芬斯勒几何中若干问题的研究

芬斯勒几何作为度量上无二次限制的黎曼几何，是现代微分几何的重要组成部分。它的理论与方法在数学、控制论、生态学、光学、天文学及其它自然科学领域中有广泛应用。本项目主要研究具有常（标量）旗曲率、爱因斯坦芬斯勒度量的例子构造、刚性和分类；芬斯勒流形上测地线的描述，球对称的芬斯勒度量，芬斯勒流形上的几何不变量（包括黎曼几何量与非黎曼几何量）。芬斯勒几何不仅与微分几何、几何分析、李群李代数等数学分支紧密联系，而且与理论物理相沟通。此项研究将对于常旗曲率芬斯勒空间的分类，爱因斯坦芬斯勒度量的构造，芬斯勒流形上测地线之刻画和描述，具有各种曲率性质的球对称芬斯勒度量的性质、构造和分类，新的芬斯勒几何不变量及其不变量之间内蕴关系的建立，Douglas度量和射影平坦芬斯勒度量的刻画和分类及其应用注入新的活力，对促进我国数学及其相关学科发展具有重要意义。

Finsler geometry is just Riemannian geometry without guadratic restriction. It is one of the principal aspects in modern differential geometry. Its theory and methods also assert themselves in applications, most notably in mathematics, control theory, ecology, optics and other natural sciences. This project study some problems in Riemann-Finsler geometry (Finsler geometry for short). We are going to do the following: 1. construction and classification of Finsler metrics of constant/scalar flag curvature, examples and characterization of real/complex Einstein Finsler metrics; 2. description of geodesics on Finsler manifolds; 3. investigation of spherically symmetric Finsler metrics with some curvature properties; 4. finding new Riemann/non-Riemann geometric quantities on a Finsler space and establishing fundamental identities for geometric quantities. Real/complex Finsler geometry has close relation with differential equations, geometric analysis, Lie group and Lie algebra, theoretical physics and so on. This project will promote development of mathematics and other natural sciences in China as well as world.