Page - 178 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Z˜1τ(A∗,A∗) B˜1τ(A∗,A∗) = Hes(M,D∗) Hyp(M,D∗). Another outstanding result of Koszul is the non rigidity of compact positive hyperbolicmanifolds [2]. Thenonrigiditymeans that everyopenneighborhoodof apositiveHyperbolic locallyflatmanifold (M,D,δKVθ) containanotherpositivehyperbolic locallyflat structurewhich isnot isomorphic to (M,D). Thisnonrigidity propertymaybeexpressedwith theMaurer–Cartanpolynomial functionPAMCof(M,D) ( see the local convexity theoremin [29]. In thenext sub-subsectionwerevisit thenotionofdualpair of foliationsas in [18]. 5.1.1. StatistcalReductions Thestatistical reductiontheoremis the followingstatement. Theorem8 ([18]). Let (M,g,D,D∗)beaduallyflatpair and letNbeasubmanifoldofM.AssumethatN is eitherD-geodesic orD∗-geodesic. ThenN inherits a structureofduallyflatpairwhich is either (N,gN,D,D∗N) or (N,gN,DN,D∗)). Thefoliationcounterpartofthereductiontheoremisofgreat interest inthedifferential topologyof statisticalmodelssee [18]. In theprecedingsectionswehaveaddressedacohomologicalaspectof this purpose. Thematterwillbemoreextensivelystudied ina forthcomingpaper (See theAppendixA). Inmathematicalphysicsaprincipalconnection1-formiscalledagaugefield. In thedifferentialgeometryaprincipalconnection1-forminabundleof linear frames iscalleda linearconnection. In the category of vector bundle Koszul connections are algebroid counterpart of principal connection1-forms. Ina tangentbundleTM,dependingonconcernsandneedsKoszulconnectionsmaycalled linear connectionsor lineargauges. Definition33. LetD,D∗ ∈LC(M). Avectorbundlehomomorphism ψ :TM→TM iscalledagaugehomomorphismof (M,D) in (M,D∗) if for all pairs ofvectorfields (X,Y)onehas D∗Xψ(Y)=ψ(DXY). The vector space of gauge homomorphisms of (M,D) in (M,D∗) is denoted byM(D,D∗). ThevectorspaceM(D,D∗) isnotaC∞(M)-module. 5.1.2.AUselfulComplex In this subsubsectionwefixaduallyflatpair (M,g,D,D∗)whoseKValgebrasaredenotedby A andbyA∗. The tangent bundleTM is endowed the structure leftmodule of the anchoredKV algebroids (TM,D,1)and (TM,D∗,1). Thismeans thateachof theKValgebrasAorA∗ is regardedas a leftmoduleof itself. Weconsider the tensorproduct C=C∗τ(A∗,A∗)⊗C∗τ(A,R). WeendowCwiththeZbi-grading. Ci,0=Ciτ(A∗,A∗)⊗C∞(M), C0,j=A∗⊗Cjτ(A,R), 178