Bumpy Finsler Research Articles

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

Let M be a compact simply connected manifold satisfying H⁎(M;Q)≅Td,n+1(x) for integers d≥2 and n≥1. If all prime closed geodesics on (M,F) with an irreversible bumpy Finsler metric F are elliptic, then either there exist exactly dn(n+1)2 (when d≥2 is even) or (d+1) (when d≥3 is odd) distinct closed geodesics, or there exist infinitely many distinct closed geodesics.

Two closed geodesics on compact bumpy Finsler manifolds

In this paper, we prove there are at least two closed geodesics on any compact bumpy Finsler $n$-manifold with $n\ge 2$. Thus generically there are at least two closed geodesics on compact Finsler manifolds. Furthermore, there are at least two closed geodesics on any compact Finsler $2$-manifold, and this lower bound is achieved by the Katok 2-sphere, cf. \cite{Kat}.

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 $n≥2$, $Γ$ 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 prove the existence of at least two distinct non-contractible closed geodesics of the class $[h]$ on $(M, F)$ with a bumpy Finsler metric, which improves a result of Taimanov in [39] by removing some additional conditions. Also our results extend the resonance identity and multiplicity results on $\mathbb{R}P^n$ in [25] to general compact space forms.

Resonance identity and multiplicity of non-contractible closed geodesics on Finsler [formula omitted

In this paper, we establish first the resonance identity for non-contractible homologically visible prime closed geodesics on Finsler n-dimensional real projective space (RPn,F) when there exist only finitely many distinct non-contractible closed geodesics on (RPn,F), where the integer n≥2. Then as an application of this resonance identity, we prove the existence of at least two distinct non-contractible closed geodesics on RPn with a bumpy and irreversible Finsler metric. Together with two previous results on bumpy and reversible Finsler metrics in [14] and [39], it yields that every RPn with a bumpy Finsler metric possesses at least two distinct non-contractible closed geodesics.

Multiplicity of periodic orbits for dynamically convex contact forms

We give a sharp lower bound for the number of geometrically distinct contractible periodic orbits of dynamically convex Reeb flows on prequantizations of symplectic manifolds that are not aspherical. Several consequences of this result are obtained, like a new proof that every bumpy Finsler metric on $$S^n$$ carries at least two prime closed geodesics, multiplicity of elliptic and non-hyperbolic periodic orbits for dynamically convex contact forms with finitely many geometrically distinct contractible closed orbits and precise estimates of the number of even periodic orbits of perfect contact forms. We also slightly relax the hypothesis of dynamical convexity. A fundamental ingredient in our proofs is the common index jump theorem due to Y. Long and C. Zhu.

Bumpy metrics on spheres and minimal index growth

The existence of two geometrically distinct closed geodesics on an n-dimensional sphere \(S^n\) with a non-reversible and bumpy Finsler metric was shown independently by Duan and Long [7] and the author [25]. We simplify the proof of this statement by the following observation: If for some \(N \in \mathbb {N}\) all closed geodesics of index \(\le \)N of a non-reversible and bumpy Finsler metric on \(S^n\) are geometrically equivalent to the closed geodesic c, then there is a covering \(c^r\) of minimal index growth, i.e., $$\begin{aligned} \mathrm{ind}(c^\mathrm{rm})=m \,\mathrm{ind}(c^r)-(m-1)(n-1), \end{aligned}$$ for all \(m \ge 1\) with \(\mathrm{ind}\left( c^\mathrm{rm}\right) \le N.\) But this leads to a contradiction for \(N =\infty \) as pointed out by Goresky and Hingston [13]. We also discuss perturbations of Katok metrics on spheres of even dimension carrying only finitely many closed geodesics. For arbitrarily large \(L>0\), we obtain on \(S^2\) a metric of positive flag curvature carrying only two closed geodesics of length \(<L\) which do not intersect.

Two closed geodesics on compact simply connected bumpy Finsler manifolds

We prove the existence of at least two distinct closed geodesics on a compact simply connected manifold $M$ with a bumpy and irreversible Finsler metric, when $H^* (M; \mathbf{Q}) \cong T_{d,h+1} (x)$ for some integer $h \geq 2$ and even integer $d \geq 2$. Consequently, together with earlier results on $S^n$, it implies the existence of at least two distinct closed geodesics on every compact simply connected manifold $M$ with a bumpy irreversible Finsler metric.

Multiplicity of closed geodesics on Finsler spheres with irrationally elliptic closed geodesics

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 the existence of three distinct closed geodesics on bumpy Finsler (S 3, F) if any prime closed geodesic has non-zero Morse index.

Two closed geodesics on $${\mathbb {R}}P^{2n+1}$$ R P 2 n + 1 with a bumpy Finsler metric

In this paper we show the existence of at least two distinct non-contractible closed geodesics on $${\mathbb {R}}P^3$$ endowed with a bumpy and irreversible Finsler metric. If the bumpy metric F is reversible or Riemannian, there exist at least two geometrically distinct non-contractible closed geodesics on $${\mathbb {R}}P^{2n+1}$$ .

Non-hyperbolic closed geodesics on Finsler spheres

In this paper, we prove that for every bumpy Finsler $2k$-sphere $(S^{2k}, F)$ with reversibility $\lambda$ and flag curvature $K$ satisfying the pinching condition $(\frac{\lambda}{\lambda + 1}) \lt K \leq 1$ either there exist infinitely many closed geodesics or there exist at least $2k$ non-hyperbolic closed geodesics. Due to the example of A. B. Katok, this estimate is sharp.

On a conjecture of Anosov

In this paper, we prove that for every bumpy Finsler n-sphere (Sn,F) with reversibility λ and flag curvature K satisfying (λλ+1)2<K≤1, there exist 2[n+12] prime closed geodesics. This gives a confirmed answer to a conjecture of Anosov [1] in 1974 for a generic case.

Multiplicity and stability of closed geodesics on bumpy Finsler 3-spheres

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.

Multiple closed geodesics on bumpy Finsler n-spheres

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