Finsler Metrics of Scalar Flag Curvature

Flag曲率是黎曼几何中截面曲率的自然推广。Riemann-Finsler几何的基本问题之一是研究和刻画具有标量/常旗曲率的Finsler空间。在这一计划中，PI将研究这类Finsler度量，并提出以下研究计划：（1）构造常/标量旗曲率的Finsler度量的新例子，（2）确定常/标量旗曲率的Finsler度量的局部结构。Riemann-Finsler几何是研究正则（不一定可逆）度量空间的一门学科。Finsler度量是没有二次约束的黎曼度量。现代微分几何提供了概念和工具，以直接和优雅的方式处理黎曼-芬斯勒几何。Riemann-Finsler几何在数学和物理学的其他领域也有许多应用，如数学心理学、广义相对论、偏微分方程、导航问题、成像过程和模空间等，它将在世界各地许多几何学家的努力下继续发展。

AbstractNSF proposal 0810159Flag 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 necessarilyreversible) 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.