A Strong Rigidity Result on Complete Finsler Manifolds via a Second Order Differential Equation

We provide an extension of the Obata's theorem to Finsler geometry and establish some rigidity results based on a second order differential equation. Mainly, we prove that: Every complete connected Finsler manifold of positive constant flag curvature is isometrically homeomorphic to an Euclidean sphere endowed with a certain Finsler metric and vice versa.