In this paper, we  study the Landsberg and mean Landsberg curvatures of two-dimensional Finsler manifolds. First, we prove that a two-dimensional Finsler metric  is a generalized Landsberg metric if and only if it is a stretch metric. Then, we study Finsler surfaces with  isotropic main scalar and find the necessary and sufficient condition under which these spaces has vanishing ${bf tilde J}$-curvature.

Keywords: Landsberg curvature, mean Landsberg curvature, generalized Landsberg metric, stretch metric.

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