The Hodge Theorem

Abstract

Throughout this chapter, M will be a compact oriented Riemannian manifold of dimension n unless otherwise indicated. We will see that the ordinary Laplacian $( - 1)\sum\limits_i {{\partial ^2}/\partial {x_i}^2}$ has a generalization to an operator Δ on differential forms, known as the Laplace-Beltrami operator. Our main objective in this chapter is a proof of the Hodge decomposition theorem, which says that the equation Δω = α has a solution co in the smooth p-forms on M if and only if the p-form α is orthogonal (in a suitable inner product on E ^p(M)) to the space of harmonic p-forms (those for which Δφ = 0). From the Hodge decomposition theorem we will conclude that there exists a unique harmonic form in each de Rham cohomology class. As another simple application we will obtain the Poincaré duality theorem for de Rham cohomology and, from it, the Poincaré duality theorem for real singular cohomology. To prove the Hodge theorem, we shall give a complete self-contained exposition of the local theory of elliptic operators, using Fourier series as our basic tool. The eigenfunctions of the Laplace-Beltrami operator and their use in a proof of the Peter-Weyl theorem are discussed in the exercises at the end of this chapter.