Seite - 232 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 AppendixA.4. The InformationGeometry Let (Ξ,Ω)beatransitivemeasurableandlet M=[E,π,M,D,p] beanobjectofGM(Ξ,Ω). LetgbetheFisher informationofM. Let {∇α, α∈R} bethe familyofα-connectionsofM.Wedefinethe followingnumerical invariant rb(M)=min [α∈R] { dim(M)−rb(∇α) } . TheoremA4. InM the followingassertionsare equivalent. (1) rb(M)=0, (2) M is anexponential family. CorollaryA1. AssumethatM is regular, viz g ispositivedefinite, then the followingassertionsare equivalent (1) rb(M)=0, (2) rb(M,g)=0, AppendixA.5. TheDifferentialTopologyof aRiemannianManifold ARiemannianmanifold(M,g) , (whoseLevi-Civitaconnectionisdenotedby∇∗), iscalledspecial if J∇∗ =0 TheoremA5. AspecialpositiveRiemannianmanifold(M,g)has the followingproperties (1) (M,g)admitsageodesicflatHessian foliation [F,g|F,∇∗]. (2) The leavesofF are theorbits of abi-invariantaffineCartan-Liegroup (G˜,∇˜). (3) Thebi-invariantaffineCartan-Liegroup(G˜,∇˜) isgeneratedbyaneffective infinitesimalactionofasimply connectedbi-invariantaffineLiegroup (G,∇). References 1. Faraut, J.;Koranyi,A.Analysis onSymmetricCones;OxfordUniversityPress:Oxford,UK,1994. 2. Koszul, J.-L.Déformationdesconnexions localementplates.Ann. Inst. Fourier1968,18, 103–114. (InFrench) 3. Vinberg,E.B.Thetheoryofhomogeneousconvexcones.Trans.Moscow.Math. Soc. 1963,12, 303–358. 4. Barbaresco,F.GeometricTheoryofHeat fromSouriauLieGroupsThermodynamicsandKoszulGeometry: Applications in InformationGeometry.Entropy2016,doi:10.3390/e18110386. 5. Nguiffo Boyom,M.; Wolak, R. Foliations in affinely flatmanifolds: InformationGeometric. Science of Information. InGeometricScienceof Information; Springer: Berlin/Heidelberg,Germany,2013;pp. 283–292. 6. Gindikkin, S.G.; Pjateckii, I.I.;Vinerg,E.B.HomogeneousKahlermanifolds. InGeometry ofHomogeneous BoundedDomains;Cremonese: Roma, Italy,1968;pp. 3–87. 7. Kaup,W.HyperbolischekomplexeRume.Ann. Inst. Fourier1968,18, 303–330. 8. Koszul, J.-L.Variété localementplateetconvexité.Osaka J.Math. 1965,2, 285–290. (InFrench) 9. NguiffoBoyom,M.TheCohomologyofKoszul-VinbergAlgebras.Pac. J.Math. 2006,225, 119–153. 10. NguiffoBoyom,M.RéductionsKahlériennesdanslesgroupesdeLieRésolublesetApplications.OsakaJ.Math. 2010,47, 237–283. (InFrench) 232