We recall and improve the correspondence theorem of 

Etnyre-Ghrist [1] between a positively rescaled Reeb 

field for a contact 1-form and a rotational Beltrami 

field for a Riemannian metric on a closed oriented 

3-manifold. Given a contact form, we associate it 

with the space of Riemannian metrics for which the 

Reeb field is a Beltrami field with certain additional 

properties. We obtain a product structure on this space 

of metrics and then, by applying it, we characterize 

certain geometric structures on $3$-manifolds.