On E-curvature of homogeneous Finsler manifolds

In this paper, we study mean Berwald curvature of the class of homogeneous Finsler manifolds and prove three rigidity theorems. First, we prove that a homogeneous Finsler surface with isotropic mean Berwald curvature and relatively isotropic mean Landsberg curvature must be Riemannian or locally Minkowskian. Then, we show that every homogeneous weakly Berwald surface is Riemannian or locally Mikowskian. This yields an extension of the celebrated Szabó’s rigidity theorem for Berwald surfaces. Also, we conclude that every homogeneous Einstein Finsler surface must be Riemannian or locally Minkowskian. It generalizes a result of Deng–Yan that was proved for \((\alpha , \beta )\) $(\alpha ,\beta )$ -metrics. Finally, we study homogeneous weakly Berwald metrics of dimension \(n\ge 3\) $n\ge 3$ with scalar flag curvature. It turns out that homogeneous reversible weakly Berwald metrics of scalar flag curvature must be Riemannian or locally Minkowskian.