Reductions: precontact versus presymplectic

TL;DR

This paper establishes a unified framework for contact reductions by relating them to symplectic reductions, viewing contact structures as homogeneous symplectic structures, and extending the approach to precontact and presymplectic cases.

Contribution

It introduces a novel perspective that describes contact reductions via symplectic reductions on symplectic covers, including general contact structures and precontact settings.

Findings

Contact reductions can be described using symplectic reduction techniques.

The approach includes non-cooriented contact structures and extends to precontact and presymplectic cases.

Contact Hamiltonians are defined on the symplectic cover, not directly on the contact manifold.

Abstract

We show that contact reductions can be described in terms of symplectic reductions in the traditional Marsden-Weinstein-Meyer as well as the constant rank picture. The point is that we view contact structures as particular (homogeneous) symplectic structures. A group action by contactomorphisms is lifted to a Hamiltonian action on the corresponding symplectic manifold, called the symplectic cover of the contact manifold. In contrast to the majority of the literature in the subject, our approach includes general contact structures (not only co-oriented) and changes the traditional view point: contact Hamiltonians and contact moment maps for contactomorphism groups are no longer defined on the contact manifold itself, but on its symplectic cover. Actually, the developed framework for reductions is slightly more general than purely contact, and includes a precontact and presymplectic setting which is based on the observation that there is a one-to-one correspondence between isomorphism classes of precontact manifolds and certain homogeneous presymplectic manifolds.