Page - 183 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Thisyieldsa linearmap H1τ(A∗,A∗) [ψ]→ [ψ⊗ωψ]∈H1,2(C). Nowlet (M,g,∇,∇∗)beaduallyflatpairwhoseKValgebrasaredenotedbyAandA∗. Weidentify thevectorspaceΓ(Hom(TM,TM))withthespaceC1τ(A∗,A∗). WekeepthenotationD∇∇∗,Cij,δij andqψ. Therefore,wecanrephraseLemma4as it follows. Proposition5. ForeverysectionψofHom(TM,TM) the followingassertionsare equivalent. (1) : D∇∇ ∗ (ψ)=0, (2) : δ12(qψ)=0 Here isan interestingfeature. Inaduallyflatpair (M,g,∇,∇∗)wecombine thedoublecomplex{ Cij,δij } withthecorrespondence ψ→ qψ. Thatallowustoreplace thedifferentialequation FE∇∇ ∗ : D∇∇ ∗ (ψ)=0 bythehomologicalequation δ12(ψ)=0. That isarelevant impactontheglobalanalysisofcombinationsof theKVcohomologicalmethods withmethods in the informationgeometry. 5.1.5.ComputationalRelations. RiemannianFoliations. SymplecticFoliations:Continued Wecontinuetorelate thevectorspaceofgaugehomomorphismsandthedifferential topology. Thetoolsweuseare theKVcohomologyandtheAmaridualistic relation. Let (M,g,D,D∗)beadualpair. Thevector subspaceof g-preservingelementsofM(D,D∗) is denotedbyM(g,D,D∗). Thuseveryψ∈M(g,D,D∗)satisfies the identity g(ψ(X),Y)+g(X,ψ(Y))=0. NowwefixaKoszulconnectionD0 andwedefinethemap Rie(M) g→Dg∈LC(M). bysetting g(DgXY,Z)=Xg(Y,Z)−g(Y,D0XZ). Wedefinethenonnegative integers nx(D0)=dim[ Mx(D0,Dg) Mx(g,D0,Dg)], n(DO)=min x∈M dim[ Mx(D0,Dg) Mx(g,D0,Dg)]. 183