  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 $M^3$.

Differential Geometry