A Hofer-type norm of Hamiltonian maps on regular Poisson manifold. (English) Zbl 1406.53082

Sun, Dawei; Zhang, Zhenxing

J. Appl. Math. 2014, Article ID 879196, 9 p. (2014).

Summary: 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 \((N,\{\cdot,\cdot\})\) by a compact Lie group Hamiltonian action is also compared.

MSC:

53D17	Poisson manifolds; Poisson groupoids and algebroids
53D05	Symplectic manifolds (general theory)