Finsler surfaces with vanishing $T$-tensor

In this paper, for Finsler surfaces, we prove that the T-condition and $\sigma T$-condition coincide. For higher dimensions $n\geq 3$, we illustrate by an example that the T-condition and $\sigma T$-condition are not equivalent. We show that the non-homothetic conformal change of a Berwald (resp. a Landsberg) surface is Berwaldian (resp. Landsbergian) if and only if the $\sigma T$-condition is satisfied. By solving the Landsberg's PDE, we classify all Finsler surfaces satisfying the T-condition, or equivalently the $\sigma T$-condition. Some examples are provided and studied.