On the first Pontryagin form of a surface. (English) Zbl 1110.53063

A universal Pontryagin form is a closed differential form on the first jet bundle of the bundle of Riemannian metrics of a manifold \(M\). By pulling back such a form by means of the first jet prolongation \(j^1g\) of a Riemannian metric \(g\), one obtains a closed differential form on \(M\), whose cohomology class does not depend on \(g\), and which is the corresponding Pontryagin class of \(M\). There are non-zero universal Pontryagin forms of degree greater than the dimension of \(M\), and the authors’ aim is to give a geometric interpretation of these forms. In the case where \(M\) is a compact orientable surface, the first universal Pontryagin form of \(M\) determines a canonical pre-symplectic structure on the space of Riemannian metrics on \(M\). The authors also introduce equivariant Pontryagin forms. The first equivariant Pontryagin form determines a canonical moment map for the first Pontryagin form. They study the corresponding symplectic reduction. They state the result that the Marsden-Weinstein quotient is the Teichmüller space of the surface, and hence they obtain a natural pre-symplectic structure on it. They state that this pre-symplectic structure is the Weil-Petersson form. The results are given without proof.


53D20 Momentum maps; symplectic reduction
53D30 Symplectic structures of moduli spaces
58D17 Manifolds of metrics (especially Riemannian)
57R20 Characteristic classes and numbers in differential topology