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K-contact manifolds with minimal closed Reeb orbits

Hui Li

TL;DR

This work analyzes Boothby–Wang fibrations to construct simply connected K-contact manifolds and derives precise criteria for when the total space is (integral or real) cohomology equivalent to odd spheres. It shows how Hamiltonian torus actions on the base lift to the total space and, via perturbations of the contact form, yields K-contact structures with closed Reeb orbits in bijection with fixed points of base actions. The paper provides both positive results—sphere-like total spaces under specific cohomology and Chern class conditions—and a range of concrete counterexamples where minimal closed Reeb orbits occur without the total space being a sphere. It also demonstrates how to realize Sasakian structures in the Kähler base setting and constructs families of simply connected examples with minimal or near-minimal Reeb dynamics that are not real cohomology spheres, enriching the landscape of contact and symplectic geometry.

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.

K-contact manifolds with minimal closed Reeb orbits

TL;DR

This work analyzes Boothby–Wang fibrations to construct simply connected K-contact manifolds and derives precise criteria for when the total space is (integral or real) cohomology equivalent to odd spheres. It shows how Hamiltonian torus actions on the base lift to the total space and, via perturbations of the contact form, yields K-contact structures with closed Reeb orbits in bijection with fixed points of base actions. The paper provides both positive results—sphere-like total spaces under specific cohomology and Chern class conditions—and a range of concrete counterexamples where minimal closed Reeb orbits occur without the total space being a sphere. It also demonstrates how to realize Sasakian structures in the Kähler base setting and constructs families of simply connected examples with minimal or near-minimal Reeb dynamics that are not real cohomology spheres, enriching the landscape of contact and symplectic geometry.

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.
Paper Structure (8 sections, 11 theorems, 57 equations)

This paper contains 8 sections, 11 theorems, 57 equations.

Table of Contents

  1. introduction
  2. proof of Theorem \ref{['thm1']}
  3. proof of Theorem \ref{['thm2']}
  4. K-contact structures on the total spaces of principal circle bundles over symplectic manifolds
  5. lift a Hamiltonian torus action on the symplectic base to the total space of the fibration $\pi\colon M\to N$
  6. changing the K-contact structures on the total space of the Boothby-Wang fibration $\pi\colon M\to N$ --- proof of Theorem \ref{['thm3']}
  7. examples of simply connected compact K-contact manifolds with minimal closed Reeb orbits which are not homeomorphic to the spheres --- proof of Theorem \ref{['thm4']}
  8. examples of simply connected K-contact manifolds which are not real cohomology spheres

Key Result

Theorem 1

Let $N$ be a compact smooth manifold of dimension $2n$, $x$ be a primitive integral degree 2 cohomology class of $N$, and $\pi\colon M\to N$ be a principal circle bundle over $N$ with Euler class $x$. Then the following claims hold.

Theorems & Definitions (29)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 2.1
  • proof
  • Remark 2.3
  • proof : Proof of Theorem \ref{['thm1']}
  • proof : Proof of Theorem \ref{['thm2']}
  • Remark 3.1
  • ...and 19 more