Page - 166 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Warning. Throughout thispaperaRiemannianmetric tensor inamanifoldMisanon-degenerate symmetricbilinear forminM. Apositivemetric tensor is apositivedefinitemetric tensor. In thenextweuse the followingdefinitionofRiemannian foliationandsymplectic foliation. Definition19. ARiemannian foliation is anelementg∈S2(M)whichhas the followingproperties (1.1) rank(g)= constant, (1.2) LXg = 0 ∀X ∈ Γ(Ker(g)). A symplectic foliation is a ( de Rham) closed differential 2-form ω whichsatisfies (2.1) rank(ω)= constant, (2.2) LXω=0 ∀X∈Γ(Ker(ω)). Warning. When g is positive semi-definite our definition is equivalent to the classical definition of Riemannian foliation [38–40]. The complete integrability of Ker(g) and the conditions to be satisfied by the holonomy of leaves are equivalent to theProperty (2.2). The set ofRiemannian foliations inamanifoldMisdenotedbyRF(M).The last theoremaboveyields the inclusionmap H2τ(A,R) H2dR(M) ⊂RF(M). Weoftenuse thenotionof affine coordinates functions ina locallyflatmanifold. Fornonspecialistswe recall twodefinitionsand the linkbetween them. Definition20. Anm-dimensional affinelyflatmanifold is anm-dimensional smoothmanifoldMadmitting a complete atlas { (Uj,φj) } whose local coordinate changes coincidewith affine transformations of the affine spaceRm. Wedenotedanaffineatlasby A= { (Uj,φj) } . Definition21. Anaffinely flat structure (M,A) and a locally flat structure (M,∇) are compatible if local coordinate functionsof (M,A)are solutions to theHessianequation ∇2xj=O Theorem 5. For every positive integer m the relation to be compatible with a locally flat manifold is an equivalencebetweenthecategoryofm-dimensional affinelyflatmanifoldsandthecategoryofm-dimensional locallyflatmanifolds. 4.2. TheGeneralLinearizationProblemofWebs In the frameworkRF(M) the inclusion H2τ(A,R) H2dR(M) ⊂RF(M) 166