Page - 80 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 386 Dα0(q(X),q(Y))=α0(f(Y)q(X)) (140) Tosynthetize theresultprovedbyJean-LouisKoszul, ifαo andDαo are thevaluesofαandDα ato, then: αo (q(X))=Tr(f(X)) ∀X∈ g (141) Dαo (q(X),q(Y))= 〈q(X),q(Y)〉o=α0(f(X)q(Y)) ∀X,Y∈ g (142) Jean-LouisKoszulhasalsoproved that the innerproduct 〈., .〉onV, givenby theRiemannian metricgij, satisfies the followingconditions: 〈f(X)q(Y),q(Z)〉+〈q(Y), f(X)q(Z)〉= 〈f(Y)q(X),q(Z)〉+〈q(X), f(Y)q(Z)〉 (143) To make the link with Souriau model of thermodynamics, the first Koszul form α=DlogΦ=Tr(f(X))will play the role of the geometric heat Q and the second koszul form Dα=DdlogΦ= 〈q(X),q(Y)〉owillbe theequivalentofSouriau-Fishermetric that isG-invariant. Koszul theory iswiderand integrates“informationgeometry” in its corpus. Koszul [117–124] hasprovedgeneral results, for example: on a complexhomogeneous space, an invariant volume defineswith thecomplexstructure,an invariantHermitianform. If this space isaboundeddomain, thenthishermitianformispositivedefiniteandcoincideswith theclassicalBergmanmetricof this domain. Duringhis stay at Institute forAdvancedStudy inPrinceton, Koszul [117–124] has also demonstrated thereciprocal foraclassof complexhomogeneousspaces,definedbyopenorbitsof complexaffine transformationgroups.KoszulandVey[137,138]havealsodevelopedextendedresults with the followingtheoremforconnectedhessianmanifolds: Theorem3(Koszul-VeyTheorem).LetMbeaconnectedhessianmanifoldwithhessianmetric g. Suppose thatMadmits a closed1-formα such thatDα= gand there exists agroupGof affineautomorphismsofM preservingα: • If M/G isquasi-compact, thentheuniversal coveringmanifoldofMisaffinely isomorphic toa convexdomainΩofanaffinespacenotcontaininganyfull straight line. • IfM/Giscompact, thenΩ is a sharpconvexcone. Onthisbasis,KoszulhasgivenaLiegroupconstructionof ahomogeneouscone thathasbeendeveloped andapplied in informationgeometrybyShimaandBoyomin the frameworkofHessiangeometry. The results of Koszul arealso fundamental in the frameworkofSouriau thermodynamics. 7. SouriauLieGroupModelandKoszulHessianGeometryAppliedintheContextof InformationGeometryforMultivariateGaussianDensities We will enlighten Souriau model with Koszul hessian geometry applied in information geometry[117–124],recentlystudiedin[3,9,139].Wehavepreviouslyshownthatinformationgeometry couldbefoundedonthenotionofKoszul-VinbergcharacteristicfunctionψΩ(x)= Ω∗ e−〈x,ξ〉dξ, ∀x∈Ω whereΩ is a convex cone andΩ∗ the dual cone with respect to Cartan-Killing inner product 〈x,y〉=−B(x,θ(y)) invariant by automorphisms ofΩ, with B(., .) theKilling formand θ(.) the Cartan involution.WecandeveloptheKoszulcharacteristic function: ψΩ(x+λu)=ψΩ(x)−λ〈x∗,u〉+λ 2 2 〈K(x)u,u〉+ ... (144) withx∗= dΦ(x) dx ,Φ(x)=−logψΩ(x)andK(x)= d 2Φ(x) dx2 (145) 80