Page - 170 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 The nonnegative integers r(M) and s(M) are global geometric invariants. They connect the totalKVcohomology togeodesicRiemannian foliations. By thisviewpoint thepropositionhasan interestingcorollary. Corollary1. Inanm-dimensionalmanifoldMsuppose that the following inequalities are satisfied 0< s(M)<m. Then themanifoldMadmits a locallyflat structure (M,D∗)which supports anon trivialD∗-geodesic Riemannian foliation. The integer s(M) isa local characteristic invariantof someclassof2-webs inHessianmanifolds. Let (M,D)bea locallyflatmanifoldwhoseKValgebra isdenotedbyA. we recall that aHessian metric tensor in (M,D) isa inversiblecocycleg∈Z2KV(A,R). Theorem 6. Let (M,D,g) and (M∗,D∗,g∗) be m-dimensional Hessian manifolds. We assume that the following inequalitieshold 0< s(M,D)= s(M∗,D∗)= s<m. ThenMandM∗ admit linearizable2-webswhichare locally isomorphic. Proof. Theproof isbasedonmethodsof the informationgeometry. LetAandA∗be theKValgebrasof (M,D)andof (M∗,D∗) respectively. Bythehypothesis there existsapairofgeosicRiemannianfoliations (B,B∗)∈SA2 ×SA ∗ 2 suchthat rank(B)= rank(B∗)= s. By the dualistic relation both M and M∗ admit locally flat structures (M,D˜) and (M∗,D˜∗) definedby g(Y,D˜XZ)=Xg(Y,Z)−g(DXY,Z), g∗(Y,D˜∗XZ)=Xg∗(y,Z)−g∗(D∗XY,Z). TheirKValgebrasaredenotedby A˜and A˜∗. Stepa Thereexistsa1-cocycle ψ∈Z1τ(A˜,A˜) suchthat B(X,Y)= g(ψ(X),Y), Ker(B)=Ker(ψ). By thedefinitionof D˜wehave TM=Ker(ψ)⊕ im(ψ). Further im(ψ) is D˜-geodesicandKer(B) isD-geodesic. Therefore, thepair (Ker(ψ),im(ψ)) 170