Flag curvature is a natural extension of the sectional curvature in Riemannian geometry. One of the fundamental problems in Riemann-Finsler geometry is to study and characterize Finsler spaces of scalar/constant flag curvature. In this proposal, the PI will investigate such Finsler metrics and propose the following research plans: (1) Construct new examples of Finsler metrics of constant/scalar flag curvature, and (2) Determine the local structure of Finsler metrics of constant/scalar flag curvature.
Riemann-Finsler geometry is a subject studying regular (not necessarily reversible) metric spaces. Finsler metrics are just Riemannian metrics without quadratic restriction. Modern differential geometry provides the concepts and tools to effect a treatment of Riemann-Finsler geometry in a direct and elegant way. Riemann-Finsler geometry has many applications to other areas in mathematics and physics, such as mathematical psychology, general relativity, partial differential equations, navigation problems, imaging process, and module spaces, etc. It will continue to develop through the efforts of many geometers around the world.
