Page - 164 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 In Part B of this paper we will be addressing this fundamental problem. A reading of the McCullaghpaperwouldbeuseful fordrawingacomparisonbetweenourapproachand[15,16,30]. 4.TheKVTopologyofLocallyFlatManifolds 4.1. TheTotalCohomologyandRiemannianFoliations InthissectionwefocusontheKValgebroidswhicharedefinedbystructuresof locallyflatmanifolds. Tofacilitateacontinuousreadingof thispaperwerecall fundamentalnotionswhichareneeded. Definition18. A locally flatmanifold is a pair (M,D). HereD is a torsion freeKoszul connectionwhose curvature tensorRDvanishes identically. Thepair (M,D)definesaKoszul-Vinbergalgebroid A=(TM,D,1) Theanchormapis the identitymapofTM. Themultiplicationofsections isdefinedbyD, viz X ·Y=DXY forallX,Y∈X(M). TheKValgebraof (M,D) is thealgebra A :=(X(M),D). The cotangent bundle T∗M is a left module of the KV algebroid (TM,D,1). For every (X,Y,θ)∈X(M)×X(M)×Γ(T∗M) thedifferential1-formX ·θ isdefinedby [X ·θ](Y)= [d(θ(Y))](X)−θ(X ·Y). Intherighthandmemberof theequalityaboved(θ(Y)) is theexteriorderivativeof therealvalued functionθ(Y). LetS2(T∗M)bethevectorbundleofsymmetricbi-linear forms inM. ThevectorspaceofsectionsofS2(T∗M) isdenotedbyS2(M),viz S2(M)=Γ(S2(T∗M)). Thevector spaceS2(M) is a leftmoduleof theKValgebraA. The left actionofA inS2(M) is definedby (X ·g)(Y,Z)= [dg(Y,Z)](X)−g(X ·Y,Z)−g(Y,X ·Z). Weput Ω1(M)=Γ(T∗M). TheT∗M-valuedcohomologyof theKValgebroid (TM,D,1) isbut thecohomologyofAwith coefficients inΩ1(M). TheKVcohomologyandthe total cohomologyaredenotedby H∗KV(A,Ω1(M)), H∗τ(A,Ω1(M)). 164