arXiv:2610.33261v1 Announce Type: new Abstract: We generalize Zoll families of hypersurfaces to Besse families of hypersurfaces, which are defined differential-topologically and may contain non-embedded and exceptional hypersurfaces. A contact-geometric discussion presents them as double Legendrian fibrations and shows that regular hypersurfaces are either spheres or real projective spaces. In analogy with the Berger conjecture on simply connected manifolds all of whose geodesics are closed, we show that every hypersurface of a Besse family on a sphere other than the 4-sphere is regular. On even-dimensional spheres other than the 4-sphere we obtain a stronger conclusion, generalizing a theorem of Gromoll and Grove about 2-spheres all of whose geodesics are closed: every hypersurface of a Besse family is regular and embedded. Moreover, building on work by Ambrozio and Guajardo, we provide metrics of non-constant curvature carrying Besse families of non-embedded minimal hypersurfaces in every odd dimension at least three, and Besse families of minimal hypersurfaces with exceptional hypersurfaces on orbifolds in every dimension at least two. We also give a short proof that manifolds admitting Zoll families of embedded hypersurfaces are homeomorphic to spheres or real projective spaces, recovering a recent result of Martins up to homeomorphism.
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