Page - 181 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Ahighlightingconsequence is the identity D∗Xψ(Y)−ψ(DXY)=D∗Yψ(X)−ψ(DYX). (15) ToeveryvectorfieldXweassignthe linearmap Y→SX(Y)=D∗XY−ψ(DXY). ThenwerewriteEquations (14)and(15)as g(SX(Y),Z)+g(Y,SX(Z))=0, SX(Y)=SY(X). Weconsider the last identities in the frameworkof theSternberggeometry [50,51]. Since theapplication (X,Y)→SX(Y) is C∞(M)-bi-linear it belongs to the first Kuranishi-Spencer prolongation of the orthogonal Lie algebra so(g). TherebySX(Y)vanishes identically. Inotherwordswehave ψ∈M(D,D∗). Thisends theproofofTheorem AComment. The Sternberg geometry is the algebraic counterpart of the global analysis onmanifolds. It has been introduced by Shlomo Sternberg andVictor Guillemin. It is an algebraic model for transitive differential geometry[50]. In thatapproachtheRiemanniangeometry isageometryof typeone.Allof itsKuranishi-Spencer prolongationsare trivial. Theunique relevantgeometrical invariantof theRiemnniangeometry is the curvature tensor of theLevi-Civita connection. Except the connectionofLevi-Civita the othermetric connectionshave beenof few interest. Reallyothermetric connectionsmayhaveoutstanding impactson thedifferential topology. I shall address thispurpose ina forthcomingpaper. 5.1.4. TheHomologicalNatureof theEquationFE∇∇∗ Beforeproceedingweplantodiscusssomehomological ingredientswhichareconnectedto the differentialequation FE∇∇ ∗ : D∇∇ ∗ (ψ)=0. Letusconsideraduallyflatpair (M,g∗,D,D∗)andtheKVcomplex ψ∈C1,0=C1τ(A∗,A∗). Lemma4yields the followingcorollary. Corollary3. Wekeep thenotationused theprecedingsub-subsection.Givenagaugemorphismψ the following statementsare equivalent. (1) ψ⊗qψ is anexact (1,2)-cocyle, (2) ψ∈B1τ(A∗,A∗). Proof. Assumethat theassertion(2)holds. Thenthere isξ∈A∗ satisfyingthecondition ψ(X)=D∗Xξ. 181