来考虑第二类标志流形。推广了与几何结构相关的超几何方程组。3. (2)研究了非完整系统（不可积分布）上规范域（半平坦部分连接）的广义实例。

An aspect of the twistor theory is to see a relation between different geometric structures defined by a double fibration.1. (1) A pure spinor structure has the model space of the orthonormal frame bundle of the unit sphere. Considering the null plane bundle, we can define the twistor space, which has a neutral conformal structure. 1. (2) A Lagrangian structure has the model space of the Lagrangian Grassmann manifold. Considering the null plane bundle, we can define the twistor space, which has a projective contact structure.2. (1) We consider twistor integral representations of solutions of the sub-Laplacian defined on a space equipped with a contact structure. The twistor space is all the null spaces associated with a Heisenberg group, 2. (2) Restricting Goursat equations to finite type, we consider local equivalence problems in terms of curvatures of normal Cartan connections.3. (1) We consider flag manifolds of second kind by various double fiberings defined from a vector space equipped with an inner product or a symplectic form. We generalize the system of hypergeometric equations associated with geometric structures. 3. (2) We consider generalized instantons of gauge fields (half-flat partial connections) on non-holonomic systems (non-integrable distributions).