Bi-invariant metric on the strict contactomorphism group

TL;DR

This paper introduces a right-invariant metric on the contactomorphism group of a contact manifold, which becomes bi-invariant when restricted to strict contactomorphisms, paralleling the Hofer metric in symplectic geometry.

Contribution

It defines a novel right-invariant metric on contactomorphism groups and demonstrates its bi-invariance on the subgroup of strict contactomorphisms.

Findings

The metric generalizes the Hofer metric concept to contact geometry.

The metric is bi-invariant on the strict contactomorphism subgroup.

Provides a new geometric tool for studying contactomorphism groups.

Abstract

A right-invariant metric $\rho_{\alpha}$ on the compactly supported identity component $Cont_0(M,\alpha)$ of the group of contactomorphisms of an arbitrary contact manifold $(M,\alpha)$ is introduced in a similar way that the Hofer metric was defined on the group of Hamiltonian symplectomorphisms of a symplectic manifold. The restriction of $\rho_{\alpha}$ to the subgroup $G(M,\alpha)$ of all strict contactomorphisms in $Cont_0(M,\alpha)$ is bi-invariant.