A symmetric property in the enhanced common index jump theorem with applications to the closed geodesic problem (Q2078363)

In the present paper, the authors prove a symmetric property for the indices of symplectic paths in the enhanced common index jump theorem. Further, they prove that every compact Finsler manifold \((M,F)\) with reversibility \(\lambda\) and flag curvature \(K\) satisfying \((\frac{\lambda}{\lambda+1})^2<K\leq 1\), there exist two elliptic closed geodesics whose linearized Poincaré map has an eigenvalue of the form \(e^{\sqrt{-1\theta}}\) with \(\frac{\theta}{\pi}\notin Q\) provided the number of closed geodesics on \(M\) is finite. All the work given by them is useful in development of the differential geometry.