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

我们把绕线理论看作是通过双纤维的对应，即不同空间上的几何结构之间的对偶性。从这个角度出发，我们应用于研究方程本身的本质和构造，以及各种微分方程的解和可积场理论。推广Hessian=const的蒙日-安培方程。和高斯曲率=const。研究了接触流形上具有拉格朗日对的蒙日-安培系统。在5维和7维的情况下，通过勒让德对偶，我们看到一般几何解分别有4种和11种奇点。Goursat方程是抛物型的二阶偏微分方程，其蒙日特征是完全可积的。它有扭曲理论的解释。我们用拉格朗日-格拉斯曼对偶构造方程，用卡坦-勒让德对偶构造解。我们看到克劳特方程只不过是扭曲理论的本质。我们用扭转理论的方法考虑了各种扩展。非简并型型（4,7）分布是有限型，与接触结构不同。我们构造了法卡坦连接它们有两个曲率不变量。双曲型分布通过扭扭理论与勒让德测地线有关。在具有SU(3)结构的6维流形上定义了SU(3)型U(1)实例。特别地，在近Kahler结构的情况下，我们在分别为S^4和CP^2的扭转空间的CP^3和F_{12}的Hopf束上构造了SU(3)型U(1)实例。

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.