Page - 230 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Onanother side, thenotionofVanishingTheoremisuseful in linkingHSM(Ξ,Ω)withboth GM(Ξ,Ω)andLM(Ξ,Ω). (1) TheGlobalVanishingTheoremis the functor HSM(Ξ,Ω)→GM(Ξ,Ω). (2) TheLocalVanishingTheoremis the functor HSM(Ξ,Ω)→LM(Ξ,Ω). 13.5.HomologicalModels andHessianGeometry In the categoryHSM(Ξ,Ω) the Hessian functor is the functor fromHEHSM(Ξ,Ω) to the categoryof randonHessianmanifolds. Furthermore,everystructureofprobabilityspace (Ξ,Ω,p∗)givesrise toacanonical functor from HEHSM(Ξ,Ω) to thecategoryofHessianmanifolds. Thecanonical functor isdefinedby [E,π,M,D,Q]→ ∫ F p∗Q Acknowledgments: The author gratefully thanks the referees for number of comments and suggestions. Theircriticismshavebeenhelpful to improvepartsof theoriginalmanuscript. Conflictsof Interest:Theauthordeclaresnoconflictof interest. AppendixA Usuallytheappendixisdevotedtooverviewthenotionswhichareusedinapaper. Inthisappendix weannouncea fewoutstanding impactsofHessiandifferentialoperatorsofKoszulconnections. In the introduction a pair of Koszul connections (∇,∇∗) is used for defining three differentialoperators X→D∇(X)= ιXR∇−LX∇ ∀X∈Γ(TM). ThedifferentialoperatorD∇ isellipticandinvolutive in thesenseof theglobalanalysis [50,51,64]. LetJ∇bethesheafofgermofsolutions to theequation FE∗∗(∇) :D∇(X)=0. If∇ torsionfree thenFE∗∗(∇) isaLieequation. Thenonnegative integers rb(∇)and rb(M)aredefinedby rb(∇)= min [x∈M] { dim(J∇(x) } , rb(M)= min [∇∈SLC(M)] { dim(M)−rb(∇) } . HereSLC(M) is the convexsetof torsion freeKoszul connections inM. Weset the following notation:Rie(M) is thesetofRiemannianmetric tensors inM.LF(M) is thesetof locallyflatKoszul connection inM. Atonesideeveryg∈Rie(M)givesrise to themap LF(M) ∇→∇∗∈LC(M) which isdefinedby g(Y,∇∗XZ)=Xg(Y,Z)−g(∇XY,Z). 230