Bezdaf Najafi, Akbar Tayebi : Some curvature properties of $(\alpha,\beta)$-Metrics, 277-291


We solve two open problems in Finsler geometry which have been proposed by Z. Shen about Finsler metrics with relatively isotropic Landsberg curvature and weakly Landsberg metrics. We define a new quantity which is closely related to the S-curvature. Then, we find some conditions for $(\alpha,\beta)$-metrics under which the notions of relatively isotropic Landsberg curvature and relatively isotropic mean Landsberg curvature are equivalent. It extends Cheng-Shen's well-known theorem that proves the equality for the Randers metrics. As an application, we prove that every weakly Landsberg $(\alpha,\beta)$-metric of non-Randers type with vanishing S-curvature is Berwaldian.

Key Words: Isotropic (mean) Landsberg curvature, $(\alpha,\beta)$-metric

2010 Mathematics Subject Classification: Primary 53B40, Secondary 53C60.