Page - 177 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Thereforeonehas g∗(φ(X),Y)=Xg∗(ξ,Y)−g∗(ξ,D0XY). Since thequadruple (M,g∗,D0,D∗) isaduallyflatpaironehas the identity g∗(φ(X),Y)= g∗(D∗Xξ,Y). Thusweget theexpectedconclusion,viz φ(X)=D∗Xξ. Conversely letusassumethat thereexistsavectorξ satisfyingthe identity φ(X)=D∗Xξ. That leads to the identity g∗(D∗Xξ,Y)=Xg∗(ξ,Y)−g(ξ,D0XY). Inotherwordsonehas gφ∈Hyp(M,D0). Thisends theproofofCorollary2. Thesetofg∗-symmetricgaugetransformation isdenotedbyΣ(g∗). Wehavethecanonical isomorphism Σ(g∗) φ→ gφ∈Rie(M). (12) Nowwedefinethesets Z˜1τ(A∗,A∗)=Σ(g∗)∩Z1τ(A∗A∗), B˜1τ(A∗,A∗)=Σ(g∗)∩B1τ(A∗,A∗). CombiningLemma3and its corollarywith the isomorphismEquation (12). Thenweobtain the identifications Z˜1τ(A∗,A∗)=Hes(M,D∗), B˜1τ(A∗,A∗)=Hyp(M,D∗). Reminder. Werecall that ahyperbolicmanifold (oraKoszulmanifold) isδKV-exactHessianmanifold (M,g,D). It is easily seen that the set ofpositivehyperbolic structures ina locallyflatmanifold (M,D) is a convex subset ofHes(M,D). So show theKoszul geometry is a vanishing theorem in the theory of KVhomology of KValgebroids. The theoryofhomological statisticalmodel (to be introduced inPartB) is another impact on the information geometryof theKVcohomology. At thepresent stepwehave the relations Hes(M,D∗) Hyp(M,D∗)⊂H 2 KV(A∗,R), 177