Hodge theorem - Imported Knowlpedia Content

Statement

Let $(M,g)$ be a compact oriented Riemannian manifold without boundary. The Hodge theorem states that every class in the real de Rham cohomology group $H^k_{\mathrm{dR}}(M;\mathbb R)$ contains a unique harmonic $k$ -form. Equivalently, the map

\mathcal H^k(M,g)\longrightarrow H^k_{\mathrm{dR}}(M;\mathbb R), \qquad \alpha\longmapsto[\alpha],

is an isomorphism of finite-dimensional real vector spaces. The harmonic representative and the displayed isomorphism depend on $g$ , whereas de Rham cohomology itself depends only on the smooth manifold de Cataldo, Corollary 2.3.7.

Why existence and uniqueness hold

The Hodge orthogonal decomposition writes every smooth $k$ -form as

\omega=h+d\beta+\delta\gamma

with $h$ harmonic. If $\omega$ is closed, orthogonality forces its coexact component $\delta\gamma$ to vanish, so $[\omega]=[h]$ . This gives existence.

If two harmonic forms represent the same class, their difference is both harmonic and exact. Exact forms are orthogonal to harmonic forms, so the difference has zero $L^2$ -norm and vanishes. This gives uniqueness. The analytic input is ellipticity of the Hodge Laplacian on the compact manifold; the cohomological conclusion is not a purely formal property of the de Rham complex Wells, Chapter IV, §2.

Consequences

Because $\mathcal H^k(M,g)$ is the kernel of an elliptic operator on a compact manifold, it is finite-dimensional. The theorem therefore proves finite-dimensionality of $H^k_{\mathrm{dR}}(M;\mathbb R)$ . It also turns cohomology questions into equations for differential forms: a cohomology class is zero exactly when its harmonic representative is zero.

The Hodge star commutes with the Hodge Laplacian and maps harmonic $k$ -forms to harmonic $(n-k)$ -forms. Together with the theorem, this supplies the analytic realization of Poincaré duality for an oriented compact manifold.

Conventions and scope

“Compact” here includes absence of boundary only because that hypothesis is stated separately. On a manifold with boundary, absolute or relative boundary conditions lead to different Hodge theorems. On a noncompact manifold, unrestricted smooth harmonic forms generally do not provide unique representatives of ordinary de Rham cohomology; $L^2$ , compactly supported, or weighted variants require additional analytic hypotheses.

References

Mark Andrea A. de Cataldo, The Hodge Theory of Projective Manifolds, Imperial College Press, 2007. Author-hosted book PDF. Relevant: Theorem 2.3.3 and Corollary 2.3.7.
Raymond O. Wells, Jr., Differential Analysis on Complex Manifolds, 3rd ed., Springer, 2008. Springer DOI record. Relevant: Chapter IV, §2 on the Hodge theorem.