ON THREE-DIMENSIONAL HOMOGENEOUS FINSLER MANIFOLDS
Abstract
In this paper, we study a long existing open problem on Landsberg metrics in Finsler geometry. For this aim, we study the Landsberg curvature of three-dimensional homogeneous Finsler manifolds. First, we express the second Matsumoto torsion of three-dimensional Finsler manifolds, explicitly. Then, we show that the mean Landsberg curvature of three-dimensional homogeneous Finsler manifolds satisfy an ODE. Finally, we prove that every homogeneous 3-dimensional L-reducible Finsler manifold has constant relatively isotropic mean Landsberg curvature if and only if it is a Landsberg metric or a Randers metric of Berwald-type.
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DOI: https://doi.org/10.22190/FUMI221028010T
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