A Special Stokes's Theorem for Complete Riemannian Manifolds
Matthew P. Gaffney · Annals of Mathematics · 1954
(The superscripts on either forms or operators denote the degree and are omitted whenever clarity permits. Unless otherwise indicated all integrals are taken over the entire manifold.) For closed (compact) manifolds the integral on the left vanishes by Stokes's theorem; the equation then states that d and 5 are adjoint operators. From this adjointness follows the symmetry of the operator A and the orthogonality of the linear spaces in the decomposition theorem. For open manifolds the integral will vanish if the domains of d and a are restricted to C1 forms with compact carrier, and the formalism goes through as before. However, it is sometimes desirable to deal with larger domains. A natural