Page - 173 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 For thosepurposeswefocusonanelementary itemwhichhasanotable impactsonourrequest. Let (M,D)bea locallyflatmanifoldwhoseKValgebra isdenotedbyA. Letg∈C2(A,C∞(M)). The leftkernelandtherightkernelofgaredenotedbyKer(g)andK0er(g) respectively. Ker(g) isdefinedby g(X,Y)=0 ∀Y∈A. K0er(g) isdefinedby g(Y,X)=0 ∀Y∈A. ThescalarKV2-cocycleshaveelementaryrelevantproperties (1) The leftkernelofeveryKV2-cocycle isclosedunder thePoissonbracketofvectorfields. (2) TherightkernelofeveryKV2-cocycle isaKVsubalgebraof theKValgebraA. Wetranslate thoseelementaryproperties in termof thedifferential topology Theorem7. Inananalytic locallyflatmanifold (M,D) (1) Thearrow Z2KV(A,C∞(M)) g→Ker(g) maps the set of analytic2-cocycles in the categoryof analytic stratified foliationsM, (2) Thearrow Z2KV(A,C∞(M)) g→K0er(g) maps the set of analytic2-cocycles in the categoryof stratified locallyflat foliations, (3) Ifa2-cocycleg isasymmetric formthenKer(g) isastratified locally flat transversallyRiemannianfoliation. Thevector subspace of symmetric 2-cocycles the kernels ofwhich areD-geodesic is denoted by Z˜2KV(A). Thecorrespondingcohomologyvectorsubspace isdenotedby H˜2KV(A)⊂H2KV(A,C∞(M)). By theexact sequence O→H2dR(M)→H2τ(A,C∞(M))→SA2 (M)→0 wehavethe inclusionmap H2τ(A,C∞(M)) H2dR(M) ⊂ H˜2KV(A)⊂RF(M). 5.TheInformationGeometry,GaugeHomomorphismsandtheDifferentialTopology Wecombinethedualistic relationwithgaugehomomorphismstorelate the total cohomologyand twoproblems. (i) Thefirst is theexistenceproblemforRiemannianfoliations. (ii) Thesecondis the linearizationofwebs. ThoserelationshipshighlightotherrolesplayedbythetotalKVcohomology. Throughthissection weuse thebrutecoboundaryoperator. 5.1. TheDualisticRelation Weare interest in the foliationcounterpartof thereduction instatisticalmodels. Thestatistical reductiontheoremisTheorem3.5as in [18].Werecall thenotionswhichareneeded. 173