A remark on the Gauss-Bonnet theorem in Finsler geometry

Here it is shown, every n-dimensional compact constant isotropic L-curvature space and J-curvature space (as well as Landsberg surface) with non-zero flag curvature is Riemannian. As a consequence of this work, not only known results on the Gauss-Bonnet theorem on Finsler surfaces become trivial, but it will be so for a larger class of n-dimensional Finsler manifolds as well, namely constant isotropic L-curvature and J-curvature spaces.