Page - 198 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 To thosequestionswehaveobtainedsubstantial solutions in thecategoryof locallyflatmanifolds. Thecohomologicalmethodswehaveusedarebasedonthesplit exact shortcohomologysequence 0→H2dR(M,R)→H2τ(A,R)→SA2 (M)→0. 7.2. TheKVCohomologyandtheGeometryofKoszul TheHessianGeometry isabyproduct localvanishingTheorems in the theoryofKVcohomology. ThegeometryofKoszul isabyproductofglobalvanishingTheoreminthesamesitting. 7.3. TheKVCohomologyandthe InformationGeometry The categoryof finite dimensional statisticalmodels for ameasurable set (Ξ,Ω) contains the subcategoryoffinitedimensionalHessianmanifolds. Fromthisviewpoint theHessian information geometry is nothing but the exponential information geometry (i.e., the geometry of exponential familiesandtheirgenelarizations). Theframeworkfor thosepurposes iscloselyrelatedtovanishing Theoremsin the theoryofKVcohomology. AtanothersidecotangentbundlesofHessianmanifoldsareKaehlerianmanifolds. Thisaspect hasbeendiscussedbymanyauthors, see [52]andthebibliographyibidem. 7.4. TheDifferentialTopologyandthe InformationGeometry Alotofoutstanding linksbetweenthedifferential topologyandthe informationgeometryare basedon thedualistic relationofAmari. This approach leads to significant results in the category ofstatisticalmanifolds. Inastatisticalmanifold (M,g,∇,∇∗)wehave introducedthesplittingshort exact sequence 0→Ω∇2 (M)→M(∇,∇∗)→S∇2 (M)→0. Here (i)Ω∇2 (M) is thespaceof∇-geodesicsymplectic foliations inM; (ii)S∇2 (M) is thespaceof ∇-geodesicRiemannianfoliations inM. Thenumerical invariantn(∇)hasoutstanding impactsonthedifferential topologyofM. Seeour resultsonorthogonal2-websandonRiemanniansubmersionsonsymplecticmanifolds. 7.5. TheKVCohomologyandtheLinearizationProblemforWebs Ina locallyflatpair (M,g,∇,∇∗)weconsider theshortexact sequence O→Ω∇2 (M)→M(∇,∇∗)→S∇2 (M)→O. The linearizationofwebsof isadifficultoutstandingprobleminthedifferential topology. Gk[Ω∇2 (M)]stands for the family formedby [ω1,...,ωk]⊂Ω∇2 (M) suchthat dim[ΣjKer(ωj)]=min[dim(M),Σjdim(Ker(ωj))]. Gk[S∇2 (M)]stands for the family formedby [B1,...,Bk]⊂S∇2 (M) suchthat dim[ΣjKer(Bj)]=min[dim(M),Σjdim(Ker(Bj))]. (i) ElementsofGp[Ω∇2 (M)]areLINEARIZABLEsymplectick-webs. (ii) ElementsofGp[S∇2 (M)]areLINEARIZABLERiemanniank-webs. 198