Page - 188 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Inastatisticalmanifold (M,g,∇,∇∗)weconsidera∇-geodesicRiemanniank-web [g˜j∈S∇2 (M); j : 1,...,k]. Thedistributions Dj=Ker(g˜j) are ingeneralposition.Weconsider the familyΨ+j ∈Σ(g)definedby g(Ψ+j (X),Y)= g˜j(X,Y). Weget the familyofg-orthogonal2-websdefinedby TM=Ker(Ψ+j )⊕ im(Ψ+j ). Mutatismutandiswecanconsidera∇-geodesicsymplecticweb [ωj∈Ω∇2 (M); j :=1,...,k]. There is familyofg-skewsymmetricgaugemorphismsΨ−j definedby g(Ψ−j (X),Y)=ωj(X,Y). Since∇and∇∗are torsionfreeKer(Ψ−j )and im(Ψ−j )arecompletely integrable. Sinceg ispositive definiteweget the2-web TM=Ker(Ψ−j )⊕ im(Ψ−j ). Furtherevery leafof im(Ψ−j ) isasymplecticmanifold. Definition37. Letω∈Ω∇2 (M)beanon trivial symplectic foliation ina statisticalmanifold (M,g,∇,∇∗). ConsiderΨ−∈M(∇,∇∗)definedby g(Ψ−(X),Y)=ω(X,Y). Thedifferential2-formω is called simple if the foliationKer(Ψ−) is simple. Inastatisticalmanifold (M,g,∇,∇∗)everynontrivial symplecticweb [ωj; j :=1,...]⊂Ω∇2 (M) givesrisetoafamilyofg-orthogonal2-webs. SointhisapproachtheroleplayedbyS∇2 (M) issimilarto theroleplayedbyΩ∇2 (M).OurconstructionofRiemannianwebsandsymplecticwebs in thecategory ofduallyflatpairsholds in thecategoryofstatisticalmanifolds. Atoneside, inaduallyflatpair(M,g,D,D∗)ourapproachyields linearizablewebs. Thisproperty doesnothold inall statisticalmanifolds. Atanotherside, inastatisticalmanifold (M,g,∇,∇∗)aRiemannianweb [g˜j, j∈ J]⊂S∇2 (M) orsymplecticweb [ωj, j∈ J]⊂Ω∇2 (M) givesrise to familiesoforthogonal2-webs. Thispropertydoesnothold inallduallyflatpairs. 188