The closed geodesics conjecture: infinitely many geometrically distinct closed geodesics on every closed Riemannian manifold

The long-standing classical conjecture, stated in the introduction of Duan–Long–Wang, 'Two closed geodesics on compact simply connected bumpy Finsler manifolds' (J. Differential Geom. 104, 2016), is that every compact Riemannian manifold carries infinitely many distinct closed geodesics, posed together with the contrasting Finsler question of when a Finsler metric admits finitely versus infinitely many closed geodesics (Katok's irreversible Finsler spheres have only finitely many). The conjecture is open. For arbitrary Riemannian metrics it is settled on surfaces (including every 2-sphere, by Bangert–Franks and Hingston), on simply connected manifolds whose rational cohomology is not singly generated (Gromoll–Meyer with Vigué-Poirrier–Sullivan), and for infinite abelian or solvable fundamental group (Bangert–Hingston, Taimanov); for generic metrics it is known far more broadly (Hingston–Rademacher on rank-one-type manifolds, Contreras–Mazzucchelli when the first Betti number is positive). The surviving open core is arbitrary — not generic or bumpy — metrics on closed manifolds of dimension at least three whose rational cohomology is that of a compact rank-one symmetric space but which are not simply connected (the finite-fundamental-group space forms with such cohomology, notably lens spaces and odd-dimensional real projective spaces), together with the general infinite non-abelian fundamental group case; the simply connected cases of this class — every Riemannian metric on $S^n$ for $n\ge2$, on $\mathbb{CP}^n$, on $\mathbb{HP}^n$, and on the Cayley plane — were announced resolved in September–October 2026 (OpenAI 2026; Chen–Deng–Zhang 2026). On the Finsler side, reversible metrics on the 2-sphere were recently shown to have infinitely many closed geodesics, and every Finsler 2-sphere has two or infinitely many; infinitude for reversible Finsler metrics beyond surfaces and the irreversible finite/infinite classification remain open.

Background

A geodesic on a Riemannian manifold $(M,g)$ is a curve that locally minimizes length, and a closed geodesic is a geodesic $\gamma:\mathbb{R}\to M$ that closes up, $\gamma(t+\tau)=\gamma(t)$ for some period $\tau>0$; it is prime if $\tau$ is its minimal period, its $m$-th iterate $\gamma^m(t)=\gamma(mt)$ is again a closed geodesic, and two closed geodesics are called geometrically distinct if their images in $M$ are different subsets. All multiplicity statements in this area count geometrically distinct closed geodesics. Since closed geodesics are the periodic orbits of the geodesic flow — the paradigmatic Hamiltonian system of classical mechanics — the question of how many closed geodesics a compact manifold must carry is simultaneously a problem of global differential geometry and of dynamics. Lyusternik and Fet proved in 1951 that every compact Riemannian manifold has at least one closed geodesic, by minimizing the energy functional $E(\gamma)=\frac{1}{2}\int_{S^{1}}\|\dot{\gamma}\|^{2}\,dt$ over the free loop space $\Lambda M$ of $H^{1}$-maps $S^{1}\to M$; the critical points of $E$ of positive energy are precisely the closed geodesics, so $\Lambda M$ carries a Morse theory whose homology controls the count. Gromoll and Meyer (1969) proved that if the Betti numbers of $\Lambda M$ are unbounded then $M$ has infinitely many closed geodesics, and Vigué-Poirrier and Sullivan (1976) showed that for simply connected $M$ the Betti numbers of $\Lambda M$ are unbounded exactly when the rational cohomology algebra $H^{*}(M;\mathbb{Q})$ is not a truncated polynomial algebra on one generator. The conjecture therefore holds for every simply connected closed manifold except those whose rational cohomology is that of a compact rank-one symmetric space (CROSS) — the sphere $S^{n}$, the complex and quaternionic projective spaces $\mathbb{CP}^{n}$ and $\mathbb{HP}^{n}$, and the Cayley projective plane $\mathrm{CaP}^{2}$ — precisely the manifolds for which the Betti numbers of $\Lambda M$ are bounded and Morse-theoretic counting produces nothing.

All surfaces are settled: for genus at least one the fundamental group forces infinitely many closed geodesics, while the 2-sphere, on which every loop is contractible, was the hardest case and was completed only in the early 1990s — Franks (1992) via the periodic-point theory of annulus homeomorphisms, combined with Bangert (1993), or alternatively Hingston (1993) by equivariant Morse theory, proved that every Riemannian metric on $S^{2}$ carries infinitely many closed geodesics, and Hingston further showed that the number of closed geodesics of length at most $\ell$ grows at least like $c\,\ell/\log\ell$. For non-simply-connected manifolds, Bangert and Hingston (1984) settled all infinite abelian fundamental groups, Taimanov (1985) extended this to larger classes of groups including all infinite solvable ones, Ballmann–Thorbergsson–Ziller (1981) and Rademacher–Taimanov (2022) treated further classes including connected sums (where the number of closed geodesics of length at most $t$ grows at least like the prime numbers on compact 3-manifolds with infinite fundamental group), and Shelukhin and Zhang (2023) record that infinitely many conjugacy classes in the fundamental group force infinitely many closed geodesics for every Riemannian metric.

For generic metrics much more is known. A Finsler or Riemannian metric is called bumpy if every closed geodesic is nondegenerate, meaning the linearized Poincaré map — the derivative of the flow transverse to the orbit after one period — has no eigenvalue equal to 1; bumpy metrics are generic in the $C^{k}$ topology. Hingston (1984), with a different proof by Rademacher (1989), proved that a $C^{4}$-generic Riemannian metric on any simply connected CROSS-cohomology manifold has infinitely many closed geodesics, and Contreras and Mazzucchelli (2024) proved that a $C^{\infty}$-generic metric on any closed manifold with nonzero first Betti number (i.e. with $H^{1}(M;\mathbb{R})\neq0$) admits closed geodesics of arbitrarily large length. None of these generic statements covers every metric, and the open problem concerns arbitrary, possibly highly degenerate metrics.

The conjecture is Riemannian in an essential way, which is what the source paper's contrast makes precise. A Finsler metric on $M$ is a function $F:TM\to[0,\infty)$, smooth away from the zero section, with $F(x,\lambda y)=\lambda F(x,y)$ for $\lambda>0$ and with the Hessian of $F^{2}$ in $y$ positive definite; it is reversible if $F(x,-y)=F(x,y)$, and it is Riemannian exactly when $F(x,y)^{2}$ is quadratic in $y$. The variational theory of closed geodesics carries over to Finsler metrics, but the multiplicity changes drastically: Katok (1973; see Ziller 1983) constructed irreversible Finsler metrics on spheres and other CROSSs with only finitely many prime closed geodesics — exactly $2\lfloor (n+1)/2\rfloor$ on $S^{n}$ — all of them nondegenerate, so in the Finsler category even bumpy metrics need not have infinitely many closed geodesics. The source, Duan–Long–Wang (2016), works exactly at this frontier: it proves that every compact simply connected manifold with a bumpy irreversible Finsler metric carries at least two prime closed geodesics, and its introduction states the classical Riemannian infinitude conjecture as the outstanding open problem, with the finite-versus-infinite classification for Finsler metrics as the contrasting open area.

Problem Statement

Prove or refute the classical closed geodesics conjecture: every closed (compact, without boundary) Riemannian manifold $M$ of dimension at least two carries infinitely many geometrically distinct closed geodesics — equivalently, for every such $M$ and every Riemannian metric on it, the number of geometrically distinct closed geodesics of length at most $t$ tends to infinity as $t\to\infty$.

In light of the cases already settled (surfaces; simply connected manifolds whose rational cohomology is not singly generated; infinite abelian and solvable fundamental groups; the connected-sum and conjugacy-class classes), the conjecture remains open for arbitrary — not generic, not bumpy — Riemannian metrics on:

1. closed manifolds of dimension at least three with finite nontrivial fundamental group whose rational cohomology ring is that of a compact rank-one symmetric space, notably the finite-fundamental-group space forms with such cohomology (lens spaces, odd-dimensional real projective spaces); the simply connected cases of this cohomological class — arbitrary metrics on $S^{n}$ for $n\ge2$, on $\mathbb{CP}^{n}$, on $\mathbb{HP}^{n}$, and on the Cayley plane $\mathrm{CaP}^{2}$ — were announced resolved in September–October 2026 (OpenAI 2026; Chen–Deng–Zhang 2026); and
2. the still-unsettled classes of closed manifolds with infinite non-abelian fundamental group, beyond the abelian, solvable, connected-sum and conjugacy-class classes already treated.

A complete negative answer is a single explicit closed Riemannian manifold of dimension at least two with only finitely many geometrically distinct closed geodesics.

The source states this conjecture together with a companion open area: the classification, for Finsler metrics, of finiteness versus infinitude of closed geodesics (Katok's irreversible examples show that finiteness is possible there). That companion is background only and is not part of this task's completion boundary, since the source formulates it as an open area rather than as a single bounded question.

References

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