一类芬斯勒度量的非黎曼曲率的研究

Finsler geometry is just the Riemannian geometry without the quadratic restriction on the metric. The rapid progress of Riemann-Finsler geometry has been made after the great encouragement of Professor S. S. Chern. The S-curvature is a very important non-Riemannian geometric quantity and plays an important role both in local and global Finsler geometry. It was introduced by Z. Shen to derive a volume comparison theorem in Riemann-Finsler geometry. General (alpha,beta)-metrics form a rich and important class of metrics. They include(alpha,beta)-metrics, spherically symmetric Finsler metrics, part of Bryant's metrics and part of general m-th root metrics and etc. Therefore, general (alpha,beta)-metrics make up of a much large class of Finsler metrics, which make it possible to find out more Finsler metrics to be of great properties. This project mainly focuses on general (alpha,beta)-metrics with isotropic S-curvature. By making use of the theory of Lie Groups and Partial Differential Equations and the method of beta deformations, we will study the local structure of general (alpha,beta)-metrics with isotropic S-curvature with the help of Maple. The main aim of this project is to characterize and classify this class of metrics. It will contribute to making rapid progress of local and global Finsler geometry.

芬斯勒几何就是在度量上没有二次型限制的黎曼几何。它经陈省身先生大力提倡，近二十多年取得了蓬勃发展。S曲率是一个非常重要的非黎曼几何量，它无论在局部还是整体芬斯几何中都占有很重要的地位。S曲率是沈忠民教授为研究芬斯勒几何中的体积比较定理而引入的。广义(alpha,beta)度量是既丰富又重要的一类芬斯勒度量，它们包含(alpha,beta)度量、球对称度量、R. Bryant构造的部分度量和部分广义m次根度量等。因此，广义(alpha,beta)度量构成了很大一类芬斯勒度量，这有利于找出更多具有很好性质的芬斯勒度量。本项目拟以具有迷向S曲率的广义(alpha,beta)度量为研究对象，利用李群和偏微分方程的理论，通过beta形变的方法，借助Maple研究具有迷向S曲率的广义(alpha,beta)度量的局部结构。我们旨在刻划和分类此类度量。本项目的实施将促进国内外局部和整体芬斯勒几何的发展。

芬斯勒几何是在度量上没有二次型限制的黎曼几何。它经陈省身先生大力提倡，近二十年来取得蓬勃发展。芬斯勒几何中存在若干重要的问题：其一，寻找和刻划n维欧氏空间中开集上的射影平坦芬斯勒度量，这正是正则情形下希尔伯特第四问题；其二，分类常旗曲率的芬斯勒度量，这是由鲍大卫和陈省身先生提出的；其三，弄清具有非黎曼曲率性质的芬斯勒度量的结构。本项目基于目前国内外研究热点，重点研究以下内容：(1) 局部射影平坦的芬斯勒度量；(2) 常旗曲率的芬斯勒度量；(3) 具有非黎曼曲率性质的芬斯勒度量。本项目取得以下重要结果：(1) 刻画了常旗曲率球对称度量并构造出新的非射影平坦且具有常旗曲率-1和0的球对称度量的例子； (2) 刻画了局部射影平坦的球对称度量并构造出新的局部射影平坦且旗曲率为零的球对称度量；(3) 分类了一类Douglas 奇异平方度量； (4) 给出具有GDW型球对称度量的刻画方程并分类了一类具有GDW型球对称度量；(5) 对一类具有迷向S曲率的球对称度量进行了分类；(6) 对一类具有迷向Berwald曲率的广义(alpha,beta) 度量进行了分类；(7) 刻画了所有Douglas广义 (alpha,beta)度量并构造出新的例子。