Page - 184 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Lemma5. The integern(D0)doesnotdependonthe choice of g∈Rie(M). AnIdea. Wefixametric tensor g. For everyg∗ ∈Rie(M) there is auniqueg-symmetricvectorbundlemorphism φ∈Σ(g) such that g∗(X,Y)= g(φ(X),Y). Therefore,wehave φ−1◦M(D0,Dg)=M(g∗,D0,Dg∗), φ−1M(g,D0,Dg)=M(g∗,D0,Dg∗). Nowonedefines thenumerical invariantn(M). Definition34. n(M)=max{n(D)|D∈SLC(M)} . Given a Koszul connection∇ the vector space of∇-parallel differential 2-forms is denoted byΩ∇2 (M). Everydualpair (M,g,∇,∇∗)givesrise to the linear isomorphisms (1) : M(∇,∇∗) M(g,∇,∇∗) [ψ]→ qψ∈S ∇ 2 (M), (2) : M(g,∇,∇∗) ψ→ωψ∈Ω∇2 (M). The isomorphism(1)derives fromthe linearmap (1∗) : ψ→ qψ(X,Y)= 12[g(ψ(X),Y)+g(X,ψ(Y))]. The isomorphism(2) isdefinedby (2∗) : ψ→ωψ(X,Y)= 12[g(ψ(X),Y)−g(X,ψ(Y))]. Proposition6. Let (M,g,∇,∇∗)beadualpair. The inclusionmap M(g,∇,∇∗)→M(∇,∇∗) induced the split short exact sequence (∗∗∗∗∗) : 0→Ω∇2 (M)→M(∇,∇∗)→S∇2 (M)→0. Reminder. According our previous notation elements ofΩ∇2 (M) are∇-geodesic symplectic foliations. Those of S∇2 (M)are∇-geodesicRiemannian foliations. Thusweapplymethodsof the informationgeometry to relate the gaugegeometryandthedifferential topolgy. Digressions. Ourconstructionmayopen tonewdevelopments.Hereare someunexploredperspectives. (a) A∇-geodesic symplectic foliationω∈Ω∇might carryricher structures suchasKahlerianstructures. 184