Non-hyperbolic closed geodesics on Finsler spheres. (English) Zbl 1326.53110

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 \((S^{2k},F)\).


53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
53C22 Geodesics in global differential geometry
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