Global Geometrical Approach to Contact Manifolds

在本项目中，我们进行了以下研究。研究了5点接触度量流形上的CR扭转空间。通过对4-dim流形的类比，证明了CR扭转空间中存在一种几乎CR结构，并证明了在给定基接触度量5-流形的曲率条件下，这种几乎CR结构是可积的，反自对偶Weyl共形张量消失，标量曲率s=-4.2。四维几何可以应用于接触度量流形的接触子束。根据Tachibana定理和N.Tanaka关于CR几何的系统理论，紧化Sasakian （2n+1）流形上的调和k型在k<n+1时在接触子束上取值。因此sasaki接触结构中的自对偶被定义为4-dim流形理论。注：sasaki接触结构是一种普通的强伪凸CR结构，是CR几何中的一个重要课题。在Sasakian接触5流形中最小浸入的Legendrian曲面的研究是根据Hopf微分、三次微分和第二变分进行的。

In this project we studied the following researches.1. Study of CR twistor space over a 5-dim contact metric manifold was developed. In analogy of 4-dim manifold it is shown that the CR twistor space admits an almost CR structure and it is verified that this almost CR structure is integrable under the curvature conditions on a given base contact metric 5-manifold, that the anti-self-dual Weyl conformal tensor vanishes and also the scalar curvature s=-4.2. 4-dimensional geometry can be applied to the contact subbundle of contact metric manifolds. By Tachibana's theorem and also by N.Tanaka's systematic theory on CR geometry harmonic k-forms over a compact Sasakian (2n+1)-manifold take values in the contact subbundle, when k<n+1. So the self-duality in Sasakian contact structure was defined like 4-dim manifold theory. Remark that Sasakian contact structure turns out to be nothing but a normal strongly pseudo convex CR structure, a main subject in CR geometry.3. Study of Legendrian surfaces minimally immersed in a Sasakian contact 5-manifold was proceeded in terms of Hopf differential, the cubic differential and also in terms of the second variation.