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

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