Page - 4 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 370 ontheconfigurationspaceofthesystemisnotaLiealgebraofsymmetriesoftheLagrangian.Moreover in its intrinsic formthatequationuses theconceptofHamiltonianmomentummappresentedlater, inSection5. Since theEuler–Poincaréequation isnotusedin the followingsections, thereadercan skip thecorrespondingsubsectionathisorherfirst reading. 1.2.Notations Thenotationsusedaremoreorlessthosegenerallyusednowindifferentialgeometry. Thetangent andcotangentbundles toasmoothmanifoldMaredenotedbyTMandT∗M, respectively,andtheir canonicalprojectionsbyτM :TM→MandπM :T∗M→M. Thevectorspacesofk-multivectorsand k-formsonMaredenotedbyAk(M)andΩk(M), respectively,withk∈Zand,ofcourse,Ak(M)={0} andΩk(M) = {0} if k< 0 and if k> dimM, k-multivectors and k-formsbeing skew-symmetric. The exterior algebras ofmultivectors and formsof all degrees aredenotedby A(M) =⊕kAk(M) andΩ(M)=⊕kΩk(M), respectively. Theexteriordifferentiationoperatorofdifferential formsona smoothmanifoldM isdenotedbyd :Ω(M)→Ω(M). The interiorproductof adifferential form η∈Ω(M)byavectorfieldX∈A1(M) isdenotedbyi(X)η. Let f : M → N be a smoothmapdefinedon a smoothmanifold M, with values in another smoothmanifoldN. Thepull-backofa formη∈Ω(N)byasmoothmap f : M→N isdenotedby f∗η∈Ω(M). Asmooth, time-dependentvectorfieldonthesmoothmanifoldM isasmoothmapX :R×M→TM suchthat, foreach t∈Randx∈M,X(t,x)∈TxM, thevectorspace tangent toMatx.When, forany x∈M,X(t,x)doesnotdependon t∈R,X isasmoothvectorfield in theusual sense, i.e., anelement inA1(M). Ofcoursea time-dependentvectorfieldcanbedefinedonanopensubsetofR×M instead thanonthewholeR×M. Itdefinesadifferentialequation dϕ(t) dt =X ( t,ϕ(t) ) , (1) said tobeassociated toX. The (full)flowofX is themapΨX,definedonanopensubsetofR×R×M, taking itsvalues inM, suchthat foreach t0∈Randx0∈M theparametrizedcurve t →ΨX(t,t0,x0) is themaximal integral curveofEquation (1) satisfyingΨ(t0,t0,x0) = x0. When t0 and t ∈R are fixed, themapx0 →ΨX(t,t0,x0) isadiffeomorphism,definedonanopensubsetofM (whichmay beempty)andtaking itsvalues inanotheropensubsetofM,denotedbyΨX(t,t0).WhenX is in facta vectorfield in theusualsense (notdependentontime),ΨX(t,t0)onlydependson t− t0. Insteadof the fullflowofXwecanuse its reducedflowΦX,definedonanopensubsetofR×Mandtaking itsvalues inM, relatedto the fullflowΨX by ΦX(t,x0)=ΨX(t,0,x0) , ΨX(t,t0,x0)=ΦX(t− t0,x0) . For each t ∈R, themap x0 →ΦX(t,x0) =ΨX(t,0,x0) is a diffeomorphism, denotedbyΦXt , definedonanopensubsetofM (whichmaybeempty)ontoanotheropensubsetofM. When f : M→N is a smoothmapdefinedona smoothmanifoldM,withvalues inanother smoothmanifoldN, thereexistsasmoothmapTf :TM→TN calledtheprolongationof f tovectors, which foreachfixedx∈M linearlymapsTxM intoTf(x)N.When f isadiffeomorphismofMontoN, Tf isan isomorphismofTMontoTN. Thatpropertyallowsus todefinethe canonical liftsofavector fieldX inA1(M) to the tangentbundleTMandto thecotangentbundleT∗M. Indeed, foreach t∈R, ΦXt isadiffeomorphismofanopensubsetofMontoanotheropensubsetofM. ThereforeTΦ X t isa diffeomorphismofanopensubsetofTMontoanotheropensubsetofTM. It turnsout thatwhen t takesallpossiblevalues inR thesetofalldiffeomorphismsTΦXt is thereducedflowofavectorfield XonTM,which is the canonical liftofX to the tangentbundleTM. Similarly, the transpose (TΦX−t)T ofTΦX−t isadiffeomorphismofanopensubsetof thecotangent bundleT∗MontoanotheropensubsetofT∗M, andwhen t takesallpossiblevalues inR thesetofall 4