On the existence of a Hofer type metric for Poisson manifolds

2015/07/16 by Rybicki, Tomasz
#53D17 #57R17 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1507.04736

Abstract

An analogue of the Hofer metric \varrho_H on the Hamiltonian group Ham(M,Λ) of a Poisson manifold (M,Λ) can be defined but there is the problem of its non-degeneracy. First we observe that \varrho_H is a genuine metric on Ham(M,Λ) when the union of all closed leaves (as subsets of M) of the corresponding symplectic foliation is dense. Next we deal with the important class of integrable Poisson manifolds. Recall that a Poisson manifold is called integrable if it can be realized as the space of units of a symplectic groupoid. Our main result states that \varrho_H is a Hofer type metric for every Poisson manifold which admits a Hausdorff integration.

Related

• Non-degeneracy of the Hofer norm for Poisson structures
• Almost Homogeneous Poisson Spaces
• Intégration symplectique des variétés de Poisson régulières
• Fragmented Hofer's geometry on Hameomorphism groups and its application
• Poisson Manifolds of Compact Types (PMCT 1)