Infinitely many closed geodesics on closed Riemannian manifolds: the surviving compact rank-one symmetric space cohomology and generic non-abelian fundamental group regimes

Whether every closed Riemannian manifold of dimension at least two carries infinitely many geometrically distinct closed geodesics is a classical open conjecture at least half a century old, with roots in the work of Gromoll–Meyer and Vigué-Poirrier–Sullivan. Contreras and Mazzucchelli (2024; published 2025) record that the conjecture survives in two topological regimes: (i) closed manifolds of dimension at least three whose rational cohomology ring is that of a compact rank-one symmetric space ( $S^n$ , $\mathbb{C}P^n$ , $\mathbb{H}P^n$ , $\mathrm{Ca}P^2$ ), where even the every-metric statement is open; and (ii) closed manifolds with infinite non-abelian fundamental group, where even $C^4$ -generic existence of infinitely many closed geodesics is open. The problem asks for a proof or refutation of the conjecture in these two regimes. A 2026 preprint of Ndombol claiming a 'complete solution' of the closed geodesics problem covers, even if its algebraic proof is correct, only manifolds whose cohomology over some field requires at least two algebra generators, and therefore does not touch the core of regime (i) — spheres and projective spaces with singly generated cohomology.

Background

A geodesic on a Riemannian manifold $(M,g)$ is a curve satisfying the geodesic equation $\nabla_t\dot\gamma\equiv0$ ; a closed geodesic is one that returns to its initial point and velocity after some positive minimal period, and two closed geodesics are geometrically distinct if neither is a reparametrization of the other. By the Lyusternik–Fet theorem, every closed (compact, without boundary) Riemannian manifold of dimension at least two has at least one closed geodesic; the classical conjecture asserts that in fact infinitely many geometrically distinct ones always exist.

The only general mechanism known for proving this is topological. Writing $LM=\mathrm{Map}(S^1,M)$ for the free loop space of $M$ (all continuous loops on $M$ , with the compact-open topology), Gromoll and Meyer (1969) showed that if the Betti numbers of $LM$ over some field (the dimensions of its homology groups) are unbounded, then every Riemannian metric on $M$ has infinitely many closed geodesics. Vigué-Poirrier and Sullivan (1976) proved that for simply connected closed $M$ , these Betti numbers over $\mathbb{Q}$ are unbounded whenever the rational cohomology ring $H^*(M;\mathbb{Q})$ requires at least two generators as an algebra. Together this settles the conjecture for all simply connected manifolds whose rational cohomology ring needs at least two generators; the surviving simply connected cases are precisely those whose rational cohomology is singly generated. The class singled out in the source consists of manifolds whose rational cohomology is that of a compact rank-one symmetric space (CROSS): the sphere $S^n$ , complex projective space $\mathbb{C}P^n$ , quaternionic projective space $\mathbb{H}P^n$ , or the Cayley plane $\mathrm{Ca}P^2$ . (Singly generated rational cohomology is not known to force CROSS type: 'rational projective spaces' with rational cohomology $\mathbb{Q}[x]/(x^{n+1})$ in a generator degree other than $2$ , $4$ , or $8$ may exist — which parameter pairs are realizable by closed manifolds is reduced, via the Barge–Sullivan rational realization theorem, to an open number-theoretic problem (DeVito–Kennard 2020) — and such manifolds would be open for the conjecture just like the CROSS-rational ones.) For singly generated rational cohomology the free-loop Betti numbers over $\mathbb{Q}$ are bounded, so the Gromoll–Meyer mechanism fails rationally on this class and no replacement mechanism is known; for CROSS-rational manifolds with integral torsion, where the mod- $p$ cohomology can acquire a second generator, whether the free-loop Betti numbers over $\mathbb{F}_p$ become unbounded is itself part of the open problem (see the algebraic-topology program below).

For non-simply-connected manifolds, the conjecture was proved for infinite abelian fundamental group by Bangert and Hingston (1984) and extended by Taimanov (1985) to larger classes of closed manifolds including those with infinite solvable fundamental group; for surfaces, in particular the 2-sphere, it follows from Bangert (1993) and Franks (1992). There is also a parallel generic version of the question: a property of Riemannian metrics is $C^k$ -generic if it holds for a residual (dense $G_\delta$ ) subset of the space of $C^k$ metrics. Hingston (1984), reproved by Rademacher (1989), showed that a $C^4$ -generic metric on any simply connected closed manifold with CROSS rational cohomology has infinitely many closed geodesics, while for infinite non-abelian fundamental groups, $C^4$ -generic existence is known only for special classes of groups or manifolds (Ballmann–Thorbergsson–Ziller 1981; Rademacher–Taimanov 2022; Shelukhin–Zhang 2023).

Contreras and Mazzucchelli (2024; Proc. London Math. Soc. 2025) pushed the generic frontier forward: on any closed manifold of dimension at least two with non-zero first Betti number $b_1=\dim H^1(M;\mathbb{R})$ , a $C^\infty$ -generic metric has infinitely many closed geodesics of arbitrarily large length. Their proof combines Mañé's ideas from Aubry–Mather theory — where a closed geodesic is minimal if it minimizes an action built from the kinetic energy minus a closed 1-form representing a non-zero cohomology class — with a new theorem: a minimal closed geodesic forces a transverse homoclinic orbit (an orbit biasymptotic to the closed geodesic's periodic orbit, meeting its stable and unstable manifolds transversally), hence a horseshoe — an invariant set carrying a symbolic shift and therefore infinitely many periodic orbits — after a $C^\infty$ -small perturbation of the metric. The technique needs a non-zero cohomology class to minimize against, so it stops exactly at $b_1=0$ . The paper's introduction records precisely the two regimes that survive, and they form this problem.

In parallel, an algebraic-topology program addresses the underlying 'topological closed geodesics problem': characterize when the Betti numbers of $LX$ over an arbitrary field $k$ are unbounded, conjecturally exactly when $H^*(X;k)$ needs at least two algebra generators. Beyond the rational case, this was proved for $k$ -formal spaces (spaces whose homotopy type is a formal consequence of their cohomology ring; Halperin–Vigué-Poirrier 1991), mod- $p$ elliptic spaces (spaces with finite-dimensional mod- $p$ homotopy and homology; Jones–McCleary 2016), and under an auxiliary algebraic condition (Ndombol–Thomas 2010); a 2026 preprint of Ndombol claims the general case.

Problem Statement

Prove or refute the classical closed geodesics conjecture — every closed Riemannian manifold of dimension at least two carries infinitely many geometrically distinct closed geodesics — in the two regimes that its source (Contreras–Mazzucchelli 2024/2025) explicitly records as surviving:

(i) (Every-metric case, CROSS cohomology.) Determine whether every Riemannian metric on every closed manifold of dimension at least three whose rational cohomology ring is that of a compact rank-one symmetric space ( $S^n$ , $\mathbb{C}P^n$ , $\mathbb{H}P^n$ , $\mathrm{Ca}P^2$ ) has infinitely many geometrically distinct closed geodesics. No restriction on torsion in the integral cohomology, on the homotopy type beyond the rational cohomology ring, or on the metric is permitted.

(ii) (Generic case, infinite non-abelian fundamental group.) Determine whether, for every closed manifold of dimension at least two whose fundamental group is infinite and non-abelian, a $C^4$ -generic Riemannian metric has infinitely many geometrically distinct closed geodesics; an affirmative answer for every metric is stronger and also suffices.

The two regimes form one objective, comprising both (i) and (ii): settling the exceptions that the source names as surviving. Because the two regimes are statements of different kinds, refutations are of different kinds as well: regime (i), an every-metric statement, is refuted by exhibiting an explicit closed manifold of dimension at least three with CROSS rational cohomology together with a smooth Riemannian metric on it having only finitely many geometrically distinct closed geodesics; regime (ii), a genericity statement, is refuted by showing that, for some closed manifold with infinite non-abelian fundamental group, the set of $C^4$ metrics with infinitely many geometrically distinct closed geodesics is not residual in the space of $C^4$ metrics — a single explicit finite-geodesic metric does not by itself establish this, since isolated exceptions are compatible with residual genericity. Since every settled class of the full conjecture is affirmative, an explicit finite-geodesic counterexample to the full conjecture must lie outside all of them: its fundamental group is infinite and non-abelian, or finite (possibly trivial), in which case finiteness lifts to the simply connected universal cover, whose rational cohomology is then necessarily singly generated (a class containing, but not known to coincide with, the CROSS-rational manifolds of regime (i)). The objective comprises regimes (i) and (ii) jointly; a narrower sub-case, such as spheres alone, rational-homology CROSS rings alone, or specific group classes alone, leaves the question open.

References

D. Gromoll and W. Meyer, Periodic geodesics on compact riemannian manifolds, Journal of Differential Geometry 3 (1969), 493–510. DOI: 10.4310/jdg/1214429070, https://doi.org/10.4310/jdg/1214429070
M. Vigué-Poirrier and D. Sullivan, The homology theory of the closed geodesic problem, Journal of Differential Geometry 11 (1976), 633–644. DOI: 10.4310/jdg/1214433729, https://doi.org/10.4310/jdg/1214433729
V. Bangert and N. Hingston, Closed geodesics on manifolds with infinite abelian fundamental group, Journal of Differential Geometry 19 (1984), 277–282. DOI: 10.4310/jdg/1214438679, https://doi.org/10.4310/jdg/1214438679
I. A. Taimanov, Closed geodesics on non-simply-connected manifolds, Russian Mathematical Surveys 40 (1985), no. 6, 143–144 (English translation; Russian original: Uspekhi Mat. Nauk 40 (1985), no. 6(246), 157–158). DOI: 10.1070/rm1985v040n06abeh003719, https://doi.org/10.1070/rm1985v040n06abeh003719
N. Hingston, Equivariant Morse theory and closed geodesics, Journal of Differential Geometry 19 (1984), 85–116. DOI: 10.4310/jdg/1214438424, https://doi.org/10.4310/jdg/1214438424
W. Ballmann, G. Thorbergsson and W. Ziller, Closed geodesics and the fundamental group, Duke Mathematical Journal 48 (1981), 585–588. DOI: 10.1215/s0012-7094-81-04832-8, https://doi.org/10.1215/s0012-7094-81-04832-8
H. Rademacher, On the average indices of closed geodesics, Journal of Differential Geometry 29 (1989), 65–83. DOI: 10.4310/jdg/1214442633, https://doi.org/10.4310/jdg/1214442633
J. Franks, Geodesics on $S^2$ and periodic points of annulus homeomorphisms, Inventiones Mathematicae 108 (1992), 403–418. DOI: 10.1007/BF02100612, https://doi.org/10.1007/BF02100612
V. Bangert, On the existence of closed geodesics on two-spheres, International Journal of Mathematics 4 (1993), 1–10. DOI: 10.1142/S0129167X93000029, https://doi.org/10.1142/S0129167X93000029
H.-B. Rademacher and I. A. Taimanov, Closed geodesics on connected sums and 3-manifolds, Journal of Differential Geometry 120 (2022), 557–573. DOI: 10.4310/jdg/1649953350, https://doi.org/10.4310/jdg/1649953350
E. Shelukhin and J. Zhang, Remark on non-contractible closed geodesics and homotopy groups, arXiv:2307.13877 (2023). https://arxiv.org/abs/2307.13877
G. Contreras and M. Mazzucchelli, Closed geodesics and the first Betti number, Proceedings of the London Mathematical Society (3) 131 (2025), no. 3, Paper No. e70085. DOI: 10.1112/plms.70085; arXiv:2407.02995. https://arxiv.org/abs/2407.02995
S. Halperin and M. Vigué-Poirrier, The homology of a free loop space, Pacific Journal of Mathematics 147 (1991), 311–324. DOI: 10.2140/pjm.1991.147.311, https://doi.org/10.2140/pjm.1991.147.311
J. D. S. Jones and J. McCleary, String homology, and closed geodesics on manifolds which are elliptic spaces, Algebraic & Geometric Topology 16 (2016), 2677–2690. DOI: 10.2140/agt.2016.16.2677, https://doi.org/10.2140/agt.2016.16.2677
B. Ndombol and J.-C. Thomas, A contribution to the closed geodesic problem, Journal of Pure and Applied Algebra 214 (2010), 937–949. DOI: 10.1016/j.jpaa.2009.09.003, https://doi.org/10.1016/j.jpaa.2009.09.003
B. Ndombol and J.-C. Thomas, WITHDRAWN: Massey iterated products and closed geodesics, arXiv:2312.00609 (2023, withdrawn). https://arxiv.org/abs/2312.00609
B. Ndombol, On the closed geodesics problem, arXiv:2602.18926 (2026, preprint, unpublished). https://arxiv.org/abs/2602.18926
J. DeVito and L. Kennard, Cohomogeneity one manifolds with singly generated rational cohomology, Documenta Mathematica 25 (2020), 1835–1863. DOI: 10.4171/dm/786, https://doi.org/10.4171/dm/786 (arXiv:1808.05588, https://arxiv.org/abs/1808.05588)