Published November 9, 2026 | Version v3
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Prime periods of geodesics on Besse manifolds of complex projective type

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Let M²ⁿ, n ≥ 2, be a closed connected Riemannian manifold all of whose geodesics are closed, and suppose that H²(M; ℚ) ≠ 0. Such a manifold has the homotopy type of ℂℙⁿ. We prove Berger’s conjecture for this class when 2 ≤ n ≤ 8: all unit-speed geodesics have the same least period. For every n ≥ 2, the same conclusion holds if each nontrivial stabilizer of the geodesic circle action has prime-power order. More precisely, suppose that n ≥ 3 and the prime periods are unequal. Then some time map of the geodesic flow has a proper connected fixed component F on which the generic stabilizer order is divisible by two distinct odd primes. The inverse image of F in an auxiliary circle bundle over the unit tangent bundle has total rational Betti number at least eight. The proof combines index iteration with loop reversal in twisted equivariant cohomology and the fixed sets of a lift of the geodesic flow whose least period is twice that of the original flow.

First published 9 November 2026. Version 3, revised 12 November 2026 (47 pages).

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2026-11-12
Version 3

References

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