Non-hyperbolic closed geodesics on Finsler spheres (Q2343973)

A Finsler sphere \((S^n,F)\) is called bumpy if all closed geodesics are non-degenerate. Suppose that \((S^{2k},F)\) is a bumpy Finsler \(2k\)-sphere satisfying \[ (\frac{\lambda}{\lambda+1})^2< K \leq 1, \] where \(K\) and \(\lambda\) denote the flag curvature and reversibility, respectively. The author of the present paper proves that either there exist infinitely many prime closed geodesics or there exist at least \(2k\) non-hyperbolic prime closed geodesics on \((S^{2k},F)\).