A Hofer-Type Norm of Hamiltonian Maps on Regular Poisson Manifold

Abstract

We define a Hofer-type norm for the Hamiltonian map on regular Poisson manifold and prove that it is nondegenerate. We show that the $L^{1,\infty }$-norm and the $L^{\infty }$-norm coincide for the Hamiltonian map on closed regular Poisson manifold and give some sufficient conditions for a Hamiltonian path to be a geodesic. The norm between the Hamiltonian map and the induced Hamiltonian map on the quotient of Poisson manifold $(M,\{·,·\})$ by a compact Lie group Hamiltonian action is also compared.