The second closed geodesic on a complex projective plane

Abstract

We show the existence of at least two geometrically distinct closed geodesics on a complex projective plane with a bumpy and non-reversible Finsler metric.

The index growth and multiplicity of closed geodesics

“…Our main result in this section generalizes the multiplicity results in [31] on rational closed geodesics on spheres, and in [12,39] on bumpy spheres, and [40] on bumpy CP 2 to all compact simply connected manifolds. Proof.…”

“…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…”

The enhanced common index jump theorem for symplectic paths and non-hyperbolic closed geodesics on Finsler manifolds

“…For example, it was proved that a Finsler (CP n , F ) with n ≥ 7 and a bumpy Finsler metric possessing only finitely many distinct closed geodesics and satisfying 2 n+1 λ 1+λ 2 ≤ K ≤ 1, carries always at least 2n distinct closed geodesics, and at least (n − 3) of them are non-hyperbolic, where λ and K denote also the reversibility and the flag curvature of (CP n , F ) respectively. In [Rad5], Rademacher further proved the existence of two prime closed geodesics on any CP 2 with a bumpy irreversible Finsler metric. In recent preprint [DLW], the authors proved the existence of at least two prime closed geodesics on every compact simply-connected Finsler manifold (M, F ) with a bumpy irreversible Finsler metric F .…”

Two closed geodesics on compact simply connected bumpy Finsler manifolds

“…Then in Duan and Long [DuL1] and Rademacher [Rad4], the existence of at least two distinct closed geodesics on every bumpy n-sphere was proved independently. In [Rad5], Rademacher further proved there exist two prime closed geodesics on any CP 2 with a bumpy irreversible Finsler metric. Related more recent results can be found in [LoW], [Wan], and [HiR].…”