  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 connected Finsler manifold of positive constant flag
curvature is isometrically homeomorphic to an 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.
