On a class of weakly Landsberg metrics composed by a Riemannian metric and a conformal 1-form

Without the quadratic restriction, there are many non-Riemannian geometric quantities in Finsler geometry. Among these geometric quantities, Berwald curvature, Landsberg curvature and mean Landsberg curvature are related directly to the famous "unicorn problem" in Finsler geometry. In this paper, Finsler metrics with vanishing weakly Landsberg curvature (i.e., weakly Landsberg metrics) are studied. For the general $(\alpha ,\beta )$-metrics, which are composed by a Riemannian metric $\alpha$ and a 1-form $\beta$, we found that if the expression of the metric function doesn't depend on the dimension $n$, then any weakly Landsberg $(\alpha ,\beta )$-metric with a conformal 1-form must be a Landsberg metric. In the two-dimensional case, the weakly Landsberg case is equivalent to the Landsberg case. Further, we classified two-dimensional Berwald general $(\alpha ,\beta )$-metrics with a conformal 1-form.