Twistor correspondence between different geometric structures and application to differential equations and field theory

我们认为扭曲理论是通过双纤维的对应，即不同空间上几何结构之间的对偶性。从这一点出发，我们研究了方程的本质、方程本身的构造以及各种微分方程组和可积场理论的解。1.推广了Hessian=Const的Monge-Ampere方程。和高斯曲率=常数，从本质上定义和研究了接触流形上具有拉格朗日对的Monge-Ampere系统。在5，7维的情形下，通过Legendre对偶，我们看到一般几何解分别有4种和11种奇性。2.Goursat方程是一个抛物型二阶偏微分方程解，其Monge特征是完全可积的。它具有龙卷风理论的解释。我们用拉格朗日-格拉斯曼对偶构造方程本身，用Cartan-Legendre对偶构造方程。3.我们看到克莱罗方程就是扭曲理论的精髓。4.非简并型(4，7)型分布与接触结构不同，是有限型分布。我们构造了正规Cartan联络，它们有两个曲率不变量。5.在具有SU(3)结构的6维流形上定义了SU(3)型U(1)瞬子，特别地，在接近Kahler结构的情况下，我们在分别为S^4和CP^2的扭曲空间的Hopf丛上构造了SU(3)型U(1)瞬子。

We regard the twister theory as the correspondence via a double fibering, which is the duality between geometric structures on different spaces. From this point of view, we applied to the study of the essence and the construction of equations themselves and solutions for various differential equations and integrable field theory.1.Extending Monge-Ampere equations which are Hessian=const. and Gaussian curvature=const., we intrinsically defined and studied Monge-Ampere systems with Lagrangian pair on contact manifolds. In the case of 5, 7 dimensins, we saw that generic geometric solutions have four and eleven kinds of singularities respevtively via Legendre duality.2.A Goursat equation is a second order PDE whichi is of parabolic type and the Monge characteristic of which is completely integrable. It has the interpretation of twistor theory. We constructed equations themselves by Lagrange-Grassmann duality and solutions by Cartan-Legendre duality.3.We saw that Clairaut equations are nothing but the essence of twistor theory. We considered the various extention by the method of twistor theory.4.Nondegenerate type type (4,7) distributions are of finite type unlike contact structures. We constructed the normal Cartan connections and they have two curvature invariants. Hyperbolic type distributions have relation to Legendre geodesies via twistor theory.5.We define SU(3) type U(1) instantons on 6-dimensional manifolds with SU(3) structure.In particular, in the case of nearly Kahler structure, we constructed SU(3) type U(1) instantons on Hopf bundles of CP^3 and F_{12} which are the twistor space of S^4 and CP^2 respectively.