关于黎曼-芬斯勒几何的若干问题研究

黎曼-芬斯勒几何（简称：芬斯勒几何）是现代数学中的重要前沿学科，其理论和方法在自然科学的许多领域中具有广泛的应用。本项目主要研究广义（α，β）度量（包括球对称度量）的几何；球面上齐性爱因斯坦芬斯勒度量的例子和构造；芬斯勒流形上测地线的描述；芬斯勒几何中的球（面）定理。芬斯勒几何不仅与齐性空间理论、微分方程、临界点理论、拓扑学等数学分支具有密切联系，并且与控制论和物理学相沟通。这项研究将对于各种广义（α，β）度量（比如，局部射影平坦的广义（α，β）度量）的性质、例子、刚性及分类，不同维数之球面上齐性爱因斯坦芬斯勒度量的描述和构造，芬斯勒流形上各类测地线的刻画和描述，芬斯勒几何中关于Ricci曲率之球面定理的建立注入新的活力，对促进我国数学及其相关学科发展，加强国内外几何学者的交流与合作具有重要意义。

Riemann-Finsler geometry (Finsler geometry for short) is an important frontier subject in modern mathematics. Its theory and methods also assert themselves in applications, most notably in many fields of the natural sciences. This project study some problems in Finsler geometry. We are going to do the following: 1. Study on geometry of general（α，β）- metrics, in particular, spherically symmetric metrics; 2. Construction of homogeneous Einstein Finsler metrics on the spheres; 3. Description of geodesics on Finsler manifolds; 4. Finding new Finslerian sphere theorems. Riemann-Finsler geometry has close relation with control theory, theoretical physics，topology and so on. This project will promote development of mathematics and related natural sciences in China. It will strengthen the communication and cooperation among the geometers at home and abroad.