Some rigidity results on complete Finsler manifolds

We provide an extension of obata's theorem to finsler geometry and establish some rigidity results based on a second-order differential equation. mainly, we prove that every complete simply connected finsler manifold of positive constant flag curvature is isometrically homeomorphic to a euclidean sphere endowed with a certain finsler metric and vice versa. based on these results, we present a classification of finsler manifolds which admit a transnormal function. specifically, we show that if a complete finsler manifold admits a transnormal function with exactly two critical points, then it is homeomorphic to a sphere.