On Finsler Surfaces with Isotropic Main Scalar

Let (M,F) be a Finsler surface with the isotropic main scalar I=I(x). The well-known Berwald’s theorem states that F is a Berwald metric if and only if it has a constant main scalar I=constant. This ensures a kind of equality of two non-Riemannian quantities for Finsler surfaces. In this paper, we consider a positively curved Finsler surface and show that H=0 if and only if I=0. This provides an extension of Berwald’s theorem. It follows that F has an isotropic scalar flag curvature if and only if it is Riemannian. Our results yield an infrastructural development of some equalities for two-dimensional Finsler manifolds.