Finsler几何中若干问题的研究

The project mainly studies the following problems in Finsler geometry. (1) Based on the foliation theory and characteristic class theory of vector bundles, we will study the theory and applications of the Godbllion-Vey type characteristic classes and the Chern-Simons type characteristic classes of the projective sphere bundle of a Finsler manifold.(2)The natrual contact metric structure of the projective sphere bundle of a Finsler manifold allows us to study the Finsler geometry by special geometric structures in the contact catalogs, such as, Sasakian structure, B tensor and CR structure. We will discuss the Finsler manifold of constant flag curvature in this style. We will also to discuss the relations between Riemannian geometry of projective sphere bundle and Finsler geometry of the base manifold via adiabatic limit technique. (3) By using the Sasaki metric and some geometric 1 form on the projective sphere bundle, we will construct some special Randers metrics and study the geometric relationship between these metrics and the base manifold.(4) We will investigate the integral geometry of some special Finsler surfaces and its applications to geometric inequalities. (5)We will try to study the hypermetric Minkowski spaces by methods of differential geometry. (6) We are going to establish the Crofton formulas related to the mean curvature in Minkowski spaces. Clearly such researches have important theoretical value to enrich people’s understanding of a given Finsler manifolds and their nature from different sides.

本课题主要在Finsler几何中的如下几个方面开展研究：(1)通过叶状结构理论以及向量丛的示性类理论，研究Finsler流形射影球丛的Godbllion-Vey型示性类和Chern-Simons型示性类；(2)在射影球丛上应用切触几何工具，研究常旗曲率Finsler流形的性质与分类；应用绝热极限方法，探讨射影球丛Riemann几何与底流形几何的关系；(3)通过射影球丛的Sasaki度量以及某些几何1形式，构造射影球丛上的Randers度量，研究这类度量与底流形几何的关系；(4)研究特殊Finsler曲面上的积分几何及几何不等式；(5)研究hypermetric的Minkowski空间的微分几何及在Finsler几何中的应用；(6)研究Minkowksi空间中关于平均曲率的Crofton型积分公式。这些研究将丰富人们对Finsler几何的理解。

本项目对于Finsler几何中若干重要的几何与拓扑问题进行了研究，得到了若干有意义的结果: (1)应用指标理论中的方法与结果, 给出了偶数维闭Finsler流形的Euler类的几何表示, 由此得到了不依赖于外在向量场的Gauss-Bonnet-陈公式; (2)建立了Minkowski空间几何与中心仿射微分几何间的联系，得到了Minkowski空间的等价性定理; (3)将等价性定理应用于Finsler几何, 证明了平均Berwald曲率为零的Landsberg空间为Berwald空间; (4)推广了semi-C-reducible范数的概念，给出了一类semi-C-reducible Minkowski空间的强等价性定理; 由此证明了一类landsberg广义(α,β)-度量为Berwald度量; (5)给出了一类旗流形的圆丛上的Einstein-Randers度量的分类。这些研究结果丰富了人们对Finsler流形及其几何与拓扑性质的理解。