Symplectic Asphericity, Category Weight, and Closed Characteristics of K-Contact Manifolds

TL;DR

This paper investigates conditions under which the Reeb vector field on a closed K-contact manifold has a minimum number of closed characteristics, linking symplectic asphericity of the leaf space to increased closed characteristics.

Contribution

It extends Rukimbira's result by showing that symplectic asphericity of the leaf space guarantees at least 2n+1 closed characteristics.

Findings

Reeb vector field has at least n+1 closed characteristics.

Symplectic asphericity implies at least 2n+1 closed characteristics.

Connects symplectic topology with Reeb dynamics on K-contact manifolds.

Abstract

Let $M$ be a closed K-contact $(2n+1)$-manifold equipped with a quasi-regular K-contact structure. Rukimbira proved that the Reeb vector field $\xi$ of this structure has at least $n+1$ closed characteristics. We note that $\xi$ has at least $2n+1$ closed characteristics provided that the space of leaves of the foliation determined by $\xi$ is symplectically aspherical.