Infinitely many closed geodesics on every Riemannian sphere and every closed three-manifold

Theorem 1.1: every smooth Riemannian metric on $S^n$ , $n\ge2$ , has infinitely many prime closed geodesics with pairwise distinct images. Corollary 1.2: the same for every closed manifold finitely covered by a sphere (real projective spaces, spherical space forms). Corollary 1.3: the same for every closed three-manifold, orientable or not, using Rademacher-Taimanov and Perelman. It does NOT settle the problem for general closed manifolds in dimension four or more, and it gives no growth rate or length count for the geodesics.

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Prior state unknown→proved

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Infinitely many closed geodesics on every Riemannian sphere and every closed three-manifold · Closed geodesics on spheres

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VibeMathed record: Infinitely many closed geodesics on every Riemannian sphere and every closed three-manifold

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollaries 1.2 and 1.3 of the TeX source read against the infinitude problem as the introduction states it. The theorem covers every smooth metric on $S^n$ , $n\ge2$ , degenerate ones included; the two-sphere case is classical and the new content is $n\ge3$ . The three-manifold corollary is conditional on published inputs: Rademacher-Taimanov (infinite fundamental group) and Perelman's elliptization. The proof was not refereed. No Lean formalization exists for this family (no lean/docs/345.md and no matching Comparator challenge at the pinned commit). The general problem for all closed manifolds stays open; the paper disputes two steps of Charles's 2019 claimed general proof.

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Infinitely many closed geodesics on every Riemannian sphere and every closed three-manifold

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