Flag curvatures of the unit sphere in a Minkowski-Randers space

Abstract

On a real vector space $V$, a Randers norm $hat{F}$ is defined by $hat{F}=hat{alpha}+hat{beta}$, where $hat{alpha}$ is a Euclidean norm and $hat{beta}$ is a covector. We show that the unit sphere $Sigma$ in the Randers space $(V,hat{F})$ has positive flag curvature, if and only if $|hat{beta}|_{hat{alpha}} < (5-sqrt{17})/2 approx 0.43845$, thus answering a problem proposed by Prof. Zhongmin Shen. Moreover, we prove that the flag curvature of $Sigma$ has a universal lower bound $-4$.