Abstract We show that if there is a nonconstant function f ∈ C 1, α ( M ) satisfying the Obata equation on a Finsler n -manifold ( M , F ), then ( M , F ) is isometric to a Finsler n -sphere. In particular, for a Randers metric, F is determined analytically by the standard sphere metric and a homothetic field, and the S -curvature is constant under the Busemann-Hausdorff volume form. As applications, we prove Lichnerowicz-Obata type rigidity theorem: for a closed Finsler n -manifold with the weighted Ricci curvature Ric N ≥ ( N − 1) k > 0, the first eigenvalue of the Finsler Laplacian is bounded below by Nk with equality if and only if ( M , F ) is isometric to a Finsler n -sphere.
Huarong Wang (Thu,) studied this question.