University of Mohaghegh Ardabili Journal of Finsler Geometry and its Applications 2783-0500 3 1 2022 07 01 Some rigidity results on complete Finsler manifolds 100 117 1671 10.22098/jfga.2022.10415.1061 EN Azam Asanjarani Department of Statistics, The University of Auckland, Auckland, New Zealand Hengameh R. Dehkordi Center of Mathematics, Computing and Cognition - CMCC Federal University of ABC - UFABC, SP, Brazil. Journal Article 2022 02 23 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. https://jfga.uma.ac.ir/article_1671_16df84e3513a82e16e6237882007ccd7.pdf