Some rigidity results on complete Finsler manifolds

Document Type : Original Article


1 Department of Statistics, The University of Auckland, Auckland, New Zealand

2 Center of Mathematics, Computing and Cognition - CMCC Federal University of ABC - UFABC, SP, Brazil.


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.