Seite - 182 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Therebyonehas qD∗ξ⊗∈Z2τ(A,R). Soonegets D∗ξ⊗qD∗ξ = δ0,2[ξ⊗qD∗ξ]. Thereforeassumption(2) implies (1). Converselyassumethat (1)holds,viz the (1,2)-cochainψ⊗qψ isexact. Thereexists ξ⊗α⊕ψ∗⊗β∈C0,2+C1,1 suchthat ψ⊗qψ= δ0,2(ξ⊗α)+δ1,1(ψ∗⊗β). Thus forvectorfieldsZ,X,Ywehave ψ(Z)⊗qψ(X,Y)= δτξ⊗α(X,Y)+ξ⊗δτα(Z,X,Y)+δτψ∗(Z,X)⊗β(Y)+ψ∗(Z)⊗δτβ(X,Y). Since ψ⊗qψ∈C1,2=C1τ(A∗,A∗)⊗C2τ(A,C∞(M)) theexactnessofψ⊗qψ implies δτα=0, α(X,Y)=α(Y,X). Therefore ψ(Z)⊗qψ(X,Y)= δξ(Z)⊗α(X,Y)+ψ∗(Z)⊗δτβ(X,Y). Nowweobserve that δτβ(X,Y)+δτβ(Y,X)=0. Infinalweget ψ(Z)⊗qψ(X,Y)= δτξ(Z)⊗α(X,Y). Soweobtain ψ(Z)=D∗Zξ, qψ(X,Y)=α(X,Y). Thisendtheproofof thecorollary. Fromthemapping C1,0 ψ→ψ⊗qψ∈C1,2 wededuce thecanonical linearmap H1τ(A∗,A∗) [ψ]→ [ψ⊗qψ]∈H1,2(C). Wedefineanothermap C1,0→C1,2 by ψ→ψ⊗ωψ. Here thedifferential2-formω isdefinedby ωψ(X,Y)= 1 2 [g(ψ(X),Y)−g(X,ψ(Y))]. 182