Bi-invariant metric on volume-preserving diffeomorphisms group of a three-dimensional manifold

TL;DR

This paper constructs a weak bi-invariant metric on the group of volume-preserving diffeomorphisms of a 3D manifold, linking it to the manifold's eta-invariant, and explores its properties.

Contribution

It introduces a novel bi-invariant 2-form on the infinite-dimensional diffeomorphism group and relates its signature to the eta-invariant of the manifold.

Findings

Existence of a weak bi-invariant 2-form on the diffeomorphism group

Definition of the signature of the bi-invariant form

Connection between the form's signature and the eta-invariant

Abstract

We show the existence of a weak bi-invariant symmetric nondegenerate 2-form on the volume-preserving diffeomorphism group of a three-dimensional manifold and study its properties. Despite the fact that the space $\mathcal{D}_\mu(M^3)$ is infinite-dimensional, we succeed in defining the signature of the bi-invariant quadric form. It is equal to the $\eta$-invariant of the manifold $M^3$.