Ivancevic_Applied-Diff-Geom

190 Applied Differential Geometry: A Modern Introductionbe considered as a system of second–order ODEs d 2 θ = 0, θ ∈ Ω n−1 (M)having a solution represented by Z n (M) = Ker(d).Analogously let M • denote the category of chains on the smooth manifoldM. When C = M • , we have the category S • (M • ) of generalizedchain complexes A • in M • , and if A n = 0 for n < 0 we have a subcategoryS• C (M • ) of chain complexes in M •A • : 0 ← C 0 (M)∂←− C 1 (M)∂←− C 2 (M) · · ·∂←− C n (M)∂←− · · · .Here A n = C n (M) is the vector space over R of all finite chains C on themanifold M and ∂ n = ∂ : C n+1 (M) → C n (M). A finite chain C suchthat ∂C = 0 is an n−cycle. A finite chain C such that C = ∂B is ann−boundary. Let Z n (M) = Ker(∂) (resp. B n (M) = Im(∂)) denote areal vector space of cycles (resp. boundaries) of degree n. Since ∂ n+1 ∂ n =∂ 2 = 0, we have B n (M) ⊂ Z n (M). The quotient vector spaceH C n (M) = Ker(∂)/ Im(∂) = Z n (M)/B n (M)is the n−homology group. The elements of H C n (M) are equivalence setsof cycles. Two cycles C 1 , C 2 belong to the same equivalence set, or arehomologous (written C 1 ∼ C 2 ), iff they differ by a boundary C 1 − C 2 =∂B). The homology class of a finite chain C ∈ C n (M) is [C] ∈ H C n (M).The dimension of the n−cohomology (resp. n−homology) group equalsthe nth Betti number b n (resp. b n ) of the manifold M. Poincaré lemmasays that on an open set U ∈ M diffeomorphic to R N , all closed forms(cycles) of degree p ≥ 1 are exact (boundaries). That is, the Betti numberssatisfy b p = 0 (resp. b p = 0) for p = 1, . . . , n.The de Rham Theorem states the following. The map Φ: H n ×H n → Rgiven by ([C], [ω]) → 〈C, ω〉 for C ∈ Z n , ω ∈ Z n is a bilinear nondegeneratemap which establishes the duality of the groups (vector spaces) H n and H nand the equality b n = b n .3.6.3.7 Hodge Star Operator and Harmonic FormsAs the configuration manifold M is an oriented ND Riemannian manifold,we may select an orientation on all tangent spaces T m M and all cotangentspaces T ∗ mM, with the local coordinates x i = (q i , p i ) at a point m ∈ M, in aconsistent manner. The simplest way to do that is to choose the Euclideanorthonormal basis ∂ 1 , ..., ∂ N of R N as being positive.Since the manifold M carries a Riemannian structure g = 〈, 〉, we havea scalar product on each T ∗ mM. So, we can define (as above) the linear