On Finslerain surfaces with special curvature properties

We introduce a new non-Riemannian quantity named mean stretch curvature. A Finsler metric with vanishing mean stretch curvature is called weakly stretch metric. We classify complete weakly stretch surfaces and show that every complete weakly stretch surface is Riemannian or Landsbergian. This provides a natural extension of Szabó’s rigidity theorem on Berwald surfaces. .Finally, we prove that every generalized Landsberg surface is a Landsberg metric