Multiplicity of closed geodesics on bumpy Finsler manifolds with elliptic closed geodesics

The existence of two non-contractible closed geodesics on every bumpy Finsler compact space form

Let $M=S^n/ Γ$ and $h$ be a nontrivial element of finite order $p$ in $π_1(M)$, where the integer $n\geq2$, $Γ$ is a finite group which acts freely and isometrically on the $n$-sphere and therefore $M$ is diffeomorphic to a compact space form. In this paper, we establish first the resonance identity for non-contractible homologically visible minimal closed geodesics of the class $[h]$ on every Finsler compact space form $(M, F)$ when there exist only finitely many distinct non-contractible closed geodesics of the class $[h]$ on $(M, F)$. Then as an application of this resonance identity, we pr…

Resonance identity and multiplicity of non-contractible closed geodesics on Finsler $\mathbb{R}P^{n}$

In this paper, we establish first the resonance identity for non-contractible homologically visible prime closed geodesics on Finsler $n$-dimensional real projective space $(\mathbb{R}P^n,F)$ when there exist only finitely many distinct non-contractible closed geodesics on $(\mathbb{R}P^n,F)$, where the integer $n\geq2$. Then as an application of this resonance identity, we prove the existence of at least two distinct non-contractible closed geodesics on $\mathbb{R}P^{n}$ with a bumpy and irreversible Finsler metric. Together with two previous results on bumpy and reversible Finsler metrics in…

Finsler 3维球面上的多条闭测地线

Multiple closed geodesics on Finsler $3$-dimensional sphere

本文在所有素闭测地线Morse指数非零的条件下，证明了Anosov关于bumpy Finsler $S^3$至少存在四条素闭测地线的猜想。

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

In this paper, we first generalize the common index jump theorem for symplectic matrix paths proved in 2002 by Long and Zhu in [LoZ], and get an enhanced version of it. As its applications, we further prove that for a compact simply-connected manifold $(M,F)$ with a bumpy, irreversible Finsler metric $F$ and $H^*(M;{\bf Q})\cong T_{d,n+1}(x)$ for some even integer $d\ge 2$ and integer $n\ge 1$, there exist at least $\frac{dn(n+1)}{2}$ distinct non-hyperbolic closed geodesics with odd Morse indices, provided the number of distinct prime closed geodesics is finite and every prime closed geodesic…

Multiplicity and stability of closed geodesics on positively curved Finsler $4$-spheres

In this paper, we prove that for every Finsler $4$-dimensional sphere $(S^4,F)$ with reversibility $λ$ and flag curvature $K$ satisfying $\frac{25}{9}\left(\fracλ{1+λ}\right)^2<K\le 1$ with $λ<\frac{3}{2}$, either there exist at least four prime closed geodesics, or there exist exactly three prime non-hyperbolic closed geodesics and at least two of them are irrationally elliptic.

The optimal lower bound estimation of the number of closed geodesics on Finsler compact space form $S^{2n+1}/ Γ$

Let $M=S^{2n+1}/ Γ$, $Γ$ is a finite group which acts freely and isometrically on the $(2n+1)$-sphere and therefore $M$ is diffeomorphic to a compact space form. In this paper, we first investigate Katok's famous example about irreversible Finsler metrics on the spheres to study the topological structure of the contractible component of the free loop space on the compact space form $M$, then we apply the result to establish the resonance identity for homologically visible contractible minimal closed geodesics on every Finsler compact space form $(M, F)$ when there exist only finitely many dist…