  The Pontryagin forms on 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 dimensions $n=4r-2$. 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 Teichm\"{u}ller space endowed with the Weil-Petersson
symplectic form.
