单色芬斯勒流形的曲率

A Finsler manifold is called single colored if the tangent spaces are linearly isometric to each other as Minkowski spaces. The corresponding Finsler function on such a manifold is called a single colored Finsler metric. This definition recovers many classical examples, such as Berwald metrics, invariant metrics on homogeneous spaces, etc. The study of single colored Finsler manifold not only has important impact theretically, but also has great prospect in application. The applicant has published several research papers on related subjects in recent years, hence he possessed the foundation to start further research. In this program, the applicant will do a series of studies concerning the curvature of single colored Finsler metrics, especially on the flag curvature, Ricci curvature and Landsberg curvature. We will try to settle the following three kinds of problems: First, on certain conditions, find new examples of single colored Finsler manifolds with constant flag curvature or constant Ricci curvature; classify such manifolds under specific geometric conditions; Second, on a given manifold, determine the existence or nonexistence of single colored Finsler metrics with positive curvature or negative curvature; Third, find examples of Finsler manifolds whose Landsberg curvature or weakly Landsberg curvature is zero, thus provide an answer to the existence problem of Landsberg manifolds.

一个芬斯勒流形，如果各点的切空间彼此线性等距，则称它为单色芬斯勒流形。 相应的度量称为单色芬斯勒度量。这一定义包含了许多重要的芬斯勒度量，如Berwald度量， 齐性空间的不变度量，等等。研究单色芬斯勒流形既有重要的理论意义，也有广阔的应用前景。申请人在近几年已发表了多篇与此相关的研究论文，具备进一步探索的基础。在本项目中，将对单色芬斯勒度量的曲率展开一系列研究，主要是对旗曲率、里奇曲率和Landsberg曲率的研究。本项目拟解决的问题包括：在一定条件下，寻找新的旗曲率为常数或里奇曲率为常数的单色芬斯勒流形的例子，可能的情况下将予以分类；在某些特定的流形上，探索曲率定号的单色芬斯勒度量的存在性；寻找Landsberg曲率或弱Landsberg曲率为零的单色芬斯勒流形的例子，即探索Landsberg流形的存在性问题。

芬斯勒几何中，最重要的几种曲率量包括旗曲率、Ricci曲率和Landsberg曲率。在芬斯勒流形上，每一点的切空间都是Minkowski空间。当这些Minkowski空间彼此线性同构时，相应的芬斯勒流形称为单色的。对单色芬斯勒流形的曲率进行研究，有助于我们深入理解曲率的内涵，也有助于我们发现新的具有特殊曲率性质的芬斯勒流形。在所有的单色芬斯勒流形中，齐性芬斯勒流形的几何内涵最为丰富。我们导出了齐性芬斯勒流形的旗曲率、Ricci曲率以及Landsberg曲率的公式，并将它们统一用李代数中的两个张量场来表示。利用这些公式，我们证明了一系列结论，推广了J. Wolf、J. Milnor和胡志广-邓少强的结果，得到了陈省身关于爱因斯坦度量的存在性问题在齐性情形的否定答案，并在正曲率齐性流形的探索中获得重要进展。此外，对二维常曲率齐性流形，我们获得了局部标准形；找到了一大批余齐性为1的共形平坦度量。