芬斯勒几何是在其度量上无二次型限制的黎曼几何。由芬斯勒几何发展起来的几何方法对于探索理论物理、生物数学和信息科学等其他领域提出的问题相当有效。(a,β)-度量是芬斯勒度量中重要的--类度量。本项目着重研究黎曼-芬斯勒几何的旗曲率和共形变换的问题。如具有数量旗曲率的Randers度量或具有常数旗曲率的(a,β)-度量的分类等至今还是公开问题。由于旗曲率与其他非黎曼几何量有密切联系，故具有非黎曼曲率性质的芬斯勒度量的研究也非常重要，如对具有数量旗曲率的Randers度量或具有其他曲率性质的(a,β)-度量进行分类;更一般的芬斯勒度量的几何刻画及其分类;各种芬斯勒度量的整体几何与拓扑性质等的进一步研究。此外，我们将探讨芬斯勒度量的共形变换;挖掘共形闭的Berwald度量的特性;研究共形于具有S曲率为零的Randers度量的属性。

Finsler geometry is just Riemannian geometry without the quadratic restriction. Methods developed by Finsler geometry are effectively applied to the fields of probing theoretical physics, mathematical biology and information science, etc. (a,ß)-metrics is one of important classes of Finsler metrics. This research program focuses on flag curvature and conformal transformation in Riemann-Finsler geometry. We will characterize or classify (positive) complete metrics of scalar curvature, hit off the properties of metric spaces, which are of constant flag curvature. Especially, we will catalogue and describe Randers metics of scalar flag curvature; study (a,ß)-metrics with other curvature properties. Then, we will break the conformal transformation on Finsler manifolds; tap the characteristic properties of closed conformal Berwald metrics; integrally depict Randers metrics which are conformal to another Randers metrics with zero S-curvature.