Hofer metric from the contact point of view

TL;DR

This paper introduces a bi-invariant pseudo-metric on the group of strict contactomorphisms of a contact manifold, which becomes a true metric in the case of open manifolds, providing a new geometric perspective.

Contribution

It defines a novel bi-invariant pseudo-metric on the contactomorphism group and establishes conditions under which it is a genuine metric.

Findings

The pseudo-metric is well-defined on the contactomorphism group.

For open manifolds, the pseudo-metric is a true metric.

Provides a new geometric framework for contactomorphism groups.

Abstract

Given a manifold $M$ endowed with a contact 1-form $\alpha$, a bi-invariant pseudo-metric $\varrho_\alpha$ is introduced on $Cont(M,\alpha)$, the compactly supported identity component of the group of all strict contactomorphisms of $(M,\alpha)$. For $M$ open $\varrho_\alpha$ is a metric.