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

We define a Hofer-type norm for the Hamiltonian map on regular Poisson manifold and prove that it is nondegenerate. We show that theL-norm and theL-norm coincide for theHamiltonianmap on closed regular Poissonmanifold 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.


Introduction and Main Results
This paper is devoted to establishing an invariant norm for Hamiltonian maps on the Poisson manifold.When  is symplectic, a remarkable bi-invariant distance was defined on Ham().This bi-invariant distance was first discovered by Hofer on the group of compactly supported symplectic diffeomorphisms of (R 2 ,  0 ) (where  0 is the standard symplectic form) [1].Viterbo defined a bi-invariant metric by generating functions [2], Polterovich generalized Hofer's metric to more symplectic manifold [3], and finally Lalonde and McDuff extended it to the group of compactly supported Hamiltonian diffeomorphisms on any symplectic manifold [4].This norm plays an important role in studying symplectic topology and has close relationship with symplectic capacity and symplectic rigidity; many mathematicians have great work in this field, but there is few work on the Poisson case; this is because the lack of variational formulation in the Poisson case, it is not easy to prove the nondegenerate.In this paper, we define a Hofer-type norm on a class of Poisson manifolds, that is, regular Poisson manifolds; with the help of Casimir functions and the decomposition of Poisson manifold, we can prove the nondegenerate.Let (, {⋅, ⋅}) be a Poisson manifold; that is, there exists a Poisson bracket {⋅, ⋅} on the smooth functions  ∞ ().For any , , ℎ ∈  ∞ () it satisfies the following: (1) {, } = −{, }, (2) {, ℎ} = {, ℎ} + ℎ{, }, (3) {, {, ℎ}} + {ℎ, {, }} + {, {ℎ, }} = 0. Definition 1.A smooth diffeomorphism  :  →  is called a Poisson diffeomorphism if for all , ℎ ∈  ∞ (), one has  * {, ℎ} = { * (),  * (ℎ)}.
If   is a Hamiltonian flow with some Hamiltonian function ℎ  (), one defines its length to be length one can generalize the bi-invariant metric on Ham() to the Poisson case.
Definition 2. Now one can define the energy of  ∈ Ham(), is a Hamiltonian flow ended with } . ( So one can define : for ,  ∈ Ham().
Let (, {⋅, ⋅}) be a Poisson manifold and let  be a Lie group acting canonically, freely, and properly on  via the map Φ :  ×  → .Let J :  → /   be the corresponding optimal momentum map.Then the orbit space / is a Poisson manifold with Poisson bracket {⋅, ⋅} / uniquely characterized by the relation
For a -invariant smooth function on , the Hamiltonian flow   of  ℎ induces a Hamiltonian flow  /  , so one has a well-defined homomorphism where Ham()  denotes the -invariant Hamiltonian maps.Now one can give a similar result as stated in [5].
Organization of This Paper.The organization is as follows.
First we will introduce the definition of the distance and give some properties.Next we will show the proofs of Theorems 3 and 4. Then we introduce the Poisson reduction.And last we give the proof of Theorem 5.
To prove this, we need to investigate the Hamiltonian functions of the Hamiltonian flows.Similar to the symplectic case, for the symplectic case, see page 144 of [6].
and  ∈ Ham() one defines the functions ℎ, ℎ# and ℎ  as follows: Proposition 12.If ℎ,  are smooth functions the following formulate hold true: where To prove this, we need the following fact.
Lemma 13 (see [7]).If  is a Poisson map, and Proof.For any function  ∈  ∞ ([0, 1] × , R), we have Since  is a Poisson map, we have So we have Proof of Lemma 13.The third formula is just the transition law of Hamiltonian vector.We know that if  is a Poisson map then Now we prove the second formula; we abbreviate the notation and observe that We need to show that   ∘   is the flow of  ℎ# , By the property of the Poisson diffeomorphism, we get that the second term is  ∘(  ) −1 .This finishes the proof of the second formula.We can obtain the first formula from the second.From the first two we can get the last one.
We are now ready to prove Proposition 10.
Proof of Proposition 10.From Proposition 12 and the Haminvariant of our pseudonorm, we get and thus () = ( −1 ).From we find so the third equality holds.
To prove the last one, we note that   ℎ# =   ℎ ∘    , and By the assumption of () = 0, we get that inf ℎ generats  ‖ h‖   0 = 0, by the definition of Hofer metric, |   0 =  and so  = .

Now we can define 𝑑:
for ,  ∈ Ham().This distance has the following properties.

󸀠
. Let (, {⋅, ⋅}) be a Poisson manifold; the function  is a bi-invariant metric; that is, for all , ,  ∈ () it satisfies the following: Similarly, so the claim is proved.
for all ,  ∈ () with supports contained in .
for any , and this is impossible; hence we finish the proof.Now we consider the geodesic under the above norm in Ham().For the standard symplectic manifold, Hofer proved that, the Hamiltonian flow generated by the timeindependent compactly supported function is a geodesic [6].Later Bialy and Polterovich gave a sufficient and necessary condition for a path to be geodesic [9]; last Lalonde and McDuff extended it to all symplectic manifolds [4] for some leaf   0 .Hence max Since ℎ is compactly supported, we have the  2 -norm of ℎ is bounded above by some number , and moreover, so by Theorem 1.2 of [3], we can choose a constant   >  such that ‖ − ℎ ‖ ≥ | − |‖ℎ‖ when | − | ≤ 1/  and this finishes the proof.

Journal of Applied Mathematics
For the time-dependent case, we have a similar result.First we recall the definition of quasiautonomous function.
We first show the following results which will be useful in the proof.
Proof.We just show that this is true on each symplectic leaf, but in the symplectic case, the Casimir functions are constants; this finishes the proof.Proposition 31 (Banyaga, cf.Proposition 3.1.5[15]).Let ℎ , be a 2-family smooth parameters of diffeomorphisms on a smooth manifold  such that ℎ 0,0 = .Let  , , Here every   is constructed with the help of   as above; we know that the partial derivative of the Hamiltonian ℎ(, , ) with respect to   at  = 0 equals   .
Remark 33.If the Poisson manifold is symplectic, then the pseudonorm is the Hofer norm.Give a Hamiltonian action; then we can get the results in the symplectic case as in [5].

Proposition 30 .
Let   be a flow generated by a Hamiltonian function  on a closed Poisson manifold.Then there exists an arbitrary small loop   such that the Hamiltonian function   of the flow  −1    satisfies {(), ⋅} ̸ = 0 for every t.
Here a Poisson manifold is called regular if the rank of the Poisson manifold is constant for all point.If one replaces the  1,∞ -norm by the  ∞ -norm, one also gets a norm on Ham().One proves that they are equal on closed regular Poisson manifold.
. Similarly, we can define   () if we restrict  in the closed ball of radius of  of Ham() centered at .Definition 18.A map  on a closed regular Poisson manifold (, {⋅, ⋅}) is bounded if () < +∞ and unbounded otherwise.Proposition 19.For any  ∈ (0, +∞], the function   () is bi-invariant, assumes the value 0 only at , and satisfies the triangle inequality   () ≤   () +   () , Proposition 20.Let (, {⋅, ⋅}) be a closed regular Poisson manifold; if there exists some  unbounded, then the Hofer metric  does not extend to a bi-invariant metric on the groups of Poisson maps.Proof.
. Now we consider similar questions on regular Poisson manifold.Let   ℎ be a Hamiltonian flow with compactly supported time-independent Hamiltonian function ℎ on (, {⋅, ⋅}), and if the maximal and minimal point of ℎ lie in the same symplectic leaf, then   ℎ is a geodesic; that is, (  ℎ ,   ℎ ) = ( − )‖ℎ‖ for | − | sufficiently small.