Infinitely many closed geodesics on every compact Riemannian and reversible Finsler manifold (Long's Conjectures 1 and 3)

The problem is Conjectures 1 and 3 of Yiming Long's 2010 survey chapter 'Recent progress on closed geodesics in some compact simply connected manifolds': does every closed manifold carry infinitely many geometrically distinct closed geodesics for every Riemannian metric, and indeed for every reversible Finsler metric? Dimension two is completely settled. After the theorems of Gromoll-Meyer and Vigué-Poirrier-Sullivan (non-CROSS simply connected manifolds), Bangert-Hingston (infinite abelian fundamental group), Franks-Bangert-Hingston (Riemannian $S^2$), and De Philippis-Marini-Mazzucchelli-Suhr (reversible Finsler $S^2$), the conjecture remains open precisely for closed manifolds of dimension at least 3 whose universal cover has the rational cohomology of a compact rank-one symmetric space other than $S^2$: the spheres $S^n$ with $n\ge3$, the projective spaces $\mathbb{CP}^n$ and $\mathbb{HP}^n$, the Cayley plane, and their finite quotients. The problem remains open for this core.

Background

A closed geodesic on a Riemannian manifold $(M,g)$ is a geodesic that closes up smoothly, $c(t+T)=c(t)$ and $\dot c(t+T)=\dot c(t)$ for some period $T>0$; closed geodesics are the periodic orbits of the geodesic flow and, equivalently, the critical points of the energy functional on the free loop space $\Lambda M$ of smooth maps $S^1\to M$. Two closed geodesics are geometrically distinct if their images in the manifold differ, $c(S^1)\neq d(S^1)$ (equivalently, if neither is a reparametrization nor an iterate of the other), and a closed geodesic is prime if it is not an iterate of a shorter one. Since Poincaré, Birkhoff and Morse, the central multiplicity question has been: how many geometrically distinct closed geodesics must a compact manifold carry? Lyusternik and Fet proved in 1951 that every closed manifold carries at least one.

The infinitude question is answered for 'most' manifolds by a classical homological argument. Gromoll and Meyer (1969) showed that if the Betti numbers (dimensions of homology groups) of the free loop space $\Lambda M$, with coefficients in some field, are unbounded, then $M$ carries infinitely many geometrically distinct closed geodesics. Vigué-Poirrier and Sullivan (1976) proved that these Betti numbers are unbounded whenever the rational cohomology ring of a simply connected $M$ needs at least two algebra generators. The method therefore fails exactly on closed simply connected manifolds whose rational cohomology is a truncated polynomial ring on a single generator, i.e. those with the cohomology of a compact rank-one symmetric space (CROSS): the sphere $S^n$, complex projective space $\mathbb{CP}^n$, quaternionic projective space $\mathbb{HP}^n$, and the Cayley plane $\mathbb{CaP}^2$. For these spaces the free-loop-space Betti numbers are bounded, so finitely many (degenerate) critical orbits can in principle carry all of the homology, and the problem survives.

In dimension two the problem is completely understood: Lusternik and Schnirelmann (1929) proved that every Riemannian 2-sphere has three simple (embedded) closed geodesics, and the combined work of Franks (1992), Bangert (1993) and Hingston (1993) gives infinitely many geometrically distinct closed geodesics on every Riemannian $S^2$; for compact manifolds with infinite abelian fundamental group, Bangert and Hingston (1984) obtained infinitely many closed geodesics growing at least like the prime numbers. The Finsler category adds a twist: a Finsler metric is a continuous function $F:TM\to[0,\infty)$, smooth off the zero section, positively homogeneous of degree one on each tangent fiber, with positive-definite Hessian of $F^2$ off the origin; it is called reversible when $F(x,-v)=F(x,v)$. Katok (1973) constructed (irreversible) Finsler metrics on spheres with only finitely many closed geodesics, so infinitude can only be conjectured in the reversible (in particular Riemannian) category. De Philippis, Marini, Mazzucchelli and Suhr (2022) extended the full two-sphere theory: every reversible Finsler 2-sphere has three simple closed geodesics and infinitely many closed geodesics.

Yiming Long's 2010 survey chapter organizes the higher-dimensional landscape into a family of conjectures: (1) every compact manifold carries infinitely many closed geodesics for every Riemannian metric; (2) for every dimension $n$ there exist integers $0<p_n<q_n<\infty$ such that the number of prime closed geodesics of any compact $n$-manifold with any irreversible Finsler metric lies in $[p_n,q_n]\cup\{+\infty\}$; (3) the infinitude statement of (1) holds for every reversible Finsler metric on any compact manifold; (4) the existence of a single hyperbolic prime closed geodesic (hyperbolic meaning no eigenvalue of modulus one for the linearized Poincaré return map) forces infinitely many prime closed geodesics; (5) if only finitely many prime closed geodesics exist, all of them must be irrationally elliptic (non-hyperbolic with rotation angles rationally independent of $\pi$). The survey reports the then-new multiplicity results on the exceptional CROSS-type manifolds: a second closed geodesic on every Finsler sphere of dimension greater than two (Rademacher 2010) and on the complex projective plane (Rademacher 2008), and two closed geodesics on every Finsler 2-sphere (Bangert-Long 2010).

Subsequent work has strengthened multiplicity on the exceptional class — two closed geodesics on every compact simply connected bumpy (all closed geodesics nondegenerate) Finsler manifold (Duan-Long-Wang 2016; Wang 2020), second-geodesic and connected-sum results for 3-manifolds and fundamental groups (Rademacher-Taimanov 2022), two short closed geodesics on odd-dimensional spheres (Rademacher 2023), and sharp lower-bound counts on positively curved Finsler spheres (Duan-Xie 2023-2026; Duan-Qi 2025) — and has settled generic statements such as infinitude of closed geodesics for $C^\infty$-generic Riemannian metrics on manifolds with nonzero first Betti number (Contreras-Mazzucchelli 2025). None of these reaches infinitude for an arbitrary metric on any CROSS-type manifold of dimension at least 3, which is the surviving core of the classical closed geodesics problem.

Problem Statement

Prove, or refute, Long's Conjecture 1: every closed manifold carries infinitely many geometrically distinct closed geodesics for every Riemannian metric — and its Finsler strengthening, Conjecture 3: the same conclusion holds for every reversible Finsler metric (a Finsler metric $F$ with $F(x,-v)=F(x,v)$ for all $(x,v)$) on every compact manifold (a positive answer to Conjecture 3 implies Conjecture 1, since Riemannian metrics are reversible Finsler metrics). Dimension two and all non-CROSS simply connected cases being settled (see Background), a resolution must in particular settle the surviving core in dimensions at least 3: determine whether every closed manifold whose universal cover has the rational cohomology of a compact rank-one symmetric space other than $S^2$ — the spheres $S^n$ with $n\ge3$, the projective spaces $\mathbb{CP}^n$ and $\mathbb{HP}^n$ with $n\ge2$, the Cayley plane $\mathbb{CaP}^2$, and their finite quotients such as $\mathbb{RP}^{2n}$ and lens-type spaces — carries infinitely many geometrically distinct closed geodesics for every Riemannian metric, respectively for every reversible Finsler metric. No genericity, nondegeneracy (bumpy), curvature, or reversibility weakening of the metric class may be assumed: a proof must cover all metrics, and a refutation consists of a single explicit reversible Finsler (or Riemannian) metric on such a manifold with only finitely many prime closed geodesics; such a counterexample refutes Conjecture 3, and refutes Conjecture 1 as well exactly when the metric is Riemannian, so a non-Riemannian reversible Finsler counterexample would leave the Riemannian Conjecture 1 open. The source's companion Conjectures 2, 4 and 5 are related open problems and are not part of this problem's completion boundary.

References

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