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arXiv · 2406.10866

K-contact manifolds with minimal closed Reeb orbits

Hui Li
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Abstract

We use the Boothby-Wang fibration to construct certain simply connected K-contact manifolds and we give sufficient and necessary conditions on when such K-contact manifolds are homeomorphic to the odd dimensional spheres. If the symplectic base manifold of the fibration admits a Hamiltonian torus action, we show that on the total space of the fibration, other than the regular K-contact structures which have infinitely many closed Reeb orbits, there are K-contact structures whose closed Reeb orbits correspond exactly to the fixed points of the Hamiltonian torus action on the base manifold. Then we give a collection of examples of compact simply connected K-contact manifolds with minimal number of closed Reeb orbits which are not homeomorphic to the odd dimensional spheres, while having the real cohomology ring of the spheres. Finally, we give a family of examples of simply connected K-contact manifolds which have one more than the minimal number of closed Reeb orbits and which do not have the real cohomology ring of the spheres.

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Hui Li. 2024-06-16. K-contact manifolds with minimal closed Reeb orbits. https://doi.org/10.1093/imrn%2Frnaf129

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