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

• Main Authors: Perez, Roberto Ferreiro, Masque, Jaime Muñoz
• Format: Journal Article
• Language: English
• Published: 04-07-2005
• Subjects: Mathematics - Differential Geometry
• DOI: 10.48550/arxiv.math/0507076

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.