A Class of Projectively Flat Finsler Metrics

Projectively flat Finlser metrics on a convex domain U in \(\mathbb {R}^n\) are regular solutions to Hilbert’s Fourth Problem. In this paper, we study projectively flat Finlser metrics on U. We find equations that characterize these metrics with weakly orthogonal invariance, refining a theorem due to Sol\(\acute{o}\)rzano-Le\(\acute{o}\)n. As its application, we obtain infinitely many new projectively flat Finlser metrics on \(\mathbb {S}^{n+1}\) and determine their scalar flag curvature. These metrics contain Bryant’s projective spherically symmetric Finsler metric of constant flag curvature 1.