An orthogonal splitting of the space of differential k‑forms on a compact oriented Riemannian manifold into exact, coexact, and harmonic parts: every form equals dα + δβ + γ with dγ = δγ = 0.

Principle

Using the Hodge Laplacian (Δ = dδ + δd) and elliptic theory on compact manifolds, forms decompose orthogonally into the ranges of d and δ and the finite-dimensional kernel of Δ (harmonic forms), yielding analytic representatives of de Rham cohomology classes.

Demonstration

On a compact oriented Riemannian 2‑sphere, the space of 1‑forms decomposes and the harmonic 1‑forms are trivial, reflecting the vanishing of the first de Rham cohomology group H^1(S^2)=0.

Misapplication

Assuming the same orthogonal decomposition holds without a Riemannian metric, on noncompact manifolds, or ignoring domain issues for unbounded operators can produce incorrect statements about harmonic representatives.

Consequence

Provides canonical representatives for cohomology, elliptic regularity for solutions of d- and δ-equations, and underpins many analytical tools linking topology and geometry (e.g., index theorems, Hodge theory).

Reversal

In purely algebraic de Rham theory without analytic structure one has cohomology groups but not canonical harmonic representatives or L^2 orthogonal decompositions provided by the metric and elliptic operator theory.

Boundary

Requires a smooth Riemannian metric and suitable compactness (or specified boundary conditions) to ensure ellipticity and finite-dimensional harmonic spaces; on noncompact manifolds one must impose growth or L^2 conditions.

Semantic Tension

Tension between algebraic-topological descriptions of cohomology (de Rham classes) and analytic, metric-dependent harmonic representatives; Hodge decomposition sits at their intersection but adds metric data.

Synthesis

Hodge decomposition is the metric-induced orthogonal splitting of differential forms into exact, coexact and harmonic parts on compact Riemannian manifolds, producing analytic representatives of topological cohomology and linking geometry to topology.