Bi-invariant metric on contact diffeomorphisms group

TL;DR

This paper constructs and analyzes a bi-invariant metric on the contact diffeomorphisms group, exploring its properties, Euler's equation, and curvature, with connections to volume-preserving diffeomorphisms in three dimensions.

Contribution

It introduces a weak bi-invariant metric on contact diffeomorphisms and studies its geometric and dynamical properties, including curvature and Euler's equation.

Findings

Existence of a weak bi-invariant symmetric 2-form on contact diffeomorphisms group

Derivation of Euler's equation for the group

Calculation of sectional curvature and connection to volume-preserving diffeomorphisms in 3D

Abstract

We show the existence of a weak bi-invariant symmetric nondegenerate 2-form on the contact diffeomorphisms group $\mathcal{D}_\theta$ of a contact Riemannian manifold $(M,g,\theta)$ and study its properties. We describe the Euler's equation on a Lie algebra of group $\mathcal{D}_\theta$ and calculate the sectional curvature of $\mathcal{D}_\theta$. In a case $\dim M =3$ connection between the bi-invariant metric on $\mathcal{D}_\theta$ and the bi-invariant metric on volume-preserving diffeomorphisms group $\mathcal{D}_\mu$ of $M^3$ is discover.