Page - 180 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Proof. (1) implies (2). Suppose thatψ∈M(D,D∗). Thenwehave D∗Xψ(Y)=ψ(DXY) ∀(X,Y). SincebothDandD∗ are torsionfreeonehas the identity D∗X.ψ(Y)−ψ(D∗XY)−D∗Yψ(X)+ψ(D∗YX)=0. Thusψ isa (1,0)-cocycleof the totalKVcomplex (C∗∗,δ∗∗). Atanotherside therelationD∗X◦ψ=ψ◦DX leads to the identity DXqψ=0. Soqψ isa (0,2)-cocycleofcomplex (C∗∗,δ∗∗).Weconcludethat δ1,2(ψ⊗qψ)=0, QED. (2) implies (1). Werecall the formula δ1,2(ψ⊗qψ)=(δτψ)⊗qψ−ψ⊗δτqψ. By this formula δ1,2(ψ⊗qψ)∈C2,2⊕C1,3 Thus thestatement (2) isequivalent to thesystem δτψ=0, δτqψ=0. Tocontinuetheproofweperformthefollowing lemma. Lemma4 ([29]). Foreverysymmetric cochainB∈C0,2, viz B(X,Y)=B(Y,X) the following identities are equivalent δτB=0, (13a) ∇B=0, (13b) ByLemma4thebilinear formqψ isD-parallel. Therebyweget the identity Xqψ(Y,Z)−qψ(DXY,Z)−qψ(Y,DXZ)=0. Tousefully interpret this identityweinvolve thedualistic relation Xg(Y,Z)= g(DXY,Z)+g(Y,D∗XZ). Thisexpression leads to the identity g(D∗Xψ(Y)−ψ(DXY),Z)+g(Y,D∗Xψ(Z)−ψ(DXZ))=0. (14) 180