Multiple Closed Geodesics on Positively Curved Finsler Manifolds

Abstract

Abstract In this paper, we prove that on every Finsler manifold ( M , F ) {(M,F)} with reversibility λ and flag curvature K satisfying ( λ λ + 1 ) 2 < K ≤ 1 {(\frac{\lambda}{\lambda+1})^{2}<K\leq 1} , there exist [ dim ⁡ M + 1 2 ] {[\frac{\dim M+1}{2}]} closed geodesics. If the number of closed geodesics is finite, then there exist [ dim ⁡ M 2 ] {[\frac{\dim M}{2}]} non-hyperbolic closed geodesics. Moreover, there are three closed geodesics on ( M , F ) {(M,F)} satisfying the above pinching condition when dim ⁡ M = 3 {\dim M=3} .