Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 189This is written in terms of scalar products on M as〈C, dω〉 = 〈∂C, ω〉 ,where ∂C is the boundary of the p−chain C oriented coherently with C.While the boundary operator ∂ is a global operator, the coboundary operator,that is, the exterior derivative d, is local, and thus more suitable forapplications. The main property of the exterior differential,d 2 = 0 implies ∂ 2 = 0,can be easily proved by the use of Stokes’ formula〈∂ 2 C, ω 〉 = 〈∂C, dω〉 = 〈 C, d 2 ω 〉 = 0.The analysis of p−-chains and p−-forms on the finite–dimensionalsmooth manifold M is usually performed in (co)homology categories (see[Dodson and Parker (1997); Dieudonne (1988)]) related to M.Let M • denote the category of cochains, (i.e., p–forms) on the smoothmanifold M. When C = M • , we have the category S • (M • ) of generalizedcochain complexes A • in M • , and if A ′ = 0 for n < 0 we have a subcategoryS • DR (M• ) of the de Rham differential complexes in M •A • DR : 0 → Ω 0 (M)d ✲ Ω 1 (M)d ✲ Ω 2 (M) · · · (3.46)· · ·d ✲ Ω n (M)d ✲ · · · .Here A ′ = Ω n (M) is the vector space over R of all p−-forms ω on M (forp = 0 the smooth functions on M) and d n = d : Ω n−1 (M) → Ω n (M) is theexterior differential. A form ω ∈ Ω n (M) such that dω = 0 is a closed form orn–cocycle. A form ω ∈ Ω n (M) such that ω = dθ, where θ ∈ Ω n−1 (M), is anexact form or n–coboundary. Let Z n (M) = Ker(d) (resp. B n (M) = Im(d))denote a real vector space of cocycles (resp. coboundaries) of degree n.Since d n+1 d n = d 2 = 0, we have B n (M) ⊂ Z n (M). The quotient vectorspaceH n DR(M) = Ker(d)/ Im(d) = Z n (M)/B n (M)is the de Rham cohomology group. The elements of HDR n (M) representequivalence sets of cocycles. Two cocycles ω 1 , ω 2 belong to the sameequivalence set, or are cohomologous (written ω 1 ∼ ω 2 ) iff they differ by acoboundary ω 1 − ω 2 = dθ. The de Rham cohomology class of any formω ∈ Ω n (M) is [ω] ∈ HDR n (M). The de Rham differential complex (3.46) can