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Prime periods of geodesics on Besse manifolds of complex projective type

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Abstract

Let M2nM^{2n}, n≥2n\ge 2, be a closed connected Riemannian manifold all of whose geodesics are closed, and suppose that H2(M;ℚ)≠0H^2(M;\mathbb{Q})\ne 0. Such a manifold has the homotopy type of ℂℙn\mathbb{CP}^n. We prove Berger’s conjecture for this class when 2≤n≤82\le n\le 8: all unit-speed geodesics have the same least period. For every n≥2n\ge 2, the same conclusion holds if each nontrivial stabilizer of the geodesic circle action has prime-power order. More precisely, suppose that n≥3n\ge 3 and the prime periods are unequal. Then some time map of the geodesic flow has a proper connected fixed component FF on which the generic stabilizer order is divisible by two distinct odd primes. The inverse image of FF 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.

MSC 2020

53C22Primary
Geodesics in global differential geometry
58E10Secondary
Variational problems in applications to the theory of geodesics (problems in one independent variable)
55N91Secondary
Equivariant homology and cohomology in algebraic topology

Mathematics Subject Classification 2020

Record

Version DOI
10.5281/zenodo.24151012
All versions
10.5281/zenodo.24068623
Subjects
Closed geodesics; Besse manifolds; complex projective spaces; equivariant Morse theory; periodic geodesic flow
Earlier title
Prime geodesic periods on manifolds of complex projective type
Language
English

Cite this paper

Anass Nifa. Prime periods of geodesics on Besse manifolds of complex projective type. Preprint, Zenodo, v3 (2026). doi:10.5281/zenodo.24151012

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