Vanishing of Killing vector fields on compact Finsler manifolds

Abstract

In this paper, we define a new Ricci curvature on Finsler manifold named the mean Ricci curvature, which is useful in the study of different symmetric fields on manifolds. By presenting a Bochner type formula of Killing vector fields on general Finsler manifolds, we prove the vanishing theorem of the Killing vector fields on any compact Finsler manifold with a negative mean Ricci curvature. This result involves the vanishing theorem of Killing vector fields in the Riemannian case.