Multiple Closed Geodesics on Positively Curved Finsler Manifolds

Abstract

In this paper, we prove that on every Finsler manifold (M,F){(M,F)} with reversibility λ and flag curvature K satisfying (λλ+1)2<K≤1{(\frac{\lambda}{\lambda+1})^{2}<K\leq 1}, there exist [dim⁡M+12]{[\frac{\dim M+1}{2}]} closed geodesics. If the number of closed geodesics is finite, then there exist [dim⁡M2]{[\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}.

Keywords

• finsler manifolds
• closed geodesics
• index iteration
• morse theory
• 53c22
• 53c60
• 58e10