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Hodge star operator

The Hodge star operator, often denoted by \star or *, is a fundamental linear map in differential geometry that acts on the exterior algebra of differential forms on an oriented pseudo-Riemannian manifold (M, g) of dimension n, transforming a k-form \alpha \in \Omega^k(M) into an (n-k)-form \star \alpha \in \Omega^{n-k}(M) such that for any \beta \in \Omega^k(M), \alpha \wedge \star \beta = \langle \alpha, \beta \rangle_g \, \mathrm{vol}_g, where \langle \cdot, \cdot \rangle_g is the inner product induced by the metric g and \mathrm{vol}_g is the volume form.^[1] This operator encodes the duality between complementary-degree forms, relying on the manifold's metric tensor and orientation to define the pairing.^[2] Originally introduced by Vito Volterra in 1889 as a tool for studying systems of partial differential equations and harmonic functions in the context of exterior calculus, the operator was later integrated into modern differential geometry by Élie Cartan and others in the early 20th century, with William Vallance Douglas Hodge popularizing its use in the 1930s through his theory of harmonic integrals on complex manifolds.^[3] In Euclidean space \mathbb{R}^n with the standard metric, the Hodge star simplifies significantly; for instance, in \mathbb{R}^3, it maps 1-forms to 2-forms in a way that corresponds to the cross product via \star (v^\flat \wedge w^\flat) = (v \times w)^\flat for vectors v, w, bridging vector calculus identities like curl and divergence to exterior derivatives.^[2] Key properties include its involutivity up to sign: \star^2 \alpha = (-1)^{k(n-k)} \alpha in positive-definite cases, and more generally \star^2 \alpha = (-1)^{k(n-k) + s} \alpha where s accounts for the metric signature (p, q) with p + q = n.^[1] The operator plays a central role in Hodge theory, where it facilitates the decomposition of the space of forms into harmonic, exact, and co-exact components via the Hodge Laplacian \Delta = d d^\star + d^\star d, with d^\star = (-1)^{nk + n + 1} \star d \star being the codifferential.^[4] In physics, it is indispensable for formulating Maxwell's equations in terms of differential forms on spacetime manifolds, such as d F = 0 and d \star F = J, generalizing classical electromagnetism to curved geometries and higher dimensions in theories like Kaluza-Klein or string theory.^[2] Extensions to non-Riemannian settings, including Galilean and Carrollian geometries, have been explored in recent work to model non-relativistic limits of physical laws.^[5]

Foundations

Formal Definition

The Hodge star operator, denoted by \star, is a linear map \star: \Lambda^k(V) \to \Lambda^{n-k}(V) defined on an n-dimensional oriented real vector space V equipped with a nondegenerate symmetric bilinear form \langle \cdot, \cdot \rangle. It satisfies the defining relation

\langle \alpha, \beta \rangle \, \mathrm{vol} = \alpha \wedge \star \beta

for all k-forms \alpha, \beta \in \Lambda^k(V), where \mathrm{vol} \in \Lambda^n(V) is the volume form determined by the orientation and inner product, normalized so that \langle \mathrm{vol}, \mathrm{vol} \rangle = 1.^[6] This equation extends linearly to define \star on the entire space of k-forms.^[1] To construct \star explicitly, select a pseudo-orthonormal basis \{e_1, \dots, e_n\} of V that respects the given orientation, where \langle e_i, e_j \rangle = \epsilon_i \delta_{ij} with \epsilon_i = \pm 1. For the basis element e_{i_1} \wedge \cdots \wedge e_{i_k} with $1 \leq i_1 < \cdots < i_k \leq n, let \{j_1 < \cdots < j_{n-k}\} be the complementary indices, so that \{i_1, \dots, i_k, j_1, \dots, j_{n-k}\} = \{1, \dots, n\}. Then,

\star(e_{i_1} \wedge \cdots \wedge e_{i_k}) = \operatorname{sgn}(\sigma) \left( \prod_{m=1}^{n-k} \epsilon_{j_m} \right) e_{j_1} \wedge \cdots \wedge e_{j_{n-k}},

where \sigma is the permutation (i_1, \dots, i_k, j_1, \dots, j_{n-k}) of (1, \dots, n) and \operatorname{sgn}(\sigma) \in \{ \pm 1 \} is its sign; this ensures the defining relation holds, as the wedge product yields \pm \mathrm{vol}.^[1] The operator extends by linearity to arbitrary k-forms.^[7] The Hodge star operator is uniquely determined by the orientation and inner product on V. This follows from the fact that the defining equation establishes a perfect pairing \Lambda^k(V) \times \Lambda^{n-k}(V) \to \Lambda^n(V) \cong \mathbb{R} via the inner product, yielding a canonical isomorphism \Lambda^k(V) \cong (\Lambda^{n-k}(V))^\vee.^[1]^[6] Among its basic algebraic properties, in the positive-definite case, applying the operator twice yields \star^2 = (-1)^{k(n-k)} \mathrm{Id} on \Lambda^k(V). In general, for a metric of signature (p, q) with p + q = n, \star^2 = (-1)^{k(n-k) + q} \mathrm{Id}.^[6]

Orientation and Inner Product

In the context of vector spaces underlying differential geometry, an orientation on an n-dimensional real vector space V is an equivalence class of ordered bases, where two bases (e_1, \dots, e_n) and (f_1, \dots, f_n) belong to the same class if the change-of-basis matrix has positive determinant.^[8] This equivalence relation partitions the set of all ordered bases into two classes, corresponding to the two possible orientations of V.^[9] Such an orientation selects a preferred volume form \omega \in \Lambda^n V^*, defined up to multiplication by a positive scalar, which encodes the "handedness" of the space and ensures consistent notions of positive volume.^[8] A Riemannian metric on V is a positive-definite symmetric bilinear form \langle \cdot, \cdot \rangle: V \times V \to \mathbb{R}, which induces a norm \|v\| = \sqrt{\langle v, v \rangle} on vectors and defines orthogonality via \langle u, v \rangle = 0.^[10] This inner product extends naturally to the exterior algebra \Lambda^\bullet V of multivectors; for decomposable k-vectors u = v_1 \wedge \cdots \wedge v_k and w = w_1 \wedge \cdots \wedge w_k, the induced inner product is given by

\langle u, w \rangle = \det \bigl( \langle v_i, w_j \rangle \bigr)_{1 \leq i,j \leq k},

with linearity extending it to general multivectors.^[11] In local coordinates where the metric tensor is g = (g_{ij}), this structure determines magnitudes and angles consistently across the space. The Riemannian metric further induces a canonical volume element on V, expressed in coordinates as

\mathrm{vol} = \sqrt{|\det g|} \, dx^1 \wedge \cdots \wedge dx^n,

which is a top-degree form invariant under coordinate changes and compatible with the inner product on \Lambda^n V.^[12] The orientation fixes the sign of this volume form, ensuring it aligns with the chosen equivalence class of bases, while the metric governs its scaling through the determinant. Together, the orientation and metric provide the structural foundation for duality in exterior algebras: the orientation determines the sign convention in pairings between complementary-degree forms, and the metric sets the magnitudes via induced inner products on multivectors.^[13] These elements enable the formal definition of operators that map between such dual spaces.

Geometric Interpretation

Action on Forms

The Hodge star operator, denoted ⋆, acts on differential forms in the exterior algebra of \mathbb{R}^n by mapping a k-form \alpha to an (n-k)-form \star \alpha, such that for any k-form \beta, the wedge product \alpha \wedge \star \beta = \langle \alpha, \beta \rangle \, \mathrm{vol}, where \langle \cdot, \cdot \rangle is the inner product induced by the Euclidean metric and \mathrm{vol} is the standard volume form on \mathbb{R}^n.^[14] This action geometrically rotates and scales the form to its complement in the ambient space, effectively encoding the duality between subspaces of complementary dimensions.^[1] In this visualization, the Hodge star applied to a k-form associated with a k-plane selects the oriented (n-k)-plane orthogonal to it, with the magnitude scaled by the volume of the original plane.^[2] For instance, the operator identifies the "perpendicular" direction in the Grassmannian sense, transforming the infinitesimal volume element of the k-plane into that of its orthogonal complement.^[14] This complementarity arises naturally from the inner product pairing, briefly referencing the formal definition where the star ensures the wedge product recovers the oriented volume up to the pairing scalar.^[1] The sign of \star \alpha depends on the chosen orientation of \mathbb{R}^n, typically the standard positive orientation from the ordered basis. For an oriented orthonormal basis where the indices of the k-form and its complement form an even permutation of \{1, \dots, n\}, the sign is positive; it becomes negative for odd permutations, preserving the overall orientation of the volume form.^[14] This convention ensures consistency in the duality, such that \star aligns with the right-hand rule in positively oriented frames.^[1] In three dimensions, this action motivates the cross product as a special case, where \star on a 1-form (corresponding to a vector) yields a 2-form representing the oriented area of the perpendicular plane, linking multivector duality to vector operations.^[2]

Duality Mechanism

The Hodge star operator plays a central role in establishing an algebraic duality on the space of differential forms. On a Riemannian manifold M of dimension n, it induces an L^2-inner product on the space of k-forms \Omega^k(M) defined by \langle \alpha, \beta \rangle_{L^2} = \int_M \alpha \wedge \star \beta for \alpha, \beta \in \Omega^k(M). This is equivalent to \int_M \langle \alpha, \beta \rangle_g \, \mathrm{vol}_g, where \langle \cdot, \cdot \rangle_g is the pointwise inner product on forms induced by the Riemannian metric g and \mathrm{vol}_g is the volume form. Locally, on an oriented inner product vector space V of dimension n, the operator \star: \Lambda^k V^* \to \Lambda^{n-k} V^* similarly defines \langle \alpha, \beta \rangle = \alpha \wedge \star \beta \in \Lambda^n V^* \cong \mathbb{R}.^[15]^[16] This construction yields a canonical isomorphism \Lambda^k V^* \cong \Lambda^{n-k} V^* via the Hodge star, which extends fiberwise to bundles of forms on M. With respect to the L^2-inner product, the Hodge star enables self-adjointness relations between differential operators. Specifically, the exterior derivative d satisfies \langle d\alpha, \beta \rangle_{L^2} = \langle \alpha, \delta \beta \rangle_{L^2} for appropriate \alpha, \beta, where \delta is the codifferential (formal adjoint of d), defined using the Hodge star as \delta = (-1)^{n(k+1)+1} \star d \star on k-forms. This adjointness underpins variational principles in Hodge theory.^[17]^[16] The duality mechanism is foundational to the Hodge decomposition theorem, which asserts that on a compact oriented Riemannian manifold, every k-form decomposes orthogonally as \alpha = d\beta + \delta \gamma + h with respect to the L^2-inner product, where \beta \in \Omega^{k-1}(M), \gamma \in \Omega^{k+1}(M), and h is harmonic (dh = 0 = \delta h). The space of harmonic forms is finite-dimensional and isomorphic to the de Rham cohomology H^k_{dR}(M), with the decomposition enabling global analysis of forms via elliptic operators.^[16]^[18]

Examples in Euclidean Spaces

Two Dimensions

In two-dimensional Euclidean space \mathbb{R}^2 equipped with the standard inner product and positive orientation, the Hodge star operator \star acts on the exterior algebra of differential forms as follows. On the scalar 0-form, \star 1 = dx \wedge dy, which is the oriented volume form. For the basis 1-forms, \star dx = dy and \star dy = -dx. On the top-degree 2-form, \star (dx \wedge dy) = 1.^[14]^[19] For a general 1-form \alpha = a \, dx + b \, dy, the Hodge star is \star \alpha = -b \, dx + a \, dy. This action corresponds to a counterclockwise rotation by 90 degrees when identifying 1-forms with vectors via the metric.^[14]^[20] Geometrically, this rotation property links the Hodge star to operations in two-dimensional vector calculus, where applying \star to the differential of a scalar function (the gradient) yields a form associated with the curl structure through subsequent exterior differentiation.^[20] The defining property of the Hodge star can be verified for 1-forms: for any \alpha, \alpha \wedge \star \alpha = \langle \alpha, \alpha \rangle \, dx \wedge dy, where \langle \cdot, \cdot \rangle is the inner product induced by the metric. For the example \alpha = dx, dx \wedge \star dx = dx \wedge dy = \langle dx, dx \rangle \, dx \wedge dy, since \langle dx, dx \rangle = 1.^[19]

Three Dimensions

In three-dimensional Euclidean space \mathbb{R}^3 equipped with the standard positive definite metric and orientation, the Hodge star operator \star maps k-forms to (3-k)-forms. On the scalar 0-form, it yields the oriented volume form:

\star 1 = dx \wedge dy \wedge dz.

Applying \star to the basis 1-forms gives

\star dx = dy \wedge dz, \quad \star dy = -dx \wedge dz, \quad \star dz = dx \wedge dy,

while on the dual basis 2-forms,

\star (dy \wedge dz) = dx, \quad \star (dz \wedge dx) = dy, \quad \star (dx \wedge dy) = dz.

These actions follow cyclic permutations adjusted by signs to ensure consistency with the metric and orientation.^[21] The Hodge star facilitates the identification of 1-forms and 2-forms with vectors in \mathbb{R}^3. For a 1-form \alpha = v_x \, dx + v_y \, dy + v_z \, dz associated to the vector \mathbf{v} = (v_x, v_y, v_z), the operator produces the corresponding 2-form

\star \alpha = v_x \, (dy \wedge dz) - v_y \, (dx \wedge dz) + v_z \, (dx \wedge dy).

The inverse application \star on this 2-form recovers \alpha, establishing an isomorphism between the spaces of 1-forms and 2-forms via vectors. Under this duality, the composition \star^2 = \mathrm{Id} on 1-forms.^[22] This identification links the Hodge star to the vector cross product. For 1-forms \alpha and \beta corresponding to vectors \mathbf{u} and \mathbf{v}, the wedge product \alpha \wedge \beta is a 2-form, and

\star (\alpha \wedge \beta)

yields the 1-form associated to \mathbf{u} \times \mathbf{v}. This relation embeds the cross product within the exterior algebra, highlighting the Hodge star's role in translating vector operations to forms.^[23] A key application arises in vector calculus identities. Consider a vector field \mathbf{v} with associated 1-form \alpha = v^\flat. The exterior derivative d\alpha is a 2-form corresponding to \nabla \times \mathbf{v}. Applying the Hodge star gives \star (d\alpha), the 1-form dual to \nabla \times \mathbf{v}. Conversely, the divergence \nabla \cdot \mathbf{v} is recovered as the scalar \star (d (\star \alpha)), linking the Hodge star to the classical divergence via the exterior derivative. For instance, if \mathbf{v} = (y, -x, 0), then \alpha = y \, dx - x \, dy, d\alpha = -2 \, dx \wedge dy, \star (d\alpha) = -2 \, dz (corresponding to \nabla \times \mathbf{v} = (0, 0, -2)), and further computation yields \nabla \cdot \mathbf{v} = 0.^[21]

Four Dimensions

In \mathbb{R}^4 with the standard Euclidean metric, the Hodge star operator \star maps 2-forms to 2-forms, and its action introduces a decomposition of the space of 2-forms into self-dual and anti-self-dual eigenspaces. Specifically, a 2-form \omega is self-dual if \star \omega = \omega and anti-self-dual if \star \omega = -\omega, corresponding to eigenvalues +1 and -1, respectively. This splitting, each of dimension 3, underscores the heightened complexity of the operator in higher even dimensions, where the middle-degree forms exhibit non-trivial eigenvalue structure unlike in lower dimensions.^[24] The explicit action on a basis of 2-forms can be computed using the Levi-Civita symbol \epsilon^{ijkl} with \epsilon^{1234} = +1. For the orthonormal coframe dx^1, dx^2, dx^3, dx^4, the Hodge star is given by

\star (dx^i \wedge dx^j) = \epsilon^{ijkl} dx^k \wedge dx^l.

Applying this yields \star (dx^1 \wedge dx^2) = dx^3 \wedge dx^4, \star (dx^1 \wedge dx^3) = -dx^2 \wedge dx^4, \star (dx^1 \wedge dx^4) = dx^2 \wedge dx^3, \star (dx^2 \wedge dx^3) = dx^1 \wedge dx^4, \star (dx^2 \wedge dx^4) = -dx^1 \wedge dx^3, and \star (dx^3 \wedge dx^4) = dx^1 \wedge dx^2. These relations facilitate computations in vector calculus identities and highlight the duality pairing.^[25] In the physical interpretation within four-dimensional spacetime, the Hodge star underlies electromagnetic duality, where applying \star to the Faraday tensor F, a 2-form encoding electric and magnetic fields, produces the dual *F that rotates the fields into each other, motivating symmetry in Maxwell's equations.^[26] A key property in \mathbb{R}^4 is that \star^2 acts as the identity on even-degree forms: \star^2 \omega = \omega for 0-forms, 2-forms, and 4-forms. This follows from the general formula \star^2 \omega = (-1)^{k(4-k)} \omega in Euclidean signature, where k(4-k) is even for even k. In contrast, odd dimensions yield \star^2 = \mathrm{Id} across all degrees due to the parity of n.^[27]

Special Properties

Conformal Invariance

A conformal transformation of the Riemannian metric is given by g' = e^{2\phi} g, where \phi is a smooth real-valued function on the manifold; this rescales lengths by e^{\phi} while preserving angles between tangent vectors.^[28] Under such a transformation, the Hodge star operator \star' associated to g' acts on a k-form \alpha by \star' \alpha = e^{(n-2k)\phi} \star \alpha, where n is the dimension of the manifold.^[28] Thus, the operator is conformally invariant precisely when k = n/2 in even-dimensional manifolds, as the scaling factor vanishes in that case.^[28] This transformation law follows from the defining property of the Hodge star, \beta \wedge \star \alpha = \langle \beta, \alpha \rangle \, \mathrm{vol}_g for all k-forms \beta, where \langle \cdot, \cdot \rangle denotes the pointwise inner product induced by g and \mathrm{vol}_g is the volume form. Under the conformal change, the volume form scales as \mathrm{vol}_{g'} = e^{n\phi} \, \mathrm{vol}_g, while the inner product on k-forms scales as \langle \beta, \alpha \rangle_{g'} = e^{-2k\phi} \langle \beta, \alpha \rangle. Substituting into the defining equation for \star' yields \beta \wedge \star' \alpha = e^{(n-2k)\phi} \, \beta \wedge \star \alpha, implying the stated relation by linearity of the wedge product.^[28] In two dimensions (n=2), the invariance for 1-forms (k=1) ensures that the Hodge star induces a complex structure on the cotangent bundle compatible with the conformal class of the metric, underpinning the metric-independent formulation of the \bar{\partial}-operator in complex analysis on Riemann surfaces. In four dimensions (n=4), the invariance for 2-forms (k=2) implies that the self-duality condition \star F = F for a 2-form F (such as the curvature in gauge theory) is preserved under conformal rescalings, which is essential for the conformal invariance of the Yang-Mills equations.

Relation to Vector Calculus

In three-dimensional Euclidean space \mathbb{R}^3 equipped with the standard metric, the Hodge star operator \star establishes a close correspondence between differential forms and the classical vector calculus operators gradient, curl, and divergence. For a smooth function f: \mathbb{R}^3 \to \mathbb{R} (a 0-form), the exterior derivative d yields the 1-form df = \frac{\partial f}{\partial x} \, dx + \frac{\partial f}{\partial y} \, dy + \frac{\partial f}{\partial z} \, dz, which corresponds precisely to the gradient vector field \nabla f under the identification of 1-forms with vectors via the metric.^[29] For a vector field v identified with its associated 1-form \alpha = v^\flat (lowering the index with the metric), the curl is given by \operatorname{[curl](/page/Curl)} v = \star d \alpha, where d \alpha is a 2-form and \star maps it back to a 1-form, again identified with the vector field. The divergence is \operatorname{div} v = \star d \star \alpha, with \star \alpha a 2-form, d (\star \alpha) a 3-form, and \star yielding a 0-form (scalar). These expressions unify the vector operators within the exterior calculus framework.^[29]^[27] A key identity in vector calculus, \operatorname{div}(\operatorname{[curl](/page/Curl)} v) = 0, follows directly from the nilpotency of the exterior derivative, d^2 = 0, and properties of the Hodge star. Substituting the expressions gives \operatorname{div}(\operatorname{[curl](/page/Curl)} v) = \star d \star (\star d \alpha) = \star d (d \alpha), since \star \star = \mathrm{id} on 1-forms in oriented \mathbb{R}^3. Thus, \star (d^2 \alpha) = \star 0 = 0. This demonstrates how the Hodge star facilitates proofs of vector identities through form duality.^[29]^[27] More generally, the Hodge star induces a duality between k-forms and (3-k)-forms, enabling the projection of differential operators like d onto vector fields and revealing the underlying geometric structure that unifies gradient (as d on scalars), curl (as \star d on vectors), and divergence (as \star d \star on vectors) as components of the de Rham complex.^[29]

Extension to Manifolds

Definition on Riemannian Manifolds

The Hodge star operator on an oriented Riemannian manifold (M, g) of dimension n extends the flat-space construction by defining a pointwise linear map \star_g: \Omega^k(M) \to \Omega^{n-k}(M) that acts on the space of smooth k-forms to produce smooth (n-k)-forms, relying on the metric g to induce inner products on the cotangent spaces and the fixed orientation to ensure consistency across the manifold.^[14] Locally, at each point p \in M, \star_g is defined on the fiber \Lambda^k T_p^* M by choosing a positively oriented orthonormal basis adapted to the inner product g_p, such that for any \alpha \in \Lambda^k T_p^* M, \star_g \alpha is the unique element satisfying \beta \wedge (\star_g \alpha) = \langle \beta, \alpha \rangle_{g_p} \vol_{g_p} for all \beta \in \Lambda^k T_p^* M, where \langle \cdot, \cdot \rangle_{g_p} denotes the induced inner product and \vol_{g_p} is the volume element on T_p^* M.^[1] This local definition extends globally to a smooth bundle map on the exterior bundle \Lambda^* T^* M because the metric g varies smoothly, implicitly incorporating parallel transport along curves to maintain compatibility with the manifold's geometry, though the operator itself is metric-dependent and alters under conformal changes to g.^[30] The orientation on M fixes the sign convention for \star_g, ensuring that the volume form \vol_g, which determines the "completion" of forms, aligns with the chosen orientation; in local coordinates (x^1, \dots, x^n), this takes the explicit form \vol_g = \sqrt{|\det g|} \, dx^1 \wedge \cdots \wedge dx^n, where g is the matrix representation of the metric, highlighting the operator's reliance on both the metric's determinant for scaling and the coordinate frame for the wedge product structure.^[14] As a map on sections of the exterior bundle, \star_g preserves the smooth structure of \Omega^*(M) and is compatible with the Levi-Civita connection induced by g, allowing it to interact covariantly with tensorial operations on the bundle without requiring explicit parallel transport in its definition.^[1] For diffeomorphisms f: N \to M between oriented Riemannian manifolds that preserve both the metric and orientation (i.e., f^* g_M = g_N), the Hodge star operators commute with pullbacks via \star_{g_N} \circ f^* = f^* \circ \star_{g_M}, ensuring that the operator respects the geometric structure under such isometries and maintains the duality between form degrees across manifolds.^[14] This compatibility underscores the Hodge star's role as a metric- and orientation-dependent isomorphism that generalizes the Euclidean case while adapting to the curvature of M through the pointwise fiberwise action.^[30]

Local Computation

In local coordinates \{x^i\} on an n-dimensional oriented Riemannian manifold (M, g), the Hodge star operator \star applied to a k-form \alpha = \sum_{i_1 < \cdots < i_k} \alpha_{i_1 \cdots i_k} \, dx^{i_1} \wedge \cdots \wedge dx^{i_k} yields the (n-k)-form \star \alpha = \sum_{j_1 < \cdots < j_{n-k}} (\star \alpha)_{j_1 \cdots j_{n-k}} \, dx^{j_1} \wedge \cdots \wedge dx^{j_{n-k}}, where the covariant components are

(\star \alpha)_{j_1 \cdots j_{n-k}} = \frac{\sqrt{|g|}}{k!} \, \varepsilon^{i_1 \cdots i_k j_1 \cdots j_{n-k}} \, \alpha_{i_1 \cdots i_k},

with the summation over i_1, \dots, i_k = 1, \dots, n, g = \det(g_{ij}), |g| = |\det(g_{ij})|, and \varepsilon^{i_1 \cdots i_n} the Levi-Civita symbol satisfying \varepsilon^{1 \cdots n} = +1.^[30] Equivalently, if the contravariant components \alpha^{m_1 \cdots m_k} of \alpha are specified (obtained by raising indices via the inverse metric g^{ab}), the contravariant components of \star \alpha are

(\star \alpha)^{j_1 \cdots j_{n-k}} = \frac{\sqrt{|g|}}{k!} \, \varepsilon^{j_1 \cdots j_{n-k} i_1 \cdots i_k} \, g_{i_1 m_1} \cdots g_{i_k m_k} \, \alpha^{m_1 \cdots m_k},

again summing over the repeated indices, with the metric tensor g_{ij} used to lower the indices in the contraction.^[30] In an orthonormal frame where g_{ij} = \delta_{ij} and thus \sqrt{|g|} = 1, these expressions simplify to the flat-space case, with (\star \alpha)_{j_1 \cdots j_{n-k}} determined by the Levi-Civita symbol alone applied to the complementary index set, yielding \star (dx^{i_1} \wedge \cdots \wedge dx^{i_k}) = \operatorname{sgn}(\sigma) \, dx^{j_1} \wedge \cdots \wedge dx^{j_{n-k}} for the permutation \sigma ordering \{i_1, \dots, i_k, j_1, \dots, j_{n-k}\} to \{1, \dots, n\}.^[27]

Codifferential Operator

The codifferential operator, denoted δ, is a key component in Hodge theory on oriented Riemannian manifolds, serving as the formal adjoint to the exterior derivative d with respect to the L² inner product induced by the metric. For a k-form α on an n-dimensional manifold M, δ acts as δα = (-1)^{n(k+1)+1} ⋆ d ⋆ α, where ⋆ denotes the Hodge star operator, which maps k-forms to (n-k)-forms using the Riemannian volume form and orientation.^[16] This definition incorporates a sign convention that ensures δ lowers the degree by one, mapping the space of k-forms Ω^k(M) to Ω^{k-1}(M), in direct analogy to how d raises the degree from (k-1) to k.^[16] The adjoint property of δ follows from integration by parts and Stokes' theorem. Specifically, for compactly supported (k-1)-forms α and k-forms β on M, the L² inner product satisfies ∫_M ⟨dα, β⟩ vol_g = ∫_M ⟨α, δβ⟩ vol_g + boundary terms, where vol_g is the Riemannian volume form and ⟨·,·⟩ is the pointwise inner product on forms induced by the metric g; on closed manifolds without boundary, the boundary terms vanish.^[31] This relation holds because the Hodge star provides the duality needed to pair forms appropriately, making δ the unique formal adjoint of d under the inner product ∫_M α ∧ ⋆ β.^[16] Algebraically, δ shares several properties with d that underpin Hodge theory. In particular, δ² = 0, mirroring the nilpotency of d, and their compositions yield the Hodge Laplacian Δ = dδ + δd, a self-adjoint elliptic operator on each Ω^k(M).^[16] These relations facilitate the Hodge decomposition of form spaces into orthogonal direct sums involving the image of d, the image of δ, and the kernel of Δ (harmonic forms).^[31] In the specific case of flat Euclidean 3-space ℝ³ with the standard metric, the codifferential recovers familiar vector calculus operators when identifying vector fields with 1-forms and 2-forms via the metric and Hodge star. Here, δ acting on 1-forms corresponds to the negative divergence operator (-div), while on 2-forms it corresponds to the curl operator (curl).^[16]

Poincaré Lemma for Codifferential

The Poincaré lemma for the codifferential provides a local analog to the classical Poincaré lemma for the exterior derivative d, but adapted to the codifferential operator \delta. On a star-shaped domain U \subset \mathbb{R}^n, every co-closed k-form \omega \in \Lambda^k(U) (i.e., \delta \omega = 0) is coexact, meaning there exists \alpha \in \Lambda^{k+1}(U) such that \omega = \delta \alpha, provided $0 < k < n. This result holds more generally on contractible open manifolds where the relevant cohomology vanishes, ensuring that co-closed forms decompose solely into the image of \delta without harmonic components. In the context of Hodge theory on compact Riemannian manifolds without boundary, the global Hodge decomposition \Lambda^k(M) = \mathrm{im}\, d \oplus \mathrm{im}\, \delta \oplus \mathcal{H}^k(M) implies that the space of co-closed k-forms satisfies \ker \delta^k = \mathrm{im}\, \delta^k \oplus \mathcal{H}^k(M), where \mathcal{H}^k(M) = \ker \Delta^k consists of the harmonic forms (both closed and co-closed). This follows from elliptic regularity of the Hodge Laplacian \Delta = d\delta + \delta d and the self-adjointness of d and \delta with respect to the L^2 inner product induced by the metric. The local Poincaré lemma for \delta corresponds to the case where \mathcal{H}^k = \{0\}, as on contractible domains. A proof of the local lemma constructs an explicit cohomotopy operator h: \Lambda^k(U) \to \Lambda^{k+1}(U) using the Hodge star operator \star and the standard homotopy operator H for d, defined as h = \eta \star^{-1} H \star (with \eta a sign factor depending on degree and dimension). This satisfies the homotopy invariance formula \delta h + h \delta = I - S_{x_0}, where S_{x_0} projects onto forms constant at a fixed basepoint x_0 \in U (and vanishes for k \geq 1). For a co-closed form \omega with \delta \omega = 0, it follows that \omega = \delta (h \omega), yielding the desired \alpha = h \omega. Unlike the Poincaré lemma for d, which relies on a direct geometric homotopy integrating along rays in star-shaped domains, the version for \delta arises from the formal adjoint relation \delta = (-1)^{nk + n + 1} \star d \star (up to sign conventions), conjugating the homotopy for d via the isomorphism \star: \Lambda^k \to \Lambda^{n-k}. This leverages the metric-dependent self-adjointness without requiring a separate geometric construction. This lemma has implications for solvability in Hodge theory: the equation \delta \beta = f admits a local solution \beta whenever f is co-closed (\delta f = 0), as f then lies in \mathrm{im}\, \delta. Globally on compact manifolds, solvability requires f to be orthogonal to the harmonic forms in its de Rham cohomology class, reflecting the identification \mathcal{H}^k(M) \cong H^k_{\mathrm{dR}}(M).

References

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[PDF] Math 396. Hodge-star operator In the theory of pseudo-Riemannian ...
In the theory of pseudo-Riemannian manifolds there will be an important operator (on differential forms) called the Hodge star; this operator will be an ...
[2]
4.8 Hodge star
The Hodge star is an operator that provides some sort of duality between k -forms and ( n − k ) -forms in . R n . It is easiest to define it in terms ...
[3]
[PDF] Differential Forms, the Early Days; or the Stories of Deahna's ...
defines the Hodge star operator * (and the associated operators, which have become so very important in the theory of harmonic forms, Yang-Mills fields, etc ...
[4]
[PDF] lecture 25: the hodge laplacian
Definition 1.1. The Hodge star operator ? : Ωk(M) → Ωm−k(M) maps any k-form η ∈ Ωk(M) to the (m - k)-form ?η ∈ Ωm−k(M) so that for any ω ∈ Ωk(M), ω ...<|control11|><|separator|>
[5]
[2206.09788] Galilean and Carrollian Hodge star operators - arXiv
Jun 20, 2022 · The standard Hodge star operator is naturally associated with metric tensor (and orientation). It is routinely used to concisely write down ...
[6]
[PDF] Manifolds and Differential Forms Reyer Sjamaar - Cornell Mathematics
A Hodge star operator corresponding to this inner product is defined ... Differential geometry textbook at advanced undergraduate level in five massive but fun to.
[7]
[PDF] A Second Course in Differential Geometry
Details can be found in any introductory differential geometry textbook. ... There is a convenient way to find a formal ajoint, via the Hodge star operator, when ...
[8]
orientation in nLab
Dec 16, 2024 · Definition 1.2. (orientation of a vector space) For V a vector space of dimension n , an orientation of V is an equivalence class of nonzero ...
[9]
[PDF] Math 396. Orientations In the theory of manifolds there will be a ...
Declare two ordered bases v and v0 to be similarly oriented precisely when the linear automorphism of V that relates them (in either direction!) has positive ...
[10]
[PDF] An introduction to Riemannian geometry - IME-USP
Riemannian manifolds. 1.1 Introduction. A Riemannian metric is a family of smoothly varying inner products on the tangent spaces of a smooth manifold.
[11]
[PDF] Differential Forms - MIT Mathematics
Feb 1, 2019 · In particular from the standard Euclidean inner product on 𝐑𝑁 one gets, by restricting this inner product to vectors in 𝑇𝑝𝑋, an inner product.
[12]
volume form in nLab
Jan 25, 2013 · In the case of a Riemannian (not pseudo-Riemannian) metric, this simplifies to vol g = det ( g ) vol_g = \sqrt{\det(g)} . (Note that local ...
[13]
[PDF] bundles III: The Hodge Star and Hodge--de Rham Laplacians - arXiv
Jul 29, 2025 · star operator requires a (non-degenerate) metric and an orientation to be defined. Given the importance of the Hodge–de Rham Laplacian in ...
[14]
[PDF] The Hodge Star Operator - Brown Math
Apr 22, 2015 · The Hodge star operator is a map from ∧k(Rn) to ∧n−k(Rn), where ∧k(Rn) is the vector space of alternating k-tensors on Rn.
[15]
[PDF] Convergence of Discrete Exterior Calculus for the Hodge-Dirac ...
Jul 28, 2025 · The Hodge star operators induce inner products on Λk(Ω). Definition 2.1. The L2 inner product on two k-forms ω and µ is given by. ⟨ω, µ⟩ ...
[16]
[PDF] Undergraduate Lecture Notes in De Rham–Hodge Theory - arXiv
May 15, 2011 · (M). 2.5 Hodge star operator. The Hodge star operator ⋆ : Ωp(M) → Ωn−p(M), which maps any p−form α into its dual. (n − p)−form ⋆α on a ...
[17]
[PDF] The Hodge Decomposition - Nikolai Nowaczyk
Then for any 0 ≤ k ≤ n, there exists precisely one isomorphism ∗ = ∗k : Λk(V ) → Λn−k(V ), ω 7→ ∗ω, which satisfies. ∀ω, η ∈ Λk(V ) : ω ∧ ∗η = hω, ηiΛk dV.
[18]
[PDF] arXiv:math/0505227v5 [math.GT] 31 Jan 2022
Jan 31, 2022 · As we recall in section 3, the two essential ingredients to the smooth Hodge star operator are Poincaré Duality and a metric, or inner product.
[19]
[PDF] On the L2-Hodge theory of Landau-Ginzburg models - arXiv
Mar 7, 2019 · It defines Hermitian inner products on all tensors bundles. We give here explicitly the inner products for differential forms to set up our ...
[20]
[PDF] arXiv:2108.07385v2 [physics.comp-ph] 22 Jun 2022
Jun 22, 2022 · The Hodge star operator may be derived from two notions of duality: the L2 inner product and Poincaré duality.
[21]
[PDF] THE HODGE DUAL OPERATOR - Oregon State University
Jan 29, 1999 · The Hodge dual operator ∗ is one of the 3 basic operations on differential forms. (The other 2 are wedge product ∧ and exterior ...
[22]
[PDF] Differential forms in two dimensions - Purdue Math
Mar 23, 2017 · Hodge star operator, denoted ∗. The Hodge star operator is defined by. ∗(F1dx + F2dy) = F1dy − F2dx. (15). 4. Page 5. It corresponds to a ...
[23]
[PDF] Numerical Method for Darcy flow derive using discrete exterior ...
The Hodge star is an isomorphism, ∗ : Ωk(M) → Ωn−k(M). For R3 with standard metric, the Hodge star satisfies the following properties: ∗1 = dx ∧ dy ∧ dz ;.
[24]
[PDF] Introduction to differential forms - Purdue Math
May 6, 2016 · To complete the picture, we can interchange 1-forms and 2-forms using the so called Hodge star operator. ... dx ∧ dy ∧ dz. Given a solid.
[25]
[PDF] Basic Concepts in Differential Geometry - OU Math
We have seen already how the cross product in R3 can be expressed via the exterior product and. Hodge star operator: u × v = ∗(u ∧ v), and also how the inner ...<|control11|><|separator|>
[26]
[PDF] Self-Duality in Four-Dimensional Riemannian Geometry
An oriented Riemannian 4-manifold is self-dual if its Weyl tensor. W=W , i.e. if W --O. Since the Weyl tensor and the star operator are conformal invariants, it ...
[27]
[PDF] Introduction to the Yang-Mills Equations Final Project for Math 581 ...
May 11, 2012 · e1 = dx1 ∧ dx2, e2 = dx1 ∧ dx3, e3 = dx1 ∧ dx4, e4 = dx2 ∧ dx3, e5 = dx2 ∧ dx4, e6 = dx3 ∧ dx4. As computed in Exercise (3.1), we have. ∗e1 = e6 ...
[28]
[PDF] On Electromagnetic Duality
Nov 14, 2018 · The Hodge Dual. For an oriented vector space V of dimension n with a metric tensor, the Hodge star opera- tor provides an isomorphism between ...Missing: Faraday | Show results with:Faraday
[29]
[PDF] class notes on hodge theory - John Etnyre
ψ is the unique element in Ωp,q−1(M) such that h∂. ∗ ψ, ηi = hψ,∂ηi for all η ∈ Ωp,q−1(M). Let ∗ denote the Hodge star operator induced by the Riemannian metric ...
[30]
https://www.math.uh.edu/~minru/Riemann08/hodgetheory.pdf
[31]
https://www.math.uh.edu/~minru/Riemann08/doperator.pdf
[32]
[PDF] 1 Hodge Theory on Riemannian Manifolds - University of Houston
The operator ∗ which sends r-forms to (m − r)-forms. It has the following properties, for any r-forms φ and ψ: (1) φ ∧ ∗ψ =<φ, ...
[33]
[PDF] 1 Differential operators - University of Houston
As the formal adjoint operator of the ex- terior differential operator d, the codifferential operator δ : Λr+1(M) →. Λr(M) is defined by, for every φ ∈ Λr(M),ψ ...