Page - 17 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 • a symplectic structure, determinedbyasymplectic formω, i.e., a2-formωwhich isbothclosed (dω=0)andnondegenerate (kerω={0}), • aPoissonstructure,determinedbyasmoothPoissonbivectorfieldΛsatisfying [Λ,Λ]=0. Definition 7. A presymplectic (resp. symplectic, resp. Poisson) diffeomorphism of a presymplectic (resp., symplectic, resp. Poisson)manifold (M,ω) (resp. (M,Λ)) is a smoothdiffeomorphism f :M→Msuch that f∗ω=ω (resp. f∗Λ=Λ). Definition8. AsmoothvectorfieldXonapresymplectic (resp. symplectic, resp. Poisson)manifold (M,ω) (resp. (M,Λ)) is said tobeapresysmplectic (resp. symplectic, resp. Poisson)vectorfield ifL(X)ω=0 (resp. if L(X)Λ=0),whereL(X)denotes theLiederivativeof formsormutivectorfieldswith respect toX. Definition9. Let (M,ω)beapresymplectic or symplecticmanifold.AsmoothvectorfieldXonMissaid to beHamiltonian if there exists a smooth functionH :M→R, calledaHamiltonian forX, such that i(X)ω=−dH . NotanysmoothfunctiononapresymplecticmanifoldcanbeaHamiltonian. Definition10. Let (M,Λ) be aPoissonmanifold. A smoothvector fieldX onMis said to beHamiltonian if there exists a smooth function H ∈ C∞(M,R), called a Hamiltonian for X, such that X = Λ (dH). Anequivalentdefinition is that i(X)dg={H,g} for anyg∈C∞(M,R) , where{H,g}=Λ(dH,dg)denotes thePoissonbracket of the functionsHandg. OnasymplecticoraPoissonmanifold,anysmoothfunctioncanbeaHamiltonian. Proposition 3. AHamiltonian vector field on a presymplectic (resp. symplectic, resp. Poisson)manifold automatically is apresymplectic (resp. symplectic, resp. Poisson)vectorfield. Theproof of this result, which is easy, can be found in anybookon symplectic andPoisson geoemetry, forexample [8–10]. 5.2. LieAlgebrasandLieGroupsActions Definition11. Anactiononthe left (resp. anactiononthe right) of aLiegroupGonasmoothmanifoldMis a smoothmapΦ :G×M→M(resp. a smoothmapΨ :M×G→M)such that • for eachfixed g∈G, themapΦg : M→MdefinedbyΦg(x)=Φ(g,x) (resp. themapΨg : M→M definedbyΨg(x)=Ψ(x,g)) is a smoothdiffeomorphismofM, • Φe= idM(resp.Ψe= idM), ebeing theneutral elementofG, • for eachpair (g1,g2)∈G×G,Φg1◦Φg2 =Φg1g2 (resp.Ψg1◦Ψg2 =Ψg2g1). An action of a Lie algebraG on a smoothmanifoldM is aLie algebrasmorphism ofG into the Lie algebraA1(M)of smoothvectorfieldsonM, i.e., a linearmapψ :G→A1(M)whichassociates to eachX∈G a smoothvectorfieldψ(X)onMsuch that for eachpair (X,Y)∈G×G,ψ([X,Y])= [ψ(X),ψ(Y)]. Proposition 4. AnactionΨ, either on the left or on the right, of a Lie groupG on a smoothmanifold M, automaticallydeterminesanactionψof itsLie algebraG onthatmanifold,whichassociates to eachX∈G the vectorfieldψ(X)onM,oftendenotedbyXMandcalled the fundamentalvectorfieldonMassociated toX. It is definedby ψ(X)(x)=XM(x)= d ds ( Ψexp(sX)(x) ) ∣∣ s=0 , x∈M , 17