Hodge Theory - Core Definitions

Hodge theory establishes a deep connection between the topology of a compact Riemannian manifold and the analysis of differential forms, showing that each de Rham cohomology class has a unique harmonic representative.

The Hodge Star Operator

Definition10.1Hodge star operator

On an oriented Riemannian $n$ -manifold $(M, g)$ , the Hodge star is the linear operator $*: \Omega^k(M) \to \Omega^{n-k}(M)$ defined by

\alpha \wedge *\beta = g(\alpha, \beta) \, \mathrm{dvol}_g

for all $k$ -forms $\alpha, \beta$ , where $g(\alpha, \beta)$ is the inner product on $k$ -forms induced by $g$ . In an oriented orthonormal coframe $\{e^1, \ldots, e^n\}$ :

*(e^{i_1} \wedge \cdots \wedge e^{i_k}) = \epsilon_{i_1 \cdots i_n} \, e^{i_{k+1}} \wedge \cdots \wedge e^{i_n},

where $\epsilon$ is the Levi-Civita symbol.

ExampleHodge star examples

On $\mathbb{R}^3$ : $*dx = dy \wedge dz$ , $*dy = dz \wedge dx$ , $*dz = dx \wedge dy$ , $*(dx \wedge dy) = dz$ , etc.
On a 2-manifold: $*1 = \mathrm{dvol}$ , $*\mathrm{dvol} = 1$ , and $*$ maps 1-forms to 1-forms (rotation by $90°$ ).
Key property: $** = (-1)^{k(n-k)}$ on $k$ -forms.

The Laplacian

Definition10.2Codifferential

The codifferential $\delta: \Omega^k(M) \to \Omega^{k-1}(M)$ is the formal $L^2$ -adjoint of $d$ :

\delta = (-1)^{nk+n+1} * d * = (-1)^{k} *^{-1} d *.

Equivalently, $\langle d\alpha, \beta \rangle_{L^2} = \langle \alpha, \delta\beta \rangle_{L^2}$ for compactly supported forms on a manifold without boundary. Note $\delta^2 = 0$ .

Definition10.3Hodge Laplacian

The Hodge Laplacian (Laplace-de Rham operator) is

\Delta = d\delta + \delta d: \Omega^k(M) \to \Omega^k(M).

This is a second-order elliptic differential operator. On functions ( $k = 0$ ), $\Delta f = \delta df = -\operatorname{div}(\nabla f)$ , recovering the Laplace-Beltrami operator (with the geometer's sign convention).

Harmonic Forms

Definition10.4Harmonic form

A $k$ -form $\omega$ is harmonic if $\Delta\omega = 0$ . On a compact manifold without boundary, this is equivalent to $d\omega = 0$ and $\delta\omega = 0$ (both closed and co-closed).

Proof

For compact $M$ without boundary: $\langle \Delta\omega, \omega \rangle = \langle d\delta\omega + \delta d\omega, \omega \rangle = \langle \delta\omega, \delta\omega \rangle + \langle d\omega, d\omega \rangle = |\delta\omega|^2 + |d\omega|^2$ . So $\Delta\omega = 0$ if and only if $d\omega = 0$ and $\delta\omega = 0$ . $\blacksquare$

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ExampleHarmonic forms on the torus

On the flat torus $T^n = \mathbb{R}^n/\mathbb{Z}^n$ , harmonic $k$ -forms are constant-coefficient forms $\omega = \sum_{|I|=k} a_I dx^I$ with $a_I \in \mathbb{R}$ . The space of harmonic $k$ -forms has dimension $\binom{n}{k}$ , matching $b_k(T^n)$ .

Remark$L^2$ inner product on forms

The $L^2$ inner product $\langle \alpha, \beta \rangle = \int_M g(\alpha, \beta) \, \mathrm{dvol}_g = \int_M \alpha \wedge *\beta$ makes $\Omega^k(M)$ into a pre-Hilbert space. Completing gives $L^2\Omega^k(M)$ , the setting for spectral theory of $\Delta$ .