Page - 168 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Example 3. What about the linearization of the 3-web defined by L1 := {(x= constant,y)} , L2 :={(x,y= constant)} ,L3 :={e−x(x+y)= constant} ,(x,y)∈R2. Uptotodaythequestionas towhether it is linearizable is subject tocontroversies, see [42]and references therein. 4.3. TheTotalKVCohomologyandtheDifferentialTopologyContinued Weimplement theKVcohomologytoaddresssomeopenproblemsinthedifferential topology. Forourpurposewerecalla fewclassicalnotionswhichareneeded. Definition26. Ametricvectorbundle overamanifoldMisavectorbundleV endowedwithanon-degenerate innerproduct<v,v∗> . AKoszulconnection inavectorbundleV isabilinearmap Γ(TM)×Γ(V) (X,v)→∇Xv∈Γ(V) whichhas theproperties ∇fXv= f∇Xv∀v,∀f ∈C∞(M), (10a) ∇Xfv=df(X)v+ f∇Xv∀v,∀f ∈C∞(M). (10b) Definition27. Ametric connection in (V,<−,>) is aKoszul connection∇whichsatisfies d(<v,v∗>)(X)−<∇Xv,v∗>−<v,∇Xv∗>=0. Definition28. Let (M,D)bea foliation in theusual sense, vizDhasconstant rankand is in involution. (1): (M,D) is transversallyRiemannian if there exists a g∈S2(M) such that D=Ker(g). (2): (M,D) is transversally symplectic if there exists a (deRham)closeddifferential2-formω such that D=Ker(ω) AtransversallyRiemannianfoliationandatransversallysymplectic foliationaredenotedby (D,g), (D,ω). Definition 29. Given aKoszul connection∇, a transversally Riemannian foliation (D,g) (respectively a transversally symplectic foliation (D,ω)) is called∇-geodesic if ∇g=0, ∇ω=0 The notions of transversally Riemannian foliation and transversally symplectic foliation are weaker than thenotionofRiemannian foliation and symplectic foliation. However if∇ a torsion freeKoszulconnectionevery∇geodesic transversallyRiemannianfoliation isaRiemannianfoliation. Every∇-geodesic transversallysymplectic foliation isasymplectic foliation. For thegeneral theoryofRiemannianfoliations thereadersarereferredto [39,40,46], seealso the monograph[38]andthereferences therein. 168