Preprint

Inserted: 9 nov 2026
Last Updated: 9 nov 2026

Pages: 47
Year: 2026
Doi: 10.5281/zenodo.24068624
Notes:

Version v1, 9 November 2026.

2020 Mathematics Subject Classification: Primary 53C22; Secondary 58E10, 55N91.

Abstract:

Let $M^{2n}$, $n\ge 2$, be a closed connected Riemannian manifold all of whose geodesics are closed, and assume $H^2(M;\mathbb{Q})\ne 0$. We prove that all prime unit-speed geodesics have the same period for $2\le n\le 8$. For every $n\ge 2$, the same conclusion holds if each nontrivial stabilizer of the geodesic circle action has prime-power order. For $n\ge 3$, unequal prime periods imply the existence of a proper connected critical manifold whose generic stabilizer order has at least two distinct odd prime divisors and whose inverse image in an auxiliary circle bundle has total rational Betti number at least eight. The period restrictions follow from reversal in equivariant Morse theory, the cohomology of fixed manifolds of a lifted geodesic flow, and index and nullity formulas under iteration.

Keywords: closed geodesics, periodic geodesic flow, Besse manifolds, complex projective spaces, equivariant Morse theory

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• Manuscript, version 1 (9 November 2026)