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

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 $M^3$.