Page - 176 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 We note Sym(g) the subset of g-symmetric gauge transformations φ such that the following assertionsareequivalent (1) φ∈Sym(g). (2) (M,gφ,D0,Dφ) isaduallyflatpair. TheLiegroupofD0-preservinggaugetransformationsofTM isdenotedbyG0. It iseasy tosee that foreveryφ∈Sym(g) the followingassertionsareequivalent (1) φ∈G0, (2) gφ∈Hes(M,D0). Henceforthwedealwithafixedg∗ ∈Hes(M,D0. The triple (M,g∗,D0) leads to theduallyflat pair (M,g,D0,Dg ∗ ).Weset D∗=Dg ∗ . The tangentbundleTM is regardedasa leftKVmoduleof theKValgebroid (TM,D∗,1). The KV algebras of (M,D0) and of (M,D∗) are denoted by A0 and by A∗ respectively. Theircoboundaryoperatorsarenotedδ0 andδ∗ respectively. Wefocusontheroleplayedbythe totalKVcohomologyof thealgebroid (M,D∗,1). Letφbeag∗-symmetricgaugetransformation. Thenφgivesrise to themetric tensorgφ which is definedby gφ(X,Y)= g∗(φ(X),Y). Lemma3. The followingassertionsare equivalent, (1) gφ∈Hes(M,D0), (2) φ∈Z1τ(A∗,A∗). Hint. Use the following formula δ0KVgφ(X,Y,Z)= g ∗(δ∗τφ(X,Y),Z). Following thepioneeringdefinitionas in [2]ahyperbolic locallyflatmanifold is apositive exactHessian manifold (M,D,δKVθ).Weextend thenotionofhyperbolicitybydeleting the condition thatδKVθ ispositive. NowdenotebyHyp(M,D0) the set of exactHessianstructures in (M,D0). Ahyperbolic structure isdefinedbya triple (M,D,θ)where (M,D) is a locallyflatmanifoldandθ is ade Rhamcloseddifferential1-formsuch that the symmetricbilinearδKVθ isdefinite. The followingstatement is a straightforwardconsequenceofLemma3. Corollary2. The followingstatementsare equivalent. (1) gφ∈Hyp(M,D0), (2) φ∈B1τ(A∗,A∗) ProofofCorollary. By(1) thereexistsa (deRham)closeddifferential1-formθ suchthat gφ(X,Y)=Xθ(Y)−θ(D0XY). Letξbetheuniquevectorfieldsuchthat θ= ιξg∗. 176