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

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