Seite - 175 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 AComment. Fromtheproposition justprovedarises a relationshipbetween theduallyflatnessand theKVcohomology. Indeed let (M,D0)beafixed locallyflatmanifoldwhoseKValgebra isdenotedbyA0. LetC∗KV(A0,R)be theKVcomplexof R˜-valuedcochainsof theKValgebroid (TM,D0,1).Weknowthat everyg∈Rie(M)yields aflatpair (M,g,D0,Dg). HereDg is theflatKoszul connectiondefinedby g(DgXY,Z)=Xg(Y,Z)−g(Y,D0XZ). Proposition4. The followingassertionsare equivalent. (1) (M,g,D0,Dg) is aduallyflatpair. (2) δ0KV(g)=0 ThescalarKVcohomologyofafixedlocallyflatmanifold (M,D0)providesawayofconstructing newlocallyflatstructures inM. Indeedletusset Hes(M,D0)=Z2KV(A0,R)∩Rie(M). For every g ∈ Hes(M,D0) there is a unique Dg ∈ LF(M) such that (M,g,D0,Dg) is a duallyflatpair. So thedualistic relation leads to themap Hes(M,D0) g→Dg∈LF(M). Werecall thatagaugemapinTM isavectorbundlemorphismofTM inTMwhichprojectson the identitymapofM. Thereaders interested inothers topological studies involvingconnectionsand gaugetransformationsarereferredto [49]. Giventwosymmetriccocyclesg,g∗ ∈Hes(M,D0) there isauniquegaugetransformation φ∗ :TM→TM suchthat g∗(X,Y)= g(φ∗(X),Y). The followingpropertiesareequivalent φ(D0XY)=D 0 Xφ(Y), (11a) Dg=Dg ∗ . (11b) Wefixametrictensorg∗∈Hes(M,D0).Agaugetransformationφ iscalledg-symmetricifwehave g(φ(X),Y)= g(X,φ(Y)) ∀(X,Y). Everyg-symmetricgaugetransformationφdefines themetric tensor gφ(X,Y)= g(φ(X),Y). Thisgivesrise to theflatpair (M,gφ,D0,Dgφ). Tosimplifyweset Dφ=Dgφ. 175