Pontryagin forms on (4r−2)(4r−2)-manifolds and symplectic structures on the spaces of Riemannian metrics

The Pontryagin forms on the 1-jet bundle of Riemannian metrics, are shown to provide in a natural way diffeomorphism-invariant pre-symplectic structures on the space of Riemannian metrics for the dimensions n≡2n≡2(mod4). The equivariant Pontryagin forms provide canonical moment maps for these structures. In dimension two, the symplectic reduction corresponding to the pre-symplectic form and its moment map attached to the first Pontryagin form, is proved to coincide with the Teichmüller space endowed with the Weil–Petersson symplectic form.