Page - 186 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 AUsefulComment. Let (M,D)bea locallyflatmanifoldwhoseKValgebra isdenotedbyA. To everydualpair (M,g,D,Dg) weassign the short split exact sequence 0→ΩA2 (M)→M(D,Dg)→SA2 (M)→0 which is canonically isomorphic to the short exact sequence 0→M(g,D,Dg)→M(D,Dg)→SA2 (M)→0. Wehavealreadydefined thegeometric invariant r(D)=dim(H2τ(A,R))−b2(M). Weobserve that the integern(D) is abyproductofmethodsof the informationgeometrywhile r(D) is a byproductofhomologicalmethods.However the split short exact sequence (****) leads to the equality n(D)= r(D). Here is a straight consequenceof the theoremwe justproved. Proposition7. Everyodd-dimensionalmanifoldMwithn(M)>0admitsageodesic symplectic foliation. Thedualistic relationofAmarihasanothersignificant impactonthedifferential topology. Definition35. Weconsideradualpair (M,g,∇,∇∗). Letψ∈M(∇,∇∗). (1) Theg-symmetricpart ofψ,ψ+ isdefinedby g(ψ+(X),Y)= 1 2 [g(ψ(X),Y)+g(X,ψ(Y))]. (2) Theg-skewsymmetricpart ofψ,ψ− isdefinedby g(ψ−(X),Y)= 1 2 [g(ψ(X),Y)−g(X,ψ(Y))]. Theorem11. Let(M,g,∇,∇∗)beadualpairwhere(M,g) isapositiveRiemannianmanifold.Letψ∈M(∇,∇∗). (1) Theg-symmetricpartψ+ is anelementM(∇,∇∗)whose rank is constant. (2) Wehave theg-orthogonaldecomposition TM=Ker(ψ+)⊕ im(ψ+). (3) If both∇and∇∗ are torsion free thenKer(ψ+)and Im(ψ+)are completely integrable. ADigression. Werecall that a statisticalmanifold is a torsion freedualpair (M,g,∇,∇∗). If thevector spaceM(∇,∇∗) isnon-trivial then itplaysanoutstandingrole in thedifferential topologyofM.Wedefineacanonicalmapof M(∇,∇∗) in the categoryof2-websby M(∇,∇∗) ψ→Ker(ψ+)⊕ im(ψ+). 186