K-contact manifolds with minimal closed Reeb orbits

TL;DR

This paper constructs and analyzes K-contact manifolds with minimal closed Reeb orbits, exploring their topological properties, and providing examples that distinguish them from spheres despite similar cohomology.

Contribution

It introduces conditions for when K-contact manifolds are homeomorphic to spheres and constructs examples with minimal Reeb orbits not homeomorphic to spheres.

Findings

Certain K-contact manifolds are homeomorphic to spheres under specific conditions.

Existence of K-contact structures with minimal Reeb orbits matching fixed points of Hamiltonian torus actions.

Examples of simply connected K-contact manifolds with minimal Reeb orbits not homeomorphic to spheres.

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.