Volume-preserving right-handed vector fields are conformally Reeb

TL;DR

This paper proves that volume-preserving right-handed vector fields on closed three-manifolds are conformally Reeb, meaning they can be transformed into Reeb vector fields, which leads to the existence of a global surface of section.

Contribution

It establishes that volume-preserving right-handed vector fields are conformally Reeb, connecting two important classes of vector fields and extending previous partial results.

Findings

Volume-preserving right-handed vector fields are contact-type.

Such vector fields are conformally Reeb.

They admit a global surface of section.

Abstract

Right-handed and Reeb vector fields are two rich classes of vector fields on closed, oriented three-manifolds. Prior work of Dehornoy and Florio-Hryniewicz has produced many examples of Reeb vector fields which are right-handed. We prove a result in the other direction. We show that the closed two-form associated to a volume-preserving right-handed vector field is contact-type. This implies that any volume-preserving right-handed vector field is equal to a Reeb vector field after multiplication by a positive smooth function. Combining our result with theorems of Ghys and Taubes shows that any volume-preserving right-handed vector field has a global surface of section.