Page - 172 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Infinal, near the p∗O∈M∗ theweb (Ker(ψ∗),im(ψ∗)) isdiffeomorphic to theaffinewebwhose leavesareparallel toadecomposition R m=Vm−s×Vs. HereVm−s andVs arevectorsubspacesofRm.Theirdimensionsarem−sand s. Conclusion. Thereexistsaunique linear transformationφofRm suchthat φ(Rm−s×0)=Vm−s, φ(0×Rs)=Vs. Therebythere isa localdiffeomorphismΦofM inM∗ subject to therequirements Φ(p0)= p∗0, (x0,y0∗)◦Φ=(x,y∗). ThedifferentialofΦ isdenotedbyΦ∗.Weexpress thepropertiesaboveby Φ(p0)= p∗0, Φ∗[Ker(ψ),im(ψ)]= [Ker(ψ∗),im(ψ∗)]. Thisends thesketchofproofofTheorem. In thenextweuse the followingdefinitions. Definition30. Afinite family { BJ, J⊂Z }⊂SA2 (M) is ingeneralposition if thedistributions { Ker(Bj), j∈ J } are ingeneralposition. Thefollowingstatement isastraightcorollaryof the theoremwejustdemonstrated. Proposition2. Ina locallyflatmanifold (M,D)withr(D)>0 everyfinite family ingeneralpositiondefinea linearizableRiemannianweb. 4.4. TheKVCohomologyandDifferentialTopologyContinued We have seen how the total cohomology and linearizable Riemannian webs are related. Moreprecisely the theoryofKVcohomologyprovidessufficientconditions fora locallyflatmanifold admitting linearizableRiemannianwebs. Thatapproach isbasedonthesplit exact sequence 0→H2dR(M)→H2τ(A,C∞(M))→SA2 (M)→0. 4.4.1.Kernelsof2-CocyclesandFoliations Notall locallyflatmanifoldsadmit locallyflat foliations. Theexistenceof locallyflat foliations is relatedto the linearholomnomyrepresentation,viz the linearcomponentof theaffineholonomy representationof the fundamentalgroup.Via thedevelopingmaptheaffineholonomyrepresentation is conjugate to the natural action of the fundamental group in the universal covering. The KV homology isuseful for investigating theexistenceof locallyflat foliations. Tosimplifywework in the analyticcategory. Soourpurposes includesingular foliations. 172