On the First Pontryagin Form of a Surface

Abstract

The first Pontryagin form of a compact orientable surface $M$ determines a canonical pre-symplectic structure on the space of Riemannian metrics on $M$. The first equivariant Pontryagin form determines a canonical moment map for it. We study the corresponding symplectic reduction and we state (Theorem 6) that the symplectic quotient is the Teichmüller space of the surface with the Weil-Petersson symplectic form.