Page - 169 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Wehavepointedout thatcriterions fordecidingwhetherasmoothmanifoldadmitsRiemannian foliations (respectivelysymplectic foliations )aremissing.Ourpurpose is toaddress thisexistence problem in the category SLC whose objects are symmetric gauge structures. Such an object is a pair (M,∇) where∇ is a torsion free Koszul connection in M. The category of locally flat structureLF isasubcategoryofSLC. The theoryofKVhomologyisuseful fordiscussinggeodesic Riemannianfoliations in thecategoryLF. Ina locallyflatmanifold (M,D)wehavebeendealingwith thedecomposition H2τ(A,R)=H2dR(M)⊕SD2 (M). HereA is theKValgebraof (M,D). Letb2(M)be thesecondBettinumberofM.Wedefinethenumericalgeometric invariant r(D)by r(D)=dim(H2τ(A,R))−b2(M). Formally r(D) is thecodimensionofH2dR(M)⊂H2τ(A,R),viz r(D)=dim( H2τ(A,R) H2dR(M) ). Weconsider theexact sequences O→H2dR(M)→H2τ(A,R)→SA2 (M)→0 and →H2τ,e(A,R)→H2τ(A,R)→H2τ,res(A,R)→H3τ,e(R,R)→ Fromthoseexact sequences,onededuces theequality H2τ(A,R) H2dR(M) = H2τ,e(A,R) H2dR(M) . Thus r(D) is formally thedimensionofSA2 (M). Thepresentapproach leads to the followingstatement Proposition1. If r(D)>0 thenMadmitsnontrivialD-geodesicRiemannian foliations. Proof. LetBbeanonzeroelementofSA2 (M)andletDbethekernelofB. (1)Suppose that 0< rank(D)<dim(M) Therefore, (M,B) isaD-geodesicRiemannianfoliation. (2)Suppose that rank(D)=O. Then (M,B) isaRiemannianmanifoldtheLevi-Civitaconnectionofwhich isD. Therefore, the propositionholds. Beforeproceedingwedefinethreenumerical invariants r(M)=max{r(D)|D∈LC(M)} , s(M,A)=max { rank(B)|B∈SA2 (M) } , s(M)=max{s(M,A)|D∈LF(M)} . 169