On the existence of a Hofer type metric for Poisson manifolds (Q2822025)

An important tool in symplectic topology is the Hofer metric on the group of compactly supported Hamiltonian diffeomorphisms of a symplectic manifold. Poisson manifolds are smooth manifolds with a singular foliation by symplectic immersed submanifolds, and as such symplectic manifolds are special cases of Poisson manifolds. The formula for the Hofer norm extends easily to the Poisson case, defining a bi-invariant pseudo-norm \(\rho_{H}\).NEWLINENEWLINEIt is now known whether it is also a norm, that is, whether it is non-degenerate. The paper gives sufficient conditions on a Poisson manifold for this to be the case. The first theorem of the paper shows that if the union of the symplectic leaves that are proper (i.e., embedded and of positive dimension) is dense, then \(\rho_{H}\) is a norm. The second theorem states that if the Poisson manifold is integrable to a Hausdorff symplectic groupoid, then \(\rho_{H}\) is a norm. An alternative version of this theorem is Corollary 4.7: if a Poisson manifold admits a complete Hausdorff symplectic realization, then \(\rho_{H}\) is a norm.