Page - 197 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 BythevirtueofTheorem3as in [2], forboth (M,g,∇)and (M,g,∇∗)beinggloballyhyperbolic it isnecessary that [g]=0∈H2KV(A,R) and [g]=0∈H2KV(A∗,R). Everychoiceof localdifferential1-formsωandω∗givesrise toauniquepairof local similarity vectorfields (H,H∗), viz ∇XH=X, ∇∗XH∗=X forallvectorfieldsX. ThevectorfieldsHandH∗areRiemanniangradientsofω∗andofω respectively. Thismeans that thosedifferential1-formsaredefinedby ω= ιHg, ω∗= ιH∗g. Here ιH stands for the innerproductbyH, viz ιHg(X)= g(H,X). Thisshortdiscussions leadto the followingstatement Theorem17. Let (M,g,∇,∇∗)bea compactduallyflatpairwhoseKValgebrasaredenotedbyAandbyA∗. The followingassertionsare equivalent (1) The locallyflatmanifold (M,∇) ishyperbolic, (2) the locallyflatmanifold (M,∇∗)admitsaglobal similarityvectorfieldH∗. Definition41. Let∇∈LC(M). (1) Thegauge structure (M,∇) is called a similarity structure if∇ admits a global similarity vector field H∈X(M). (2) Adualpair (M,g,∇,∇∗) is a similaritydualpair if either (M,∇)or (M,∇∗) is a similarity structure. Thefollowingproposition isastraightforwardconsequenceofourdefinition. Proposition10. Ifagaugestructure(M,∇) isflatandis locallyasimilaritystructure, then(M,∇) isa locally flatmanifold 7. SomeHighlightingConclusions In thisPartAouraimhasbeentoaddressvariouspurposes involvingthetheoryofKVhomology. Doingthatwehavepointedsignificant relationshipsbetweensomemajor topics inmathematicsand the local informationgeometry. Thoserelationshipsmightbesourcesofnewinvestigations. Wesummarizesomerelevant relationshipswehavebeenconcernedwith. 7.1. TheTotalKVCohomologyandtheDifferentialTopology Wehaveaddressed the existenceproblemfor a fewmajor objects of thedifferential topology. Riemannianfoliationsandsymplectic foliations.Riemannianwebsandtheir linearizationproblem. 197