Knowlpedia

Theorem

Hodge decomposition on a compact Riemannian manifold

Smooth forms on a compact oriented Riemannian manifold split orthogonally into harmonic, exact, and coexact parts.

Statement

Let (M,g)(M,g) be a compact oriented without boundary. The Hodge decomposition in degree kk is the L2L^2-orthogonal direct sum

Ωk(M)=Hk(M,g)⊕dΩk−1(M)⊕δΩk+1(M),\Omega^k(M) =\mathcal H^k(M,g) \oplus d\Omega^{k-1}(M) \oplus\delta\Omega^{k+1}(M),

where Hk(M,g)=ker⁡Δ\mathcal H^k(M,g)=\ker\Delta is the space of , dΩk−1(M)d\Omega^{k-1}(M) consists of exact forms, and δΩk+1(M)\delta\Omega^{k+1}(M) consists of coexact forms. Thus every smooth kk-form has a unique harmonic, exact, and coexact component. The decomposition depends on gg and uses compactness and the absence of a boundary de Cataldo, Theorem 2.3.3.

Orthogonality and uniqueness

Exact and coexact forms are orthogonal because

⟨dβ,δγ⟩L2=⟨d2β,γ⟩L2=0.\langle d\beta,\delta\gamma\rangle_{L^2} =\langle d^2\beta,\gamma\rangle_{L^2}=0.

Harmonic forms are closed and coclosed, so they are orthogonal to both summands. These identities show uniqueness once existence is known. Existence is the analytic part: elliptic theory for the gives a Green operator on the of its finite-dimensional kernel Wells, Chapter IV, §2.

Equivalently,

Ωk(M)=ker⁡Δ⊕im⁡Δ.\Omega^k(M)=\ker\Delta\oplus\operatorname{im}\Delta.

Indeed, im⁡Δ\operatorname{im}\Delta is the orthogonal sum of the exact and coexact subspaces in the displayed decomposition.

Cohomological meaning

If ω\omega is closed, its coexact component vanishes, leaving

ω=h+dβ.\omega=h+d\beta.

The harmonic component hh is therefore the unique harmonic representative of [ω][\omega]. This recovers the and identifies Hk(M,g)\mathcal H^k(M,g) with de Rham cohomology.

The decomposition is sometimes compared with the Helmholtz decomposition of : the exact and coexact pieces generalize gradient and curl-type contributions. The analogy concerns orthogonal splitting; the actual objects here are differential forms in every degree.

Conventions and scope

This Riemannian decomposition should not be confused with the (p,q)(p,q)-decomposition of complex differential forms or the Hodge decomposition of cohomology on a compact . Those use complex type and additional . On manifolds with boundary, boundary conditions add further summands or modify the domains; on noncompact manifolds, closures of images and the chosen L2L^2 realization become essential.

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