Page - 187 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 ThusonemayregardelementsofM(∇,∇∗)asorthogonal2-webs inM. Wekeepourpreviousnotation. Thewehave qψ(X,Y)=g(ψ+(X),Y), ωψ(X,Y)=g(ψ−(X),Y). Nowsuppose that (M,g,∇,∇∗) is aduallyflatpairwhoseKValgebrasarenotedAandA∗.We take into account the inclusion M(∇,∇∗)⊂Z1τ(A∗,A∗). WehaveamapofM(∇,∇∗) in the spaceofdeRham2-cocyleswhich isdefinedby M(∇,∇∗) ψ→ωψ. Assumethat the cocycleψ∈M(∇,∇∗) is exact. Then there existsξ∈A∗ such that ψ(X)=∇∗Xξ ∀X∈A. Bythedualistic relationoneeasily sees that ωψ=ddR(ιξg). Thereforeonegets a canonical linearmap H1τ(A∗,A∗) [ψ]→ [ωψ]∈H2dR(M,R). Thenext subsubsection isdevoted toa fewconsequencesof itemswe justdiscussed. 5.1.6. RiemannianWebs—SymplecticWebs inStatisticalManifolds WeintroduceRiemannianwebsandsymplecticwebsandwediscusstheir impactsonthetopology ofstatisticalmanifolds.Werecall thataRiemannianfoliation isasymmetricbilinear formg∈S2(M) withthe followingproperties (a) rank(g)= constant, (b) LXg=0∀X∈Γ(Ker(g)). Weput D=Ker(g). Toavoidconfusions thepair (D,g)stands for theRiemannianfoliationg. Definition 36. A Riemannian k-web is a family of k Riemannian foliations in general position (Dj,gj), j :=1,...,k.Asymplectic k-web is a familyof k symplectic foliations ingeneralposition (Dj,ωj); j :=1,...,k). Let (M,g,D,D∗)beaduallyflatpairwhoseKValgebras aredenotedbyAandA∗. We recall the inclusion M(D,D∗)⊂Z1τ(A∗,A∗). Consider a statistical manifold (M,g,∇,∇∗). By the classical theorem of Frobenius every ∇-paralleldifferential systeminM is completely integrable. 187