Finsler Metrics and the Degree of Symmetry of a Closed Manifold

In this paper, we use the representation theory of compact Lie groups to prove that the degree of symmetry of a closed manifold can be realized by a non-Riemannian Finsler metric unless the manifold is diffeomorphic to a rank-one Riemannian symmetric space. As a by product, we obtain a sufficient and necessary condition for a coset space of a Lie group to admit invariant non-Riemannian Finsler metrics.