凯勒流形的复结构与代数结构研究
摘要：本文以K¨ahler流形为研究对象。在默认读者具有一定几何基础的前提下，文章 的前半部分阐述了K¨ahler流形的研究背景、预备知识及K¨ahler几何学的基础知识， 如：K¨ahler流形的定义、相关性质及其判定方法。随后借助具体的例子详细分析 了K¨ahler流形上的三种结构――复结构、Hermite结构和辛结构，给出了其上几何结 构的具体形式。除代数结构外，整个过程中使用的方法是偏分析的。
由此，得到了K¨ahler流形的几个不同角度的等价定义：
1) K¨ahler流形是具有可积近复结构的辛流形；
2) K¨ahler流形是具有闭的Hermite形式的Hermite 流形：
3) K¨ahler流形是一类特殊的Riemann流形，一方面它的Riemann度量在典型复结构作用下保持不变，另一方面在相应的Riemann联络下它的典型复结构是平行的。
在应用方面，本文利用Newton力学以及Hamilton力学等手段，研究了K¨ahler流形上质点的运动规律，得到了一些简单的力学结果。
关键词: K¨ahler流形、复结构、代数结构、力学。
Abstract：This article takes K¨ahler manifolds as the research object. Under the premise of reader having a certain geometric base, the first half of the article expounds the research background of K¨ahler manifolds, prior knowledge and the basic knowledge of K¨ahler geometry, such as: the definition and related properties of K¨ahler manifolds and its determination methods. Then we analyzed three structures of K¨ahler manifolds in detail with the help of concrete examples and got their specific form, these are complex structure, Hermitian structure and symplectic structure. The whole process is analytic except the part of algebraic structure.
Thus, several equivalent definition of K¨ahler manifolds in different point of view have been got:
1) K¨ahler manifold is a symplectic manifold with a integrable almost complex structure;
2) K¨ahler manifold is an Hermitian manifold with closed Hermitian form;
3) K¨ahler manifold is a special Riemann manifold, on the one hand, it keeps Rieman metric invariable under the canonical complex structure, on the other hand, its canonical complex structure is parallel under the Levi-Civita connection.
In the aspect of application, this article studied the rules of particle movement and got some simple results using the method of Newton mechanics and Hamilton mechanics.
Keywords: K¨ahler manifold, complex structure, algebraic structure, mechanics.
目 录
摘 要 I
Abstract II
1 引言 1
1.1 K¨ahler流形的引入. 1
1.2 K¨ahler几何中的重要结果. 1
1.2.1 Kodaira的工作 . . . 1
1.2.2 K¨ahler-Einstein度量 . . 1
1.2.3 Yau的工作 . .