Related papers: Bumpy metrics on spheres and minimal index growth

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.

In this paper, we prove that on every Finsler $n$-sphere $(S^n, F)$ for $n\ge 6$ with reversibility $\lambda$ and flag curvature $K$ satisfying $(\frac{\lambda}{\lambda+1})^2<K\le 1$, either there exist infinitely many prime closed…

In this paper, we proved that for every Finsler metric on $S^n$ $(n\ge 4)$ with reversibility $\lambda$ and flag curvature $K$ satisfying $(\frac{2n-3}{n-1})^2 (\frac{\lambda}{\lambda+1})^2<K\le 1$ and $ \lambda<\frac{n-1}{n-2} $, there…

In this paper we prove that for every bumpy Finsler metric $F$ on every rationally homological $n$-dimensional sphere $S^n$ with $n\ge 2$, there exist always at least two distinct prime closed geodesics.

It's well known that the n-sphere $S^n$ is the universal double covering of the $n$-dimensional real projective space $\mathbb{R}P^n$ and then any Finsler metric on $\mathbb{R}P^n$ induces a Finsler metric of $S^n$. In this paper, we prove…

In this paper, we prove that on every Finsler $n$-sphere $(S^n, F)$ with reversibility $\lambda$ satisfying $F^2<(\frac{\lambda+1}{\lambda})^2g_0$ and $l(S^n, F)\ge \pi(1+\frac{1}{\lambda})$, there always exist at least $n$ prime closed…

In 1973, Katok constructed a non-degenerate (also called bumpy) Finsler metric on $S^3$ with exactly four prime closed geodesics. And then Anosov conjectured that four should be the optimal lower bound of the number of prime closed…

In this paper, we prove there are at least two closed geodesics on any compact bumpy Finsler $n$-manifold with finite fundamental group and $n\ge 2$. Thus generically there are at least two closed geodesics on compact Finsler manifolds with…

We prove that for every $\Q$-homological Finsler 3-sphere $(M,F)$ with a bumpy and irreversible metric $F$, either there exist two non-hyperbolic prime closed geodesics, or there exist at least three prime closed geodesics.

In this paper, we prove that for every Finsler $n$-dimensional sphere $(S^{n},F)$ with reversibility $\lm$ and flag curvature $K$ satisfying $\left(\frac{\lm}{1+\lm}\right)^2<K\le 1$, either there exist infinitely many closed geodesics, or…

Let $M$ be a compact simply connected manifold satisfying $H^*(M;\mathbf{Q})\cong T_{d,n+1}(x)$ for integers $d\ge 2$ and $n\ge 1$. If all prime closed geodesics on $(M,F)$ with an irreversible bumpy Finsler metric $F$ are elliptic, either…

Let $M=S^{2n+1}/ \Gamma$, $\Gamma$ 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…

For non-reversible Finsler metrics of positive flag curvature on spheres and projective spaces we present results about the number and the length of closed geodesics and about their stability properties.

We study non-reversible Finsler metrics with constant flag curvature 1 on S^2 and show that the geodesic flow of every such metric is conjugate to that of one of Katok's examples, which form a 1-parameter family. In particular, the length…

If all prime closed geodesics on $(S^n,F)$ with an irreversible Finsler metric $F$ are irrationally elliptic, there exist either exactly $2\left[\frac{n+1}{2}\right]$ or infinitely many distinct closed geodesics. As an application, we show…

In this paper, we prove that for every Finsler $n$-sphere $(S^n, F)$ for $n\ge 3$ with reversibility $\lambda$ and flag curvature $K$ satisfying $(\frac{\lambda}{\lambda+1})^2<K\le 1$, either there exist infinitely many prime closed…

In this paper, we prove that for every Finsler $4$-dimensional sphere $(S^4,F)$ with reversibility $\lambda$ and flag curvature $K$ satisfying $\frac{25}{9}\left(\frac{\lambda}{1+\lambda}\right)^2<K\le 1$ with $\lambda<\frac{3}{2}$, either…

In this paper, we prove that for every irreversible Finsler $n$-dimensional real projective space $(\mathbb{R}P^n,F)$ with reversibility $\lambda$ and flag curvature $K$ satisfying $\frac{16}{9}\left(\frac{\lambda}{1+\lambda}\right)^2<K\le…

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…

We give a lower bound for the length of a non-trivial geodesic loop on a simply-connected and compact manifold of even dimension with a non-reversible Finsler metric of positive flag curvature. Harris and Paternain use this estimate in…