On Homogeneous Projectively Flat Finsler Metrics

Abstract

Recently, Liu-Deng studied projectively flat homogeneous \((\alpha , \beta )\)-metrics and showed that if these metrics are not Riemannian nor locally Minkowskian, then the Finsler metrics are left invariant Randers metrics on the hyperbolic space \(\textbf{H}^n\) as a solvable Lie group (Liu and Deng in Forum Math 27:3149–3165, 2015). In this paper, we study homogeneous projectively flat (or projective) general Finsler metrics. First, we prove that homogeneous projectively flat Finsler metrics have vanishing \({{\bar{\textbf{E}}}}\)-curvature if and only if they have almost isotropic S-curvature if and only if they have relatively isotropic L-curvature. In any cases, the Finsler metric reduces to a locally Minkowskian metric or a Riemannian metric of constant sectional curvature. This yields a classification of homogeneous projective Finsler metrics with the above mentioned non-Riemannian curvatures properties. Finally, we show that Liu-Deng’s Randers metrics are Douglas metrics which have not isotropic S-curvature nor relatively isotropic L-curvature.