Page - 228 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 This subsection ismainlydevoted to examples of stratifiedRiemannian foliations inanalyticmanifolds. Formoredetails about thoseobject the readersare referred to [46,63]. Theorem24. LetMbeanorientable compact analyticmanifold. LetCω(M2)be the space realvaluedanalytic functions defined in M2. There exists a canonical map of Cω(M×M) in the family of analytic stratified Riemannian foliations inM. TheIdeaofConstruction. Letdvbeananalyticvolumeelement inM. InMwefixananalytic torsion freeKoszul connection∇. Toa function f ∈Cω(M2)weassign the functionP∈Cω(M2) Pf(x,x∗)= exp[f(x,x∗)]∫ Mexp[f(x,x ∗∗)]dv(x∗∗) . Wemake the following identification X(M)=X(M)×0⊂X(M2). Theanalytic bilinear formgf ∈S2(M) isdefinedby [gf(x)](X,Y)=− ∫ M Pf(x,x∗)[∇2(log(Pf))(X,Y)](x,x∗)dv(x∗). The formgf has the followingproperties. (a) gf doesnotdependonthe choice of∇, (b) gf is symmetric andpositive semi-definite, (c) IfX isa sectionofKer(gf) thenLXgf =0. Conclusion. If rank(gf)= constant thengf is aRiemannian foliationas in [38–40,46]. If rank(gf) isnot constantweapply [62]. Therebygf is ananalytic stratifiedRiemannian foliation. Reminder. The ideaof the straficationof gf. Step0 TheopensubsetU0⊂Misdefinedby x∈U0 if f rank(gf(x))= max [x∗∈M] rank(gf(x∗)). Theclosedanalytic submanifoldF1⊂Misdefinedby F1=M\U0. 228