Page - 171 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 isa2-webinM. In (M∗,D∗,g∗)weobtainsimilar2-web (Ker(ψ∗),im(ψ∗)). By thechoiceofBandB∗wehave rank(Ker(ψ))= rank(Ker(ψ∗))=m−s. Nowweperformthefollowingarguments. (a): The foliationB isD-geodesic. Inaneighbourhoodofeverypoint p0∈ (M,D)welinearizeB bychoosingappropriate localaffinecoordinate functions (x,y)=(x1,...,xm−s,y1,...,ys). The leavesofKer(ψ)aredefinedby y= constant. Therebythose leavesare locally isomorphic toaffinesub-spaces. Stepb Thedistribution im(ψ) is D˜-geodesic. Therefore,near thesamepoint p0∈ (M,D˜)welinearize im(ψbychoosingappropriate localaffinecoordiante functions (x∗,y∗)=(x∗1,...,y ∗ 1,...). The leavesof im(ψ)aredefinedby x∗= constant. Thusnear p0 the foliationdefinedbym(ψ) is isomorphic toan linear foliation. Stepc Byboth step a and stepbwe choose aneighbourhoodof p0 which is thedomainof systems of appropriate local coordinate functions (x,y) and (x∗,y∗). From those data we pick the local coordinate functions (x,y∗)=(x1,...,xm−s,y∗1,...,y ∗ s). Sowelinearize the2-web (Ker(ψ),im(ψ))withthe local coordinate functions (x,y∗). (Ker(ψ),im(ψ)). Thusnear the p0 the 2-web (Ker(ψ),im(ψ)) is isomorphic to the linear 2-web (L1,L2)which is definedinRmby R m=Rm−s×Rs. Stepd Atapointp∗0 inM∗weperformtheconstructionasinstepaandinstepsbandc, thenwelinearize (Ker(ψ∗),im(ψ∗))bychoosingappropriate local coordinate functions (x0,y0∗)=(x01,...,x 0 m−s,y0∗1 ,...,y 0∗ s ). 171