Page - 143 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 WefocusonFE(∇∇∗)andofFE∗∗(∇). Theyareequivalent to the followingsystemofpartial differentialequations [Sij:k] : ∂ψkj ∂xi − ∑ 1≤ ≤m (Γij: ψk −Γ∗i :kψ j)=0, [Θkij(X)] : ∂2Xk ∂xi∂xj +∑ α [Γkiα ∂Xα ∂xj +Γkjα ∂Xα ∂xi −Γα ij∂Xk∂xα ]+∑ α [ ∂Γkjα ∂xi +∑ β [ΓβjαΓ k iβ−ΓβijΓkβα]]Xα=0. InPartAweaddressthelinksbetweenthefollowingtopicsDTO,HGE,IGEandENT.Thosetopics arepresentedasverticesofasquarewhosecentre isdenotedbyKVH. (1) DTOstands forDifferentialTOpology. InDTO,FWEstands forFoliationsandWEbs. (2) HGE stands for Hessian GEometry. Its sources are the geometry of bounded domains, the topology of bounded domains, the analysis in bounded domains. Among the notable references are [1–3]. Hessiangeometryhas significant impacts on thermodynamics, see [4,5], About the impactsonotherrelatedtopics thereadersarereferredto [6–12]. (3) IGEstands for InformationGEometry. That is thegeometryofstatisticalmodels.Moregenerally its concern is thedifferential geometryof statisticalmanifolds. The rangeof the information geometry is large [13].Currently, the interest in informationgeometry is increasing. Thiscomes fromthe linkswithmanymajorresearchdomains [14–16].Weaddresssomesignificantaspects of those links.Non-specialist readersarereferredtosomefundamental referencessuchas [17,18]. Seealso[4,19–23]. Theinformationgeometryalsoprovidesaunifyingapproachtomanyproblems indifferentialgeometry,see[21,24,25]. Theinformationgeometryhasalargescopeofapplications, e.g.,physics, chemistry,biologyandfinance. (4) ENTstands forENTropy. Thenotionofentropyappears inmanymathematical topics, inPhysics, in thermodynamics and inmechanics. Recent interest in the entropy functionarises from its topological nature [14]. In Part B we introduce the entropy flow of a pair of vector fields. TheFisher information is thendefinedas theHessianof theentropyflow. (5) KVHstandsforKVHomology. ThetheoryofKVhomologywasdevelopedin[9]. Themotivation was the conjectureofM.Gerstenhaber in the categoryof locallyflatmanifolds. In thispaper weemphasize othernotable rolesplayedby the theoryofKVhomology. It is alsouseful for discussingaproblemraisedbyJohnMilnor in [26]. TheconjectureofGerstenhaber is the followingclaim. Everyrestricted theoryofdeformationgenerates itsproper cohomology theory [27]. Looselyspeaking, inarestrictedtheoryofdeformationonehas thenotionofboth infinitesimal deformationand trivialdeformation. Thechallenge is the search for a cochain complexadmitting infinitesimaldeformationsascocycles. In thepresentpaper,KVH isuseful foremphasizingthe links betweentheverticesDTO,HGE, IGEandENT. That isourreasonfordevotingasectiontoKVH. Warning. Wepropose tooverviewthe structureof thispaper. The readers are advised to read thispaperas through it wereawanderaroundthevertices of the square“DTO-HGE-IGE-ENT”.Thus,dependingonhis interests and his concernsareader couldwalk several timesacross the samevertex. For instance the informationgeometry appears inmanysections,dependingonthepurposeandontheaims. 1.3. Thecontentof thePaper Thispaper isdividedintoPartAandPartB. PartA:Sections1–7. Section1 is the Introduction. Section2 isdevotedtoalgebroids,modulesofalgebroidsandthe theoryofKVhomologyof theKoszul-Vinbergalgebroids. To introduce theKVcohomologywehave 143