The closed geodesics conjecture for compact rank-one symmetric spaces: does every Riemannian metric on $S^n$ ($n\ge3$), $\mathbb{CP}^m$, $\mathbb{HP}^m$, or the Cayley plane carry infinitely many geometrically distinct closed geodesics?

After the theorems of Gromoll–Meyer (1969) and Vigué-Poirrier–Sullivan (1976), the classical closed geodesics problem for simply connected closed Riemannian manifolds survives only for manifolds whose rational cohomology ring is that of a compact rank-one symmetric space: spheres $S^n$ with $n\ge3$, complex projective spaces $\mathbb{CP}^m$, quaternionic projective spaces $\mathbb{HP}^m$, and the Cayley projective plane. The conjecture is that every Riemannian metric on such a manifold admits infinitely many geometrically distinct closed geodesics. Only $S^2$ is settled (Bangert, Franks, Hingston, 1992–1993), and the generic-metric case is settled affirmatively on all these spaces; the conjecture for arbitrary metrics has been open since the 1960s–70s and remains open as of August 2026.

Background

A closed geodesic on a Riemannian manifold $(M,g)$ is a periodic orbit of the geodesic flow, i.e. a geodesic $c:S^1\to M$ that closes up; it is a critical point of the energy functional on the free loop space $\Lambda M$ of all closed curves. Because the $O(2)$-action reparametrizing and reversing loops leaves the energy invariant, and because every iterate $c^m(t)=c(mt)$ of a closed geodesic is again a closed geodesic with the same image, one counts closed geodesics up to reparametrization and iteration: two closed geodesics are geometrically distinct when their traces (images in $M$) differ. The classical closed geodesics problem, going back to Hadamard, Poincaré, Birkhoff, and Morse, asks how many geometrically distinct closed geodesics a closed Riemannian manifold must carry; every closed manifold carries at least one, and the standing conjecture is that a closed manifold of dimension at least two always carries infinitely many. The variational approach studies the Morse theory of the energy functional on $\Lambda M$: if the Betti numbers of $\Lambda M$ are unbounded, Morse theory forces infinitely many critical points, hence infinitely many geometrically distinct closed geodesics (Gromoll–Meyer, 1969). Vigué-Poirrier and Sullivan (1976) computed the loop-space homology and showed these Betti numbers are unbounded whenever the rational cohomology ring of the simply connected manifold needs at least two algebra generators (a non-monogenic ring). This pair of theorems settles the problem for every simply connected closed Riemannian manifold except those whose rational cohomology ring is generated by a single element of positive degree. The surviving open cases take their name from the central family, the compact rank-one symmetric spaces (CROSSes) — the simply connected symmetric spaces on which, for the standard metric, all geodesics are closed: the spheres $S^n$ ($n\ge3$), the complex projective spaces $\mathbb{CP}^m$ ($m\ge2$, since $\mathbb{CP}^1\cong S^2$ is itself a sphere), the quaternionic projective spaces $\mathbb{HP}^m$ ($m\ge1$, note $\mathbb{HP}^1\cong S^4$), and the Cayley projective plane $\mathbb{OP}^2$; the cohomology ring of each of these spaces is generated by a single element, and the question concerns arbitrary metrics on manifolds carrying such rings. The only fully solved case is $S^2$: Bangert (1993) proved that a Riemannian two-sphere whose Birkhoff annulus map (a Poincaré return map on the annulus of geodesics connecting two chosen points) is undefined has infinitely many closed geodesics, and Franks (1992) proved that every area-preserving annulus homeomorphism has infinitely many periodic points; together these give infinitely many geometrically distinct closed geodesics for every Riemannian metric on $S^2$, and Hingston (1993) sharpened this to prime-number-type lower bounds on the growth of the counting function $N(t)$, the number of geometrically distinct closed geodesics of length at most $t$. For generic metrics the problem is also settled on all CROSSes: a metric is called bumpy when all its closed geodesics are non-degenerate critical points, and Hingston (1984), using equivariant Morse theory — Morse theory adapted to the $O(2)$ reparametrization symmetry of the loop space — on the homotopy quotient of the free loop space, proved that a generic Riemannian metric on a simply connected closed manifold with the rational homotopy type of a rank-one symmetric space carries infinitely many closed geodesics; Rademacher (1989) completed the $C^4$-generic case through the theory of average Morse indices (the mean growth rate of the Morse index — the number of negative directions of the second variation of the energy — along the iterates of a closed geodesic), and Rademacher (1994) with Contreras (2010) strengthened the conclusion to exponential growth of the number of closed geodesics of bounded length for $C^2$-generic metrics, while Rademacher (2024) showed that for a generic metric on a compact simply connected manifold of dimension at least three all prime closed geodesics (closed geodesics that are not iterates of shorter ones) are even simple (embedded) and pairwise disjoint. None of these generic techniques applies to an arbitrary metric, and the $S^2$ proof relies on surface-specific tools — the curve shortening flow (a deformation that moves each point of a curve in the direction of its curvature normal), the Birkhoff annulus map, and one-dimensional dynamical arguments — with no known higher-dimensional analogue. De Philippis–Marini–Mazzucchelli–Suhr (2022) extended the classical $S^2$ theorems — the Lusternik–Schnirelmann theorem on three simple closed geodesics (closed geodesics whose traces are embedded circles) and the Bangert–Franks–Hingston theorem on infinitely many — to reversible Finsler metrics (a Finsler metric generalizes a Riemannian metric by allowing the norm on each tangent space to be non-Euclidean; reversible means the norm of a vector equals that of its opposite); the CROSS conjecture for Riemannian metrics remains open. The Finsler side also shows why reversibility matters: Katok's 1973 examples of non-reversible Finsler metrics on $S^2$, analyzed by Ziller (1983), carry only finitely many (in fact two) geometrically distinct closed geodesics, but no such phenomenon is known for Riemannian metrics, which are automatically reversible. On the non-simply-connected side of the problem, Bangert–Hingston (1984) settled manifolds with infinite abelian fundamental group, and recent work of Contreras–Mazzucchelli (2024) handles generic metrics on manifolds with non-trivial first Betti number (nonzero rank of the first homology group); these results leave the simply connected CROSS core untouched.

Problem Statement

Prove or refute the following conjecture. Let $M$ be a simply connected closed smooth manifold whose rational cohomology ring is that of a compact rank-one symmetric space — concretely, $M$ carries the rational cohomology ring of $S^n$ with $n\ge3$ (any smooth structure on the topological sphere, exotic spheres included), of $\mathbb{CP}^m$ with $m\ge2$ (since $\mathbb{CP}^1\cong S^2$ is the settled case), of $\mathbb{HP}^m$ with $m\ge1$ (note $\mathbb{HP}^1\cong S^4$), or of the Cayley projective plane $\mathbb{OP}^2$. These four families are the surviving open cases of the Gromoll–Meyer and Vigué-Poirrier–Sullivan dichotomy: those theorems settle all simply connected closed manifolds whose rational cohomology ring is generated by at least two algebra generators, while the cohomology ring of each of these four families is generated by a single element of positive degree. Does every smooth Riemannian metric $g$ on $M$ admit infinitely many geometrically distinct closed geodesics — that is, must the set of traces of closed geodesics of $(M,g)$ be infinite, where iterates $c^m(t)=c(mt)$ and reparametrizations of a closed geodesic are not counted separately? A positive answer for all four families, or an explicit Riemannian metric on one of these spaces with only finitely many geometrically distinct closed geodesics, resolves the question.

References

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