Page - 190 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Let [γ],[γ∗]∈π1(x∗). Thecompositionofpaths leads to the formula Q([γ].[γ∗])= f([γ])Q([γ∗])+Q([γ]). The last relation shows that the pair (f,Q) is an affine representation of π1(x∗) in Tx∗M. Thisrepresentationiscalledtheholonomyrepresentationofthelocallyflatmanifold(M,D). Thegroup (f,Q)(π1(x∗)) is calledtheaffineholonomygroupof (M,D). Its linearcomponent f(π1(x∗)) is called the linearholonomygroupof (M,D). Definition38 ([2]). Anm-dimensional locallyflat structure (M,D) is calledhyperbolic if Q˜(M˜) is a convex domainnot containinganystraight line inTx∗M. Definition39. Alocallyflatmanifold (M,D) is complete if themapQ˜ is adiffeomorphismontoTx∗M. Amongthemajoropenproblemsin the theoryofspacegroups is theconjectureofMarkus. ConjecureofMarkus: a compact locallyflatmanifold (M,D)whose linearholonomygroup is unimodular iscomplete. BeforepursuingwerecallKVcohomologicalversionofTheorem3as in [2]. Theorem13 ([2]). Anecessary condition for a locallyflatmanifold (M,D)beinghyperbolic is that (M,D) carries apositiveHessianstructurewhosemetric tensor is exact in theKVcomplexof (M,D). This condition is sufficient ifMis compact. Wehavealreadymentionedanotableconsequenceof this theoremofKoszul. In thecategoryof compact locallyflatmanifolds thesubcategoryofhyperbolic locallyflat structures is thesamethingas thecategoryofpositiveexactHessianstructures. SoThegeometryofcompacthyperbolic localflat manifolds isanappropriatevanishingtheorem. Intheprecedingsections thefamilyofHessianmetrics ina locallyflatmanifold(M,D) isdenoted byHes(M,D). Therefore,H+es(M)standsfor thesub-familyofpositiveHessianmetric tensors. It is aconvexcone inRie(M). WehavealreadyusedtheKVcomplexforexpressingtheduallyflatness.Morepreciselylet(M,D) beafixedlocallyflatmanifoldwhoseKValgebra isnotedA. Adualpair(M,g,D,D(g)) isduallyflat if andonlyifg∈Z2KV(A,R).Therefore,everyduallyflatpair(M,g,D,D∗)yieldstwocohomologyclasses [g]D∈H2KV(A,R), [g]D∗ ∈H∗KV(A∗,R). Thereby,wecanusemethodsof the informationgeometry for rephrasingTheorem3as in [2]. Theorem14. Anecessary condition for (M,D) being hyperbolic is the existence a positive dually flat pair (M,g,D,D∗) such that [g]=0∈H2KV(A,R). IfMis compact this (vanishing ) condition is sufficient. About the geometry ofKoszul the non specialists are referred to [2,7,8,52] and bibliography therein [12]. Aboutapplicationsof thegeometryofKoszul thereadersarerefereedto [12,13,54,55]. Aboutrelationshipsbetweenthetheoryofdeformationandthetheoryofcohomology, thereaders arereferredto [9,27,56]. 190