Page - 142 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 1. Introduction 1.1. TheNotation Throughout thepaperweuse thafollowingnotation.N is thesetofnonnegative integers,Z is theringof integers,R is thefieldofrealnumbers,C∞(M) is theassociativecommutativealgebraof realvaluedsmoothfunctions inasmoothmanifoldM. Let∇beaKoszulconnection inamanifoldM, R∇ is thecurvature tensorof∇. It isdefinedby R∇(X,Y)=∇X∇Y−∇Y∇X−∇[X,Y]. T∇ is the torsiontensorof∇. It isdefinedby T∇(X,Y)=∇XY−∇YX− [X,Y]. LetXbeasmoothvectorfield inM. LX∇ is theLiederivativeof∇ in thedirectionX · ι(X)R∇ is the innerproductbyX. ToapairofKoszulconnections (∇,∇∗)weassignthreedifferentialoperators. TheyaredenotedbyD∇∇∗,D∇ andD∇. (A.1) D∇∇∗ is afirst orderdifferential operator. It is defined in thevectorbundleHom(TM,TM). Itsvaluesbelongto thevectorbundleHom(TM⊗2,TM). (A.2) D∇ andD∇ are2ndorderdifferential operators. Theyaredefined in thevectorbundleTM. Theirvaluesbelongto thevectorbundleHom(TM⊗2,TM). LetXbeasectionofTMandletψ beasectionofT∗M⊗TM. Thedifferentialoperators justmentionedaredefinedby D∇∇ ∗ (ψ)=∇∗◦ψ−ψ◦∇, (1a) D∇(X)=LX∇− ι(X)R∇, (1b) D∇(X)=∇2(X). (1c) PartAof thispaper ispartiallydevotedto theglobalanalysisof thedifferentialequation FE(∇∇∗) : D∇∇∗(ψ)=O. The solutions to FE(∇∇∗) are useful for addressing the links between the KV homology, thedifferential topologyandthe informationgeometry. Thepurposeofa forthcomingpaper is thestudyof thedifferentialequations FE∗(∇) : D∇(X)=0, FE∗∗(∇) : D∇(X)=0. In theAppendixAto thispaperweoverviewtheroleplayedbythesolutions toFE∗∗(∇) in somestillopenproblems. 1.2. SomeExplicitFormulas Let x = (x1,...,xm) be a systemof local coordinate functions of M. In those coordinates the Christoffel symbols of both∇ and∇∗ are denoted by Γij:k and Γ∗ij:k respectively. We use those coordinate functions for presenting an element ψ ∈M(∇∇∗) as amatrix [ψij]. Thus by setting ∂i= ∂∂xi onehas ∇∂i∂j=Γij:k∂k. 142