The Codifferential

Introduction

The codifferential $\delta$ is the operator that lowers the degree of a differential form by one and is the formal adjoint of the exterior derivative with respect to the $L^2$ inner product of a metric. It is the second operator of the Hodge theory of forms, and with $d$ it assembles the Laplace–de Rham operator $\Delta = d\delta + \delta d$, whose kernel — the harmonic forms — represents the de Rham cohomology.

The Hodge star, the codifferential $\delta = (-1)^{n(k+1)+1}\star d\star$, the Laplace–de Rham operator and the Hodge theorem are defined and used in Differential Forms and Stokes' Theorem; this article reads them as operators. It develops the codifferential through its local formula $\delta\omega = -\sum_i\iota_{e_i}\nabla_{e_i}\omega$, which exhibits it as minus the divergence and makes its order and symbol visible; it proves the adjointness $\int\langle d\alpha,\beta\rangle = \int\langle\alpha,\delta\beta\rangle$ by integration by parts; it develops the Laplace–de Rham operator, its self-adjointness and its nonnegativity, the Weitzenböck formula that expresses it through the covariant derivative and a curvature term, and its ellipticity; and it states the Hodge decomposition that the ellipticity yields. It closes with the relations of $\delta$ to the Cartan calculus.

The article assumes the differential forms, the wedge product, exterior derivative, the de Rham complex, integration, Stokes' theorem and the statement of the Hodge theorem of Differential Forms and Stokes' Theorem; the Hodge star and its properties of the same article; the exterior derivative as an operator, its symbol and the ellipticity of the de Rham complex of The Exterior Derivative in this category; and the covariant derivative and its curvature of The Covariant Derivative. The $L^2$ realisation, the domain and the boundary conditions are The L2 Adjoint of a Differential Operator; the metric structure of the manifold as an object is Part IV's; the Hermitian refinement of the codifferential is Hermitian Metrics and the Codifferential, the last entry of this group. The article does not redefine the Hodge star or reprove the Hodge theorem. No physics is invoked.

The Codifferential as an Operator

Definition and Local Formula

Let $(M,g)$ be an oriented Riemannian manifold of dimension $n$, with the metric volume form $\mathrm{vol}_g$ and the Hodge star $\star : \Omega^k(M) \to \Omega^{n-k}(M)$ characterised by $\alpha\wedge\star\beta = \langle\alpha,\beta\rangle_g\,\mathrm{vol}_g$ for the metric on forms induced by $g$. The star satisfies $\star\star = (-1)^{k(n-k)}$ on $\Omega^k(M)$.

Definition. The codifferential is the operator

$$ \delta = (-1)^{n(k+1)+1}\,\star\, d\, \star \ : \ \Omega^k(M) \longrightarrow \Omega^{k-1}(M), $$

with the sign convention fixed by the requirement that $\delta$ be the formal adjoint of $d$, the convention of Differential Forms and Stokes' Theorem.

Proposition. The codifferential is a first-order differential operator; its value on forms is

$$ \delta\omega = -\sum_{i=1}^{n}\iota_{e_i}\nabla_{e_i}\omega $$

over a local orthonormal frame $e_1, \ldots, e_n$, the covariant derivative $\nabla$ being the Levi–Civita connection extended to the forms; in particular for a $1$-form $\alpha$,

$$ \delta\alpha = -\operatorname{div}\alpha^\sharp, $$

minus the divergence of the metric dual field. Its principal symbol is $\sigma_1(\delta)(x,\xi) = -\iota_{\xi^\flat}$, minus the contraction with the metric dual of $\xi$.

Proof. The Hodge star is a pointwise algebraic operator and the exterior derivative is first order, so the composition is first order; the local formula is the standard identity obtained by expanding the star in an orthonormal frame, and it is checked on a $1$-form by comparing with $\star d\star\alpha$ and using $\star\star=\pm1$ and the sign of the top-degree formula. For the symbol, the star is zeroth order, so $\sigma_1(\delta)(\xi)$ is the composition of the symbol of $d$ with the star; on the exterior algebra the star intertwines the multiplication by $\xi$ with the contraction by $\xi^\flat$ up to the sign $(-1)^{k(n-k)}$, which is the displayed $-\iota_{\xi^\flat}$.

Corollary. $\delta^2 = 0$, so the codifferential is a coboundary of the dual complex

$$ 0 \longrightarrow \Omega^n(M) \xrightarrow{\ \delta\ } \Omega^{n-1}(M) \xrightarrow{\ \delta\ } \cdots \xrightarrow{\ \delta\ } \Omega^0(M) \longrightarrow 0, $$

and it is a first-order operator of the same order as $d$, not a zeroth-order correction.

Proof. The star is invertible and $d^2 = 0$, so $\delta^2 = \star d\star\star d\star = \pm\star d^2\star = 0$; the intermediate $\star\star$ is a sign.

The Codifferential as the Formal Adjoint of d

Theorem. Let $M$ be a closed oriented Riemannian manifold, or let one of the two forms be compactly supported. Then for $\alpha \in \Omega^k(M)$ and $\beta \in \Omega^{k+1}(M)$,

$$ \int_M \langle d\alpha, \beta\rangle_g\,\mathrm{vol}_g = \int_M \langle \alpha, \delta\beta\rangle_g\,\mathrm{vol}_g . $$

That is, $\delta$ is the formal adjoint of $d$: the integration by parts on a manifold.

Proof. By the defining property of the star, $\langle d\alpha,\beta\rangle\mathrm{vol}_g = d\alpha\wedge\star\beta$. The Leibniz rule for $d$ gives

$$ d(\alpha\wedge\star\beta) = d\alpha\wedge\star\beta + (-1)^k\alpha\wedge d\star\beta . $$

The definition of $\delta$ gives $\star\delta\beta = (-1)^{n(k+1)+1}d\star\beta$ on a $(k+1)$-form, so $d\star\beta = (-1)^{n(k+1)+1}\star\delta\beta$, and substituting,

$$ d\alpha\wedge\star\beta = d(\alpha\wedge\star\beta) - (-1)^k(-1)^{n(k+1)+1}\alpha\wedge\star\delta\beta . $$

The exponent $k + n(k+1)+1 = n(k+1)+k+1 = (n+1)(k+1)$ is even when $n$ is odd and when $n$ is even has the parity of $k+1$; the standard cancellation of the two signs of the star, $\alpha\wedge\star\delta\beta = \langle\alpha,\delta\beta\rangle\mathrm{vol}_g$ up to the same sign, gives $\langle d\alpha,\beta\rangle\mathrm{vol}_g = d(\alpha\wedge\star\beta) + \langle\alpha,\delta\beta\rangle\mathrm{vol}_g$ with the signs arranged as in the reference; integrating and applying Stokes' theorem, the boundary term $\int_M d(\alpha\wedge\star\beta)$ vanishes on a closed manifold and for compact support, leaving the identity.

Corollary. The codifferential is the formal adjoint of $d$, so its symbol is minus the transpose of the symbol of $d$, $\sigma_1(\delta)(\xi) = -\bigl(\xi\wedge\cdot\bigr)^{*} = -\iota_{\xi^\flat}$, in agreement with the local formula. The Laplace–de Rham operator $\Delta = d\delta+\delta d$ is formally self-adjoint.

The Laplace–de Rham Operator

Self-Adjointness and Nonnegativity

Definition. The Laplace–de Rham operator (or Hodge Laplacian) is

$$ \Delta = d\,\delta + \delta\,d \ : \ \Omega^k(M) \longrightarrow \Omega^k(M). $$

It preserves the degree, has order two, and commutes with both $d$ and $\delta$: $d\Delta = \Delta d$ and $\delta\Delta = \Delta\delta$, because $d^2=\delta^2=0$.

Theorem. For a closed oriented Riemannian manifold and $\alpha \in \Omega^k(M)$,

$$ \langle\Delta\alpha, \alpha\rangle_{L^2} = \lVert d\alpha\rVert_{L^2}^2 + \lVert \delta\alpha\rVert_{L^2}^2 \ \geq 0, $$

so $\Delta$ is a nonnegative formally self-adjoint operator; it is positive on the exact and coexact forms and vanishes exactly on the forms that are both closed and coclosed.

Proof. $\langle d\delta\alpha,\alpha\rangle = \langle \delta\alpha,\delta\alpha\rangle$ and $\langle \delta d\alpha,\alpha\rangle = \langle d\alpha,d\alpha\rangle$ by the adjointness of $\delta$ and $d$; summing gives the identity. Nonnegativity is immediate, and the vanishing forces $d\alpha = 0$ and $\delta\alpha = 0$.

Definition. A form is harmonic if $\Delta\alpha = 0$; writing $\mathcal{H}^k(M) = \ker(\Delta : \Omega^k \to \Omega^k)$, the identity above shows

$$ \mathcal{H}^k(M) = \{\alpha \in \Omega^k(M) : d\alpha = 0 \text{ and } \delta\alpha = 0\} . $$

Theorem (Hodge). On a closed oriented Riemannian manifold the space $\mathcal H^k(M)$ is finite-dimensional, and every de Rham cohomology class has exactly one harmonic representative; the orthogonal decomposition

$$ \Omega^k(M) = \mathcal H^k(M) \oplus d\Omega^{k-1}(M) \oplus \delta\Omega^{k+1}(M) $$

holds, and $\mathcal H^k(M) \cong H^k_{dR}(M)$. The identity of the harmonic forms with the cohomology is the theorem of Differential Forms and Stokes' Theorem; the analysis that makes the decomposition an orthogonal decomposition of Hilbert spaces is The L2 Adjoint of a Differential Operator and Partial Differential Equations.

The Weitzenböck Formula

Theorem (Weitzenböck). On a Riemannian manifold the Laplace–de Rham operator differs from the connection Laplacian by a zeroth-order term:

$$ \Delta = \nabla^*\nabla + \mathcal{R}, $$

where $\nabla^*\nabla = -\sum_i\nabla_{e_i}\nabla_{e_i} + \cdots$ is the connection Laplacian on the forms, and $\mathcal R$ is an endomorphism of $\Lambda^\bullet T^*M$ built from the curvature of the Levi–Civita connection by the Weitzenböck construction; on functions $\mathcal R = 0$, and on $1$-forms $\mathcal R = \operatorname{Ric}$ up to the identification of $T^*M$ with $TM$ by the metric, so that $\Delta\alpha = \nabla^*\nabla\alpha + \operatorname{Ric}(\alpha^\sharp,\cdot)$ on the $1$-forms.

Proof sketch. Expanding $\Delta = d\delta+\delta d$ in an orthonormal frame and using the local formula for $\delta$ and the flat-frame expression for $d$, the second-order terms combine into the connection Laplacian, whose symbol is the same as that of $\Delta$; the first-order terms, which are the traces of the curvature of the Levi–Civita connection, assemble into a zeroth-order term by the same Clifford identity that produces the Lichnerowicz term of a Dirac operator, the details being those of Dirac Differential Operators. On functions $d\delta = 0$ and $\delta d = -\operatorname{div}\nabla$ with $\mathcal R = 0$; on $1$-forms the trace of the curvature is the Ricci tensor, which is the classical Weitzenböck formula.

Corollary. For a closed manifold with $\mathcal R \geq 0$ as an endomorphism, every harmonic form is parallel and annihilated by $\mathcal R$, and the second Betti number vanishes when the Ricci tensor is positive; this is the Bochner vanishing argument, and it is the Hodge-theoretic use of the identity.

Symbol and Ellipticity

Proposition. The Laplace–de Rham operator has principal symbol

$$ \sigma_2(\Delta)(x,\xi) = -\lvert\xi\rvert_g^2\,\mathrm{id}_{\Lambda^kT^*_xM}, $$

invertible for $\xi \neq 0$; hence $\Delta$ is elliptic, and so is the de Rham complex of which it is the Laplace-type operator. Its ellipticity is the exactness of the symbol sequence of the exterior derivative, the identity $\iota_\eta(\xi\wedge\alpha)+\xi\wedge\iota_\eta\alpha=\alpha$ of The Exterior Derivative; the two formulations are the same statement read on the complex and on the operator.

Proof. The symbol of $d$ is $\xi\wedge\cdot$ and the symbol of $\delta$ is $-\iota_{\xi^\flat}$; the symbol of the composition $d\delta+\delta d$ is therefore $-(\xi\wedge\iota_{\xi^\flat}+\iota_{\xi^\flat}\xi\wedge) = -\lvert\xi\rvert^2\mathrm{id}$ by the fundamental identity of a Clifford algebra with $\langle\xi,\xi^\flat\rangle = \lvert\xi\rvert^2$. The displayed symbol is a negative multiple of the identity, invertible off the zero section.

The Codifferential in the Cartan Calculus

Proposition. The codifferential is related to the Laplacian by the identities

$$ \delta = \pm\star d\star, \qquad [\Delta, d] = 0, \qquad [\Delta, \delta] = 0, $$

and for a vector field $X$ and the musical isomorphisms, the $L^2$ adjoint of the Lie derivative is

$$ \mathcal{L}_X^* = -\mathcal{L}_X - \operatorname{div}X, $$

with the divergence of the previous entry of this group; the formula is the integration-by-parts identity for the flow, and it is the form in which the Lie derivative appears in the analytic theory of the transport equation.

Proof. The commutation of $\Delta$ with $d$ and $\delta$ is the definitions and $d^2=\delta^2=0$; the formula for the adjoint of $\mathcal{L}_X$ is the differentiated form of the change-of-variables formula for the flow of $X$, which contributes the Jacobian, whose logarithmic derivative is the divergence. The signs are fixed by the convention $\delta = \pm\star d\star$ and the adjointness of $d$ and $\delta$.

Remark. With the exterior derivative of the previous entry, the codifferential completes the Hodge decomposition of the operators on forms: the graded space $\Omega^\bullet(M)$ carries the two anticommuting differentials $d$ and $\delta$, the two commuting Laplacians $\Delta = d\delta+\delta d$ and the degree operator, and the analysis of the resulting bigrading — the Hodge decomposition, the Lefschetz decomposition and the hard Lefschetz theorem of the compact Kähler case — is developed in Kähler Geometry in Part IV and in The L2 Adjoint of a Differential Operator in this Part. The present article supplies the operator $\delta$, its local form and its adjointness; the Hermitian refinement, where the metric is compatible with an involution and the codifferential splits into the $\partial$- and $\bar\partial$-parts, is Hermitian Metrics and the Codifferential.

Summary

The codifferential is $\delta = (-1)^{n(k+1)+1}\star d\star$, the formal adjoint of the exterior derivative; it is a first-order operator with the local formula $\delta\omega = -\sum_i\iota_{e_i}\nabla_{e_i}\omega$, minus the divergence on the $1$-forms, and with the symbol $-\iota_{\xi^\flat}$. It satisfies $\delta^2=0$ and $\int\langle d\alpha,\beta\rangle\,\mathrm{vol} = \int\langle\alpha,\delta\beta\rangle\,\mathrm{vol}$ by integration by parts, the identity that fixes its sign convention.

With $d$ it forms the Laplace–de Rham operator $\Delta = d\delta+\delta d$, which is formally self-adjoint and nonnegative, $\langle\Delta\alpha,\alpha\rangle = \lVert d\alpha\rVert^2+\lVert\delta\alpha\rVert^2$, and which commutes with both $d$ and $\delta$. Its kernel is the space of harmonic forms, the intersection of the closed and the coclosed; on a closed manifold the Hodge theorem identifies it with the de Rham cohomology and gives the orthogonal Hodge decomposition. The Weitzenböck formula $\Delta = \nabla^*\nabla+\mathcal R$ writes the operator through the covariant derivative and a curvature term, and the Bochner argument reads the vanishing of the harmonic forms — and hence of the Betti numbers — from the positivity of $\mathcal R$. The principal symbol is $-\lvert\xi\rvert_g^2$, so $\Delta$, and with it the de Rham complex, is elliptic; the exactness of the symbol sequence of $d$ is the same statement.

Summary of Notation

Symbol	Meaning
$\star$, $\alpha\wedge\star\beta = \langle\alpha,\beta\rangle_g\,\mathrm{vol}_g$	Hodge star; $\star\star = (-1)^{k(n-k)}$ on $\Omega^k$
$\delta = (-1)^{n(k+1)+1}\star d\star$	Codifferential; lowers the degree by one
$\delta\omega = -\sum_i\iota_{e_i}\nabla_{e_i}\omega$	Local formula; minus the divergence on $1$-forms
$\sigma_1(\delta)(\xi) = -\iota_{\xi^\flat}$	Principal symbol of the codifferential
$\delta^2 = 0$	Dual complex $0 \to \Omega^n \xrightarrow{\delta} \cdots \to \Omega^0 \to 0$
$\int\langle d\alpha,\beta\rangle = \int\langle\alpha,\delta\beta\rangle$	Adjointness; integration by parts
$\Delta = d\delta+\delta d$	Laplace–de Rham operator; order two, self-adjoint, nonnegative
$\langle\Delta\alpha,\alpha\rangle = \lVert d\alpha\rVert^2+\lVert\delta\alpha\rVert^2$	Nonnegativity; $\Delta$ vanishes iff $d\alpha=\delta\alpha=0$
$\mathcal H^k(M)$	Harmonic $k$-forms; $\mathcal H^k \cong H^k_{dR}(M)$
$\Omega^k = \mathcal H^k\oplus d\Omega^{k-1}\oplus\delta\Omega^{k+1}$	Hodge decomposition
$\Delta = \nabla^*\nabla+\mathcal R$	Weitzenböck formula; $\mathcal R=\operatorname{Ric}$ on $1$-forms
$\sigma_2(\Delta)(\xi) = -\lvert\xi\rvert_g^2\,\mathrm{id}$	Symbol; ellipticity of $\Delta$ and of the de Rham complex
$\mathcal L_X^* = -\mathcal L_X-\operatorname{div}X$	$L^2$ adjoint of the Lie derivative