Page - 185 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 (b) Suppose that themanifoldMiscompactandsuppose thatg∈S∇2 (M) isapositiveRiemannianfoliation,viz g(X,X)≥0 ∀X. Thenthe theoryofMolinomaybeapplied to studyg [38]. Therefore, the structure theoremofMolino tells that ggives rise toaLie foliationwhoses leavesare theadherences leavesof g¯ [39]. (c) In theprincipal bundle offirst order linear framesofMtheanalogof aKoszul connection∇ is aprincipal connection1-formωwhosecurvature formisdenotedbyΩ. Thecurvature formis involvedinconstructing characteristic classes ofM, (the formalismofChern-Weill.) At another side∇-geodesic Riemannian foliations and∇-geodesic symplectic foliations are Lie algebroids. Theyhave their extrinsic algebraic topology. Inparticular the theoryof integrable systemsmay be performed in every leaf of ω ∈Ω∇2 (M). If one considers the α-connections in a statisticalmodel thosenewinsightsmaybeof interest. Here isan interpretationof thenumerical invaraintn(∇). Theorem 10. We assume there exists ∇ ∈ SLC(M) whose linear holonomy group H(∇) is neither an orthogonal subgroup nor a symplectic subgroup. If n(∇) > 0 then the manifold M admits a couple (Fr,Fs) formedbya∇-geodesicRiemannian foliationFr anda∇-geodesic symplectic foliationFs. Proof. LetgbeaRiemannianmetric tensor inM. Since n(∇)≤dim(S∇2 (M)(x)) forallx∈M thereexistsψ∈M(∇,∇(g)suchthat qψ∈S∇2 (M)\{0} , ωψ∈Ω∇2 (M)\{0} . Theassumptionthat theholonomygroupH(∇) isneitherorthogonalnorsymlectic implies Ker(qψ) =0, Ker(ωψ) =0. BothdistributionsKer(qψ)andKer(ωψ)are∇-geodesic. Since∇ is torsionfree thosedistributions arecompletely integrable. ForallX∈Γ(Ker(qψ))wehave LXqψ=0. Mutatismutandis forallX∈Γ(Ker(ωψ))wehave LXωψ=0. Fromthosepropertiesweconclude (M,Ker(qψ),qψ) is a ∇-geodesic Riemannian foliation, (M,Ker(ωψ),ωψ) is a ∇-geodesic symplectic foliation. Thetheoremisproved. 185