On Homogeneous Weakly Stretch Finsler Metrics

In this paper, we show that every homogeneous Finsler metric is a weakly stretch metric if and only if it reduces to a weakly Landsberg metric. This yields an extension of Tayebi–Najafi’s result that proved the result for the class of stretch Finsler metrics. Let F := αφ(β/α) be a homogeneous weakly stretch (α, β)-metric on a manifold M. We show that if φ is of polynomial type, then F is a Berwald metric. Also, we prove that F is a Berwald metric if and only if it has vanishing S-curvature. Then, we show that F is a Douglas metric if and only if it reduces to a Berwald metric. In continue, we show that every homogenous weakly stretch surface is a Landsberg surface. Finally, we characterize homogeneous weakly stretch spherically symmetric Finsler metrics.