The second closed geodesic on Finsler spheres of dimension 𝑛>2

Abstract

Abstract. We show the existence of at least two geometrically distinct closed geodesics on an n-dimensional sphere with a bumpy and non-reversible Finsler metric for n > 2.

Multiple 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. In [LoD] and [DuL2] of Duan and Long, they proved there exist at least two closed geodesics on any compact simply-connected Finsler 3 and 4 manifold.…”

The second closed geodesic on a complex projective plane

“…[1] that for every non-reversible Finsler metric on S 2 , there are two geometrically distinct closed geodesics. Independently, Duan and Long [2] and the author [6] showed that on an ndimensional sphere with a bumpy non-reversible Finsler metric, there are at least two geometrically distinct closed geodesic for all n > 2. A recent survey on existence results for closed geodesics on Finsler manifolds is Ref.…”

The index growth and multiplicity of closed geodesics

“…But when the dimension of a compact simply connected manifold is greater than 2, we are not aware of any multiplicity results on the existence of at least two closed geodesics without pinching, generic or bumpy conditions even on spheres (cf. [1][2][3][4][5][12][13][14]25,[38][39][40]), except Theorem C below proved recently in [31].…”

“…Note that Claim 2 was proved in [12,39] when d is odd or h = 1, and in [40] when d = h = 2. Next we give the proof of Claim 2 in two cases for all the values of d 2 and h 1, which yields also a new proof for the results in [12,39,40]. By Lemma 2.6, for any odd integer k 2h + 1 the Betti numbers b j in this case satisfy…”