Closed geodesics on positively curved Finsler spheres

Abstract

In this paper, we prove that for every Finsler n-sphere (S n , F ) for n ≥ 3 with reversibility λ and flag curvature K satisfying λ λ+1 2 < K ≤ 1, either there exist infinitely many prime closed geodesics or there exists one elliptic closed geodesic whose linearized Poincaré map has at least one eigenvalue which is of the form exp(πiµ) with an irrational µ. Furthermore, there always exist three prime closed geodesics on any (S 3 , F ) satisfying the above pinching condition.

Multiple Closed Geodesics on Positively Curved Finsler Manifolds

“…Rademacher studied the existence and stability of closed geodesics on positively curved Finsler manifolds. In [Wan1]- [Wan3], W. Wang studied the existence and stability of closed geodesics on positively curved Finsler spheres. In [DuL1] of Duan and Long and in [Rad6] of Rademacher, they proved there exist at least two closed geodesics on any bumpy Finsler n-sphere independently.…”

“…(cf. Theorem 7.9 in [Rad2] or Theorem 5.5 in [Wan1]) Suppose that there exist only finitely many prime closed geodesics {c j } 1≤j≤p withî(c j ) > 0 for 1 ≤ j ≤ p on (S n , F ). Then the following identity holds…”

Closed Geodesics on Positively Curved Finsler 3-Spheres

“…Some similar results in the Riemannian case are obtained in [2,3]. Recently, Wang proved in [33] that for every Finsler n-dimensional sphere S n with reversibility λ and flag curvature K satisfying (λ/( + λ)) < K ≤ , either there exist infinitely many prime closed geodesics or there exists one elliptic closed geodesic whose linearized Poincaré map has at least one eigenvalue which is of the form exp( − πμ) with an irrational μ. The same author proved in [36] that for every Finsler n-dimensional sphere S n for n ≥ with reversibility λ and flag curvature K satisfying (λ/( + λ)) < K ≤ , either there exist infinitely many prime closed geodesics or there exist [n/ ] − closed geodesics possessing irrational mean indices.…”

“…Note that Wang proved in [33,Theorem 1.5] that there exist at least three distinct closed geodesics on (S , F) with flag curvature K satisfying (λ/( + λ)) < K ≤ . Motivated by the results mentioned above, in this paper, we prove the following theorem.…”

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Abstract:In [33], Wang proved that for every Finsler three-dimensional sphere (S , F) with reversibility λ and flag curvature K satisfying (λ/( + λ)) < K ≤ , there exist at least three distinct closed geodesics. In this paper, we prove that for every Finsler three-dimensional sphere (S , F) with reversibility λ and flag curvature K satisfying ( / )(λ/( + λ)) < K ≤ with λ < , if there exist exactly three prime closed geodesics, then two of them are irrationally elliptic and the third one is infinitely degenerate.

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