Page - 165 - in Differential Geometrical Theory of Statistics

Entropy2016,18, 433 Warning. Weobserve that elementsofS2(M)mayberegardedas1-cochainsofAwithcoefficients in its leftmodule Ω1(M). By [29]wehave Z2τ(A,C∞(M))=SA2 (M). (8) Atanother sidewehave the cohomolgyexact sequence →H1KVres(A,V)→H2KVe(A,V)→H2KV(A,V)→H2KVres(A,V)→ (9) ByEquations (8) and (9)weobtain the inclusionmaps SA2 (M)⊂Z1KV(A,Ω1(M))⊂Z2KV(A,R). Mutatismutandis onealsohas SA2 (M)⊂Z1τ(A,Ω1(M)∩Z2τ(A,R). Remark1 (ImportantRemarks). Wegive somesubtle consequencesof (1). (R.1)Everyexact total2-cocycleω∈C2τ(A,R) is a skewsymmetricbilinear form.Vizonehas the identity ω(X,X)=0 ∀X∈A. (R.2)EverysymmetricKV2-cocycle g∈Z2KV(A,R) is locallyanexactKVcocycle, viz inaneighbourhoodof everypoint there exists a local sectionθ∈Ω1(M) such that g= δKVθ. (R.3)Everysymmetric total2-cocycle is a left invariant cochain,viz Z2τ(A,R)∩S2(M)=SA2 (M). By(R.1)and (R.3)weobtain the inclusionmap SA2 (M)⊂H2τ(A,R). LetH2dR(M)be the secondcohomologyspaceof thedeRhamcomplexofM.The followingtheoremisuseful for relating the totalKVcohomologyandthedifferential topology. Theorem4. [29]There exists a canonical linear injectionofH2dR(M) inH 2 τ(A,R) such that H2τ(A,R)=H2dR(M)⊕SA2 (M) Thetheoremabovehighlightsafruitful linkbetweenthetotalKVcohomologyandthedifferential topology. We are particularly interested in D-geodesic Riemannian foliations in a locally flat manifold (M,D). 165