Ivancevic_Applied-Diff-Geom

186 Applied Differential Geometry: A Modern Introductionexact if this containment is equality. In the case of the de Rham complex(3.42), exactness means that a closed p−form ω, meaning that dω = 0, isnecessarily an exact p−form, meaning that there exists a (p − 1)−form θsuch that ω = dθ. (For p = 0, it says that a smooth function f is closed,df = 0, iff it is constant). Clearly, any exact form is closed, but the converseneed not hold. Thus the de Rham complex is not in general exact. Thecelebrated de Rham Theorem states that the extent to which this complexfails to be exact measures purely topological information about the manifoldM, its cohomology group.On the local side, for special types of domains in Euclidean space R m ,there is only trivial topology and we do have exactness of the de Rhamcomplex (3.42). This result, known as the Poincaré lemma, holds for star–shaped domains M ⊂ R m : Let M ⊂ R m be a star–shaped domain. Thenthe de Rham complex over M is exact.The key to the proof of exactness of the de Rham complex lies in theconstruction of suitable homotopy operators. By definition, these are linearoperators h : Ω p → Ω p−1 , taking differential p−forms into (p − 1)−forms,and satisfying the basic identity [Olver (1986)]ω = dh(ω) + h(dω), (3.44)for all p−forms ω ∈ Ω p . The discovery of such a set of operators immediatelyimplies exactness of the complex. For if ω is closed, dω = 0, then(3.44) reduces to ω = dθ where θ = h(ω), so ω is exact.3.6.3.4 Stokes Theorem and de Rham CohomologyStokes Theorem states that if α is an (n − 1)−form on an orientablen−manifold M, then the integral of dα over M equals the integral of αover ∂M, the boundary of M. The classical theorems of Gauss, Green, andStokes are special cases of this result.A manifold with boundary is a set M together with an atlas of charts(U, φ) with boundary on M. Define (see [Abraham et al. (1988)]) theinterior and boundary of M respectively asInt M = ⋃ Uφ −1 (Int (φ(U))) , and ∂M = ⋃ Uφ −1 (∂ (φ(U))) .If M is a manifold with boundary, then its interior Int M and its boundary∂M are smooth manifolds without boundary. Moreover, if f : M → Nis a diffeomorphism, N being another manifold with boundary, then f in-