Page - 167 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 mayberewrittenas theexact sequence O→H2dR(M)→H2τ(A,R)→RF(M). Let (M,∇)bea locallyflatmanifoldwhoseKValgebra isdenotedbyA. Everyfinite family in H2τ(A,R) isa familyof∇-geodesicRiemannianfoliations. TheredoesnotexistanycriteriontoknowwhetheramanifoldsupportsRiemannianfoliations. Theexactcohomologysequenceswehavebeenperformingprovideuswithacohomologicalmethod forconstructingRiemannianfoliations in thecategoryof locallyflatmanifolds. This isan impactof the theoryofKVhomologyonDTO. In thenextsectionwewill introduceothernewingredientswhichhighlight the impactsonDTO of the informationgeometry. Further we will see that those new machineries from the information geometry have a homologicalnature. Anothermajorproblemsinthedifferentialtopologyisthelinearizationofwebs.Amongreferences are [41–43]. Definition22. Consider afinite family of distributionsDj⊂TM, j := 1,2,...,k. Thosedistributions are in generalpositionatapointx∈Mif for everysubset J⊂{1,2,...,k}onehas dim(∑ j∈J Dj(x))=min { dim(M),∑ j∈J dim(Dj(x)) } . Definition23. Ak-web inMisa familyof completely integrabledistributionswhichare ingeneralposition everywhere inM. AComment. The distributions belonging to a web may have different dimensions. An example of problem is the symplectic linearizationof lagrangian2-webs. Let (Dj, j :=1,2)bea lagrangian2-web ina2n-dimensional symplecticmanifold (M,ω). Thechallenge is the searchof special localDarbouxcoordinate functions (x,y)=(x1,...,xn,y1,...,yn). Those functionsmusthave threeproperties (1):ω(x,y)=Σjdxj∧dyj;(2) : The leavesofD1 aredefinedbyx= constant; (3): The leavesofD2 aredefined byy= constant. Definition24. Anaffineweb inanaffinespace is awebwhose leavesareaffine subspaces. Definition25. Aweb inam-dimensionalmanifold is linearizable if it is locallydiffeomorphic toanaffineweb inam-dimensional affine space. Example 1. In the symplecticmanifold (R2,exydx∧dy) one considers the lagrangian 2-webwhich isdefinedby L1={(x,y)|x= constant} , L2={(x,y)|y= constant} . This lagrangian2-webisnotsymplectic linearizable. Example 2. Wekeep (L1,L2) as in example.1. It is symplectic linearizable in (R2,(ex+ey)dx∧dy). Thelinearizationproblemforlagrangian2-websiscloselyrelatedtothelocallyflatgeometry[10,44,45]. 167