The explicit construction of projectively Ricci-flat Finsler metrics on double hemispheres

We give an explicit construction of an $(n+1)$-parameter family of projectively Ricci-flat Finsler metrics on an $n$-dimensional double hemisphere by using the spherical harmonics of degree 1. We also show that all locally projectively flat Randers metrics with isotropic $S$-curvature on the standard unit sphere are projectively Ricci-flat.