Hodge star operator

Formal definition

Let V be an n-dimensional oriented vector space with a symmetric bilinear form \langle \cdot,\cdot \rangle, referred to here as an inner product. (In more general contexts such as pseudo-Riemannian manifolds and Minkowski space, the bilinear form may not be positive-definite.) This induces an inner product on k-vectors \alpha, \beta\in \bigwedge^{!k}V, for 0 \le k \le n, by defining it on simple k-vectors \alpha = \alpha_1 \wedge \cdots \wedge \alpha_k and \beta = \beta_1 \wedge \cdots \wedge \beta_k to equal the Gram determinant : \langle \alpha, \beta \rangle = \det \left( \left\langle \alpha_i, \beta_j \right\rangle _{i,j=1}^k\right) extended to \bigwedge^{!k}V through linearity. The Gram matrix G of Gram determinants is a 2^n \times 2^n matrix allowing the inner product on the whole of \bigwedge^{!k}V to be expressed as a^{\mathrm T}Gb, where a and b are arbitrary multivectors represented by 2^n \times 1 column matrices with entries corresponding to a fixed ordering of the 2^n basis elements.

The unit n-vector \omega\in{\textstyle\bigwedge}^{!n}V is defined in terms of an oriented orthonormal basis {e_1,\ldots,e_n} of V as: : \omega := \pm e_1\wedge\cdots\wedge e_n, where the sign is free to be chosen and fixed as plus or minus. (Note: In the general pseudo-Riemannian case, orthonormality means \langle e_i,e_j\rangle\in{\delta_{ij},-\delta_{ij}} for all pairs of basis vectors.) With respect to \omega, the right complement of a basis element m is defined as the quantity \overline{m} such that m \wedge \overline{m} = \omega, and this is extended to \bigwedge^{!k}V through linearity.

The Hodge star operator is a linear operator on the exterior algebra of V, mapping k-vectors to (n–k)-vectors, for 0 \le k \le n. It is defined for an arbitrary multivector \alpha by the constructive formula : \star\alpha = \overline{G\alpha}, which applies the Gram matrix and takes the right complement. The Hodge star has the following property, which can be derived from the definition: : \alpha \wedge \star\beta = \langle \alpha,\beta \rangle ,\omega for all k-vectors \alpha,\beta\in {\textstyle\bigwedge}^{!k}V.

Dually, in the space {\textstyle\bigwedge}^{!n}V^* of n-forms (alternating n-multilinear functions on V^n), the dual to \omega is the volume form \det, the function whose value on v_1\wedge\cdots\wedge v_n is the determinant of the n\times n matrix assembled from the column vectors of v_j in e_i-coordinates. Applying \det to the above equation, we obtain the dual property : \det(\alpha \wedge {\star} \beta) = \langle \alpha,\beta \rangle for all k-vectors \alpha,\beta\in {\textstyle\bigwedge}^{!k}V.

Equivalently, taking \alpha = \alpha_1 \wedge \cdots \wedge \alpha_k, \beta = \beta_1 \wedge \cdots \wedge \beta_k, and {\star}\beta = \beta_1^\star \wedge \cdots \wedge \beta_{n-k}^\star: : \det\left(\alpha_1\wedge \cdots \wedge\alpha_k\wedge\beta_1^\star\wedge \cdots \wedge\beta_{n-k}^\star\right) \ = \ \det\left(\langle\alpha_i, \beta_j\rangle\right).

This means that, writing an orthonormal basis of k-vectors as e_I \ = \ e_{i_1}\wedge\cdots\wedge e_{i_k} over all subsets I = {i_1 of [n]={1,\ldots,n}, the Hodge dual is the (n – k)-vector corresponding to the complementary set \bar{I} = [n] \smallsetminus I = \left{\bar i_1 : : \star e_I = s\cdot t\cdot e_\bar{I} , where s\in{1,-1} is the sign of the permutation i_1 \cdots i_k \bar i_1 \cdots \bar i_{n-k} and t\in{1,-1} is the product \langle e_{i_1},e_{i_1}\rangle\cdots \langle e_{i_k},e_{i_k}\rangle. In the Riemannian case, t=1.

The Hodge dual of a multivector m can be calculated with the geometric product using the identity : \star m = \widetilde{m} I, where the tilde denotes the reverse operation, and I is the volume element, or pseudoscalar.

When the bilinear form is nondegenerate, the Hodge star operator takes an orthonormal basis to an orthonormal basis. In this case, it is an isometry on the exterior algebra \bigwedge V.

Geometric explanation

The Hodge star is motivated by the correspondence between a subspace W of V and its orthogonal subspace (with respect to the scalar product), where each space is endowed with an orientation and a numerical scaling factor. Specifically, a non-zero decomposable k-vector w_1\wedge\cdots\wedge w_k\in \textstyle\bigwedge^{!k} V corresponds by the Plücker embedding to the subspace W with oriented basis w_1,\ldots,w_k, endowed with a scaling factor equal to the k-dimensional volume of the parallelepiped spanned by this basis (equal to the Gramian, the determinant of the matrix of scalar products \langle w_i, w_j \rangle). The Hodge star acting on a decomposable vector can be written as a decomposable (n − k)-vector: : {\star}(w_1\wedge\cdots\wedge w_k) ,=, u_1\wedge\cdots\wedge u_{n-k}, where u_1,\ldots,u_{n-k} form an oriented basis of the orthogonal space U = W^\perp!. Furthermore, the (n − k)-volume of the u_i-parallelepiped must equal the k-volume of the w_i-parallelepiped, and w_1,\ldots,w_k,u_1,\ldots,u_{n-k} must form an oriented basis of V.

A general k-vector is a linear combination of decomposable k-vectors, and the definition of Hodge star is extended to general k-vectors by defining it as being linear.

Examples

Two dimensions

In two dimensions with the normalized Euclidean metric and orientation given by the ordering (x, y), the Hodge star on k-forms is given by \begin{align} {\star} , 1 &= dx \wedge dy \ {\star} , dx &= dy \ {\star} , dy &= -dx \ {\star} ( dx \wedge dy ) &= 1 . \end{align}

Three dimensions

A common example of the Hodge star operator is the case , when it can be taken as the correspondence between vectors and bivectors. Specifically, for Euclidean R3 with the basis dx, dy, dz of one-forms often used in vector calculus, one finds that \begin{align} {\star} ,dx &= dy \wedge dz \ {\star} ,dy &= dz \wedge dx \ {\star} ,dz &= dx \wedge dy. \end{align}

The Hodge star relates the exterior and cross product in three dimensions: {\star} (\mathbf{u} \wedge \mathbf{v}) = \mathbf{u} \times \mathbf{v} \qquad {\star} (\mathbf{u} \times \mathbf {v}) = \mathbf{u} \wedge \mathbf{v} . Applied to three dimensions, the Hodge star provides an isomorphism between axial vectors and bivectors, so each axial vector a is associated with a bivector A and vice versa, that is: \mathbf{A} = {\star} \mathbf{a}, \ \ \mathbf{a} = {\star} \mathbf{A}.

The Hodge star can also be interpreted as a form of the geometric correspondence between an axis of rotation and an infinitesimal rotation (see also: 3D rotation group#Lie algebra) around the axis, with speed equal to the length of the axis of rotation. A scalar product on a vector space V gives an isomorphism V\cong V^! identifying V with its dual space, and the vector space L(V,V) is naturally isomorphic to the tensor product V^!!\otimes V\cong V\otimes V. Thus for V = \mathbb{R}^3, the star mapping \textstyle {\star} : V\to\bigwedge^{!2}! V \subset V\otimes V takes each vector \mathbf{v} to a bivector {\star} \mathbf{v} \in V\otimes V, which corresponds to a linear operator L_{\mathbf{v}} : V\to V. Specifically, L_{\mathbf{v}} is a skew-symmetric operator, which corresponds to an infinitesimal rotation: that is, the macroscopic rotations around the axis \mathbb{v} are given by the matrix exponential \exp(t L_{\mathbf{v}}). With respect to the basis dx, dy, dz of \R^3, the tensor dx\otimes dy corresponds to a coordinate matrix with 1 in the dx row and dy column, etc., and the wedge dx\wedge dy ,=, dx\otimes dy - dy\otimes dx is the skew-symmetric matrix \scriptscriptstyle\left[\begin{array}{rrr} ,0!! & !!1 & !!!!0!!!!!! \[-.5em] ,!-1!!&!!0!!&!!!!0!!!!!! \[-.5em] ,0!! & !!0!! & !!!!0!!!!!! \end{array}!!!\right], etc. That is, we may interpret the star operator as: \mathbf{v} = a,dx + b,dy + c,dz \quad\longrightarrow \quad {\star}{\mathbf{v}} \ \cong\ L_{\mathbf{v}} \ = \left[\begin{array}{rrr} 0 & c & -b \ -c & 0 & a \ b & -a & 0 \end{array}\right]. Under this correspondence, cross product of vectors corresponds to the commutator Lie bracket of linear operators: L_{\mathbf{u}\times\mathbf{v}} = L_{\mathbf{v}} L_{\mathbf{u}} - L_{\mathbf{u}} L_{\mathbf{v}}=-\left[L_{\mathbf{u}}, L_{\mathbf{v}}\right]. 3(R)](clifford-algebra-examples-real-and-complex-clifford-algebras), (the vectors {eℓ} are an orthonormal basis in three-dimensional Euclidean space) according to the relations \mathbf{A} = \mathbf{a}i,\quad\mathbf{a} = - \mathbf{A} i.

The dual of a vector is obtained by multiplication by i, as established using the properties of the geometric product of the algebra as follows: \begin{align} \mathbf{a}i &= \left(a_1 \mathbf{e}_1 + a_2 \mathbf{e}_2 + a_3 \mathbf{e}_3\right) \mathbf{e}_1 \mathbf{e}_2 \mathbf{e}_3 \ &= a_1 \mathbf{e}_2 \mathbf{e}_3 (\mathbf{e}_1)^2 + a_2 \mathbf{e}_3 \mathbf{e}_1(\mathbf{e}_2)^2 + a_3 \mathbf{e}_1 \mathbf{e}_2(\mathbf{e}_3)^2 \ &= a_1 \mathbf{e}_2 \mathbf{e}_3 + a_2 \mathbf{e}_3 \mathbf{e}_1 + a_3 \mathbf{e}_1 \mathbf{e}_2 \ &= ({\star} \mathbf{a}) \end{align} and also, in the dual space spanned by {eℓem}: \begin{align} \mathbf{A} i &= \left(A_1 \mathbf{e}_2\mathbf{e}_3 + A_2 \mathbf{e}_3\mathbf{e}_1 + A_3 \mathbf{e}_1\mathbf{e}_2\right) \mathbf{e}_1 \mathbf{e}_2 \mathbf{e}_3 \ &= A_1 \mathbf{e}_1 (\mathbf{e}_2 \mathbf{e}_3)^2 + A_2 \mathbf{e}_2 (\mathbf{e}_3 \mathbf{e}_1)^2 + A_3 \mathbf{e}_3(\mathbf{e}_1 \mathbf{e}_2)^2 \ &= -\left( A_1 \mathbf{e}_1 + A_2 \mathbf{e}_2 + A_3 \mathbf{e}_3 \right) \ &= -({\star} \mathbf{A}) \end{align}

In establishing these results, the identities are used: (\mathbf{e}_1\mathbf{e}_2)^2 = \mathbf{e}_1\mathbf{e}_2\mathbf{e}_1\mathbf{e}_2= -\mathbf{e}_1\mathbf{e}_2\mathbf{e}_2\mathbf{e}_1 = -1 and: \mathit{i}^2 = (\mathbf{e}_1\mathbf{e}_2\mathbf{e}_3)^2 = \mathbf{e}_1\mathbf{e}_2\mathbf{e}_3\mathbf{e}_1\mathbf{e}_2\mathbf{e}_3 = \mathbf{e}_1\mathbf{e}_2\mathbf{e}_3\mathbf{e}_3\mathbf{e}_1\mathbf{e}_2 = \mathbf{e}_1\mathbf{e}_2\mathbf{e}_1\mathbf{e}_2 = -1.

These relations between the dual {\star} and i apply to any vectors. Here they are applied to relate the axial vector created as the cross product to the bivector-valued exterior product of two polar (that is, not axial) vectors u and v; the two products can be written as determinants expressed in the same way: \mathbf a = \mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{e}1 & \mathbf{e}2 & \mathbf{e}3\u_1 & u_2 & u_3\v_1 & v_2 & v_3 \end{vmatrix},,\quad\mathbf A = \mathbf{u} \wedge \mathbf{v} = \begin{vmatrix} \mathbf{e}{2}\mathbf{e}{3} & \mathbf{e}{3}\mathbf{e}{1} & \mathbf{e}{1}\mathbf{e}_{2} \ u_1 & u_2 & u_3 \ v_1 & v_2 & v_3 \end{vmatrix}.

These expressions show these two types of vector are Hodge duals: {\star} (\mathbf{u} \wedge \mathbf{v}) = \mathbf{u \times v},,\quad{\star} (\mathbf{u} \times \mathbf {v}) = \mathbf{u} \wedge \mathbf{v}, as a result of the relations: {\star} \mathbf{e}\ell = \mathbf{e}\ell \mathit{i} = \mathbf{e}_\ell \mathbf{e}_1\mathbf{e}_2\mathbf{e}_3 = \mathbf{e}_m \mathbf{e}n ,, with ℓ, m, n cyclic, and: {\star} ( \mathbf{e}\ell \mathbf{e}m ) = -( \mathbf{e}\ell \mathbf{e}m ) \mathit{i} = -\left( \mathbf{e}\ell \mathbf{e}_m \right)\mathbf{e}_1\mathbf{e}_2\mathbf{e}_3 = \mathbf{e}_n also with ℓ, m, n cyclic.

Using the implementation of {\star} based upon i, the commonly used relations are: \mathbf{u \times v} = -(\mathbf{u} \wedge \mathbf{v}) i ,,\quad \mathbf{u} \wedge \mathbf{v} = (\mathbf{u \times v} ) i \ .

Four dimensions

In case n=4, the Hodge star acts as an endomorphism of the second exterior power (i.e. it maps 2-forms to 2-forms, since ). If the signature of the metric tensor is all positive, i.e. on a Riemannian manifold, then the Hodge star is an involution. If the signature is mixed, i.e., pseudo-Riemannian, then applying the operator twice will return the argument up to a sign – see ** below. This particular endomorphism property of 2-forms in four dimensions makes self-dual and anti-self-dual two-forms natural geometric objects to study. That is, one can describe the space of 2-forms in four dimensions with a basis that "diagonalizes" the Hodge star operator with eigenvalues \pm 1 (or \pm i, depending on the signature).

For concreteness, we discuss the Hodge star operator in Minkowski spacetime where n=4 with metric signature (− + + +) and coordinates (t,x,y,z). The volume form is oriented as \varepsilon_{0123} = 1. For one-forms, \begin{align} {\star} dt &= -dx \wedge dy \wedge dz ,, \ {\star} dx &= -dt \wedge dy \wedge dz ,, \ {\star} dy &= -dt \wedge dz \wedge dx ,, \ {\star} dz &= -dt \wedge dx \wedge dy ,, \end{align} while for 2-forms, \begin{align} {\star} (dt \wedge dx) &= - dy \wedge dz ,, \ {\star} (dt \wedge dy) &= - dz \wedge dx ,, \ {\star} (dt \wedge dz) &= - dx \wedge dy ,, \ {\star} (dx \wedge dy) &= dt \wedge dz ,, \ {\star} (dz \wedge dx) &= dt \wedge dy ,, \ {\star} (dy \wedge dz) &= dt \wedge dx ,. \end{align}

These are summarized in the index notation as \begin{align} {\star} (dx^\mu) &= \eta^{\mu\lambda} \varepsilon_{\lambda\nu\rho\sigma} \frac{1}{3!} dx^\nu \wedge dx^\rho \wedge dx^\sigma ,,\ {\star} (dx^\mu \wedge dx^\nu) &= \eta^{\mu\kappa} \eta^{\nu\lambda} \varepsilon_{\kappa\lambda\rho\sigma} \frac{1}{2!} dx^\rho \wedge dx^\sigma ,. \end{align}

Hodge dual of three- and four-forms can be easily deduced from the fact that, in the Lorentzian signature, {\star}^2=1 for odd-rank forms and {\star}^2=-1 for even-rank forms. An easy rule to remember for these Hodge operations is that given a form \alpha, its Hodge dual {\star}\alpha may be obtained by writing the components not involved in \alpha in an order such that \alpha \wedge ({\star} \alpha) = dt \wedge dx \wedge dy \wedge dz . An extra minus sign will enter only if \alpha contains dt. (For (+ − − −), one puts in a minus sign only if \alpha involves an odd number of the space-associated forms dx, dy and dz.)

Note that the combinations (dx^\mu \wedge dx^\nu)^{\pm} := \frac{1}{2} \big( dx^\mu \wedge dx^\nu \mp i {\star} (dx^\mu \wedge dx^\nu) \big) take \pm i as the eigenvalue for Hodge star operator, i.e., {\star} (dx^\mu \wedge dx^\nu)^{\pm} = \pm i (dx^\mu \wedge dx^\nu)^{\pm} , and hence deserve the name self-dual and anti-self-dual two-forms. Understanding the geometry, or kinematics, of Minkowski spacetime in self-dual and anti-self-dual sectors turns out to be insightful in both mathematical and physical perspectives, making contacts to the use of the two-spinor language in modern physics such as spinor-helicity formalism or twistor theory.

Conformal invariance

The Hodge star is conformally invariant on n-forms on a 2n-dimensional vector space V , i.e. if g is a metric on V and \lambda 0 , then the induced Hodge stars {\star}g, {\star}{\lambda g} : \Lambda^n V \to \Lambda^n V are the same.

Example: Derivatives in three dimensions

The combination of the {\star} operator and the exterior derivative d generates the classical operators grad, curl, and div on vector fields in three-dimensional Euclidean space. This works out as follows: d takes a 0-form (a function) to a 1-form, a 1-form to a 2-form, and a 2-form to a 3-form (and takes a 3-form to zero). For a 0-form f = f(x,y,z), the first case written out in components gives: df = \frac{\partial f}{\partial x} , dx + \frac{\partial f}{\partial y} , dy + \frac{\partial f}{\partial z} , dz.

The scalar product identifies 1-forms with vector fields as dx \mapsto (1,0,0), etc., so that df becomes \operatorname{grad} f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right).

In the second case, a vector field \mathbf F = (A,B,C) corresponds to the 1-form \varphi = A,dx + B,dy + C,dz, which has exterior derivative: d\varphi = \left(\frac{\partial C}{\partial y} - \frac{\partial B}{\partial z}\right) dy\wedge dz + \left(\frac{\partial C}{\partial x} - \frac{\partial A}{\partial z}\right) dx\wedge dz + \left({\partial B \over \partial x} - \frac{\partial A}{\partial y}\right) dx\wedge dy.

Applying the Hodge star gives the 1-form: {\star} d\varphi = \left({\partial C \over \partial y} - {\partial B \over \partial z} \right) , dx - \left({\partial C \over \partial x} - {\partial A \over \partial z} \right) , dy + \left({\partial B \over \partial x} - {\partial A \over \partial y}\right) , dz, which becomes the vector field \operatorname{curl}\mathbf{F} = \left( \frac{\partial C}{\partial y} - \frac{\partial B}{\partial z},, -\frac{\partial C}{\partial x} + \frac{\partial A}{\partial z},, \frac{\partial B}{\partial x} - \frac{\partial A}{\partial y} \right).

In the third case, \mathbf F = (A,B,C) again corresponds to \varphi = A,dx + B,dy + C,dz. Applying Hodge star, exterior derivative, and Hodge star again: \begin{align} {\star}\varphi &= A,dy\wedge dz-B,dx\wedge dz+C,dx\wedge dy, \ d{\star\varphi} &= \left(\frac{\partial A}{\partial x}+\frac{\partial B}{\partial y}+\frac{\partial C}{\partial z}\right)dx\wedge dy\wedge dz, \ {\star} d{\star}\varphi &= \frac{\partial A}{\partial x}+\frac{\partial B}{\partial y}+\frac{\partial C}{\partial z} = \operatorname{div}\mathbf{F}. \end{align}

One advantage of this expression is that the identity , which is true in all cases, has as special cases two other identities: (1) , and (2) . In particular, Maxwell's equations take on a particularly simple and elegant form, when expressed in terms of the exterior derivative and the Hodge star. The expression {\star}d{\star} (multiplied by an appropriate power of −1) is called the codifferential; it is defined in full generality, for any dimension, further in the article below.

One can also obtain the Laplacian in terms of the above operations: \Delta f = {\star}d{\star}d f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}.

The Laplacian can also be seen as a special case of the more general Laplace–deRham operator \Delta = d\delta + \delta d where in three dimensions, \delta = (-1)^k {\star} d{\star} is the codifferential for k-forms. Any function f is a 0-form, and \delta f = 0 and so this reduces to the ordinary Laplacian. For the 1-form \varphi above, the codifferential is \delta = - {\star} d{\star} and after some straightforward calculations one obtains the Laplacian acting on \varphi.

Duality

When the bilinear form is nondegenerate, applying the Hodge star twice leaves a k-vector unchanged up to a sign: for \eta\in {\textstyle\bigwedge}^k V in an n-dimensional space V, one has : {\star} {\star} \eta = (-1)^{k(n-k)} s, \eta , where s is the parity of the signature of the scalar product on V, that is, the sign of the determinant of the matrix of the scalar product with respect to any basis. For example, if and the signature of the scalar product is either (+ − − −) or (− + + +) then . For Riemannian manifolds (including Euclidean spaces), we always have .

The above identity implies that the inverse of {\star} can be given as : \begin{align} {\star}^{-1}: ~ {\textstyle\bigwedge}^{!k} V &\to {\textstyle\bigwedge}^{!n-k} V \ \eta &\mapsto (-1)^{k(n-k)} !s, {\star} \eta \end{align}

If n is odd then k(n − k) is even for any k, whereas if n is even then k(n − k) has the parity of k. Therefore: : {\star}^{-1} = \begin{cases} s, {\star} & n \text{ is odd} \ (-1)^k s, {\star} & n \text{ is even} \end{cases} where k is the degree of the element operated on.

On manifolds

For an n-dimensional oriented pseudo-Riemannian manifold M, we apply the construction above to each cotangent space \text{T}^_p M and its exterior powers \bigwedge^k\text{T}^_p M, and hence to the differential k-forms \zeta\in\Omega^k(M) = \Gamma\left(\bigwedge^k\text{T}^!M\right), the global sections of the bundle \bigwedge^k \mathrm{T}^! M\to M. The Riemannian metric induces a scalar product on \bigwedge^k \text{T}^*_p M at each point p\in M. We define the Hodge dual of a k-form \zeta , defining {\star} \zeta as the unique (n – k)-form satisfying \eta\wedge {\star} \zeta \ =\ \langle \eta, \zeta \rangle , \omega for every k-form \eta , where \langle\eta,\zeta\rangle is a real-valued function on M, and the volume form \omega is induced by the pseudo-Riemannian metric. Integrating this equation over M, the right side becomes the L^2 (square-integrable) scalar product on k-forms, and we obtain: \int_M \eta\wedge {\star} \zeta \ =\ \int_M \langle\eta,\zeta\rangle\ \omega.

More generally, if M is non-orientable, one can define the Hodge star of a k-form as a (n – k)-pseudo differential form; that is, a differential form with values in the canonical line bundle.

Computation in index notation

We compute in terms of tensor index notation with respect to a (not necessarily orthonormal) basis \left{\frac{\partial}{\partial x_1}, \ldots, \frac{\partial}{\partial x_n}\right} in a tangent space V = T_p M and its dual basis {dx_1,\ldots,dx_n} in V^* = T^*p M, having the metric matrix (g{ij}) = \left(\left\langle \frac{\partial}{\partial x_i}, \frac{\partial}{\partial x_j}\right\rangle\right) and its inverse matrix (g^{ij}) = (\langle dx^i, dx^j\rangle). The Hodge dual of a decomposable k-form is: {\star}\left(dx^{i_1} \wedge \dots \wedge dx^{i_k}\right) \ =
\frac{\sqrt{\left|\det [g_{ij}]\right|}}{(n-k)!} g^{i_1 j_1} \cdots g^{i_k j_k} \varepsilon_{j_1 \dots j_n} dx^{j_{k+1}} \wedge \dots \wedge dx^{j_n}.

Here \varepsilon_{j_1 \dots j_n} is the Levi-Civita symbol with \varepsilon_{1 \dots n} = 1, and we implicitly take the sum over all values of the repeated indices j_1,\ldots,j_n. The factorial (n-k)! accounts for double counting, and is not present if the summation indices are restricted so that j_{k+1} . The absolute value of the determinant is necessary since it may be negative, as for tangent spaces to Lorentzian manifolds.

An arbitrary differential form can be written as follows: \alpha \ =\ \frac{1}{k!}\alpha_{i_1, \dots, i_k} dx^{i_1}\wedge \dots \wedge dx^{i_k} \ =\ \sum_{i_1

The factorial k! is again included to account for double counting when we allow non-increasing indices. We would like to define the dual of the component \alpha_{i_1, \dots, i_k} so that the Hodge dual of the form is given by {\star}\alpha = \frac{1}{(n-k)!}({\star} \alpha){i{k+1}, \dots, i_n} dx^{i_{k+1}} \wedge \dots \wedge dx^{i_n}.

Using the above expression for the Hodge dual of dx^{i_1} \wedge \dots \wedge dx^{i_k}, we find: ({\star} \alpha){j{k+1}, \dots, j_n} = \frac{\sqrt{\left|\det [g_{ab}]\right|}}{k!} \alpha_{i_1, \dots, i_k},g^{i_1 j_1}\cdots g^{i_k j_k} ,\varepsilon_{j_1, \dots, j_n}, .

Although one can apply this expression to any tensor \alpha, the result is antisymmetric, since contraction with the completely anti-symmetric Levi-Civita symbol cancels all but the totally antisymmetric part of the tensor. It is thus equivalent to antisymmetrization followed by applying the Hodge star.

The unit volume form \omega = {\star} 1\in \bigwedge^n V^* is given by: \omega = \sqrt{ \left| \det [g_{ij}] \right| };dx^1 \wedge \cdots \wedge dx^n .

Codifferential

The most important application of the Hodge star on manifolds is to define the codifferential \delta on k-forms. Let \delta = (-1)^{n(k + 1) + 1} s\ {\star} d {\star} = (-1)^{k}, {\star}^{-1} d {\star} where d is the exterior derivative or differential, and s = 1 for Riemannian manifolds. Then d:\Omega^k(M)\to \Omega^{k+1}(M) while \delta:\Omega^k(M)\to \Omega^{k-1}(M).

The codifferential is not an antiderivation on the exterior algebra, in contrast to the exterior derivative.

The codifferential is the adjoint of the exterior derivative with respect to the square-integrable scalar product: \langle!\langle\eta,\delta \zeta\rangle!\rangle \ =\ \langle!\langle d\eta,\zeta\rangle!\rangle, where \zeta is a k-form and \eta a (k!-!1)-form. This property is useful as it can be used to define the codifferential even when the manifold is non-orientable (and the Hodge star operator not defined). The identity can be proved from Stokes' theorem for smooth forms: 0 \ =\ \int_M d (\eta \wedge {\star} \zeta) \ =
\int_M \left(d \eta \wedge {\star} \zeta + (-1)^{k-1}\eta \wedge {\star} ,{\star}^{-1} d, {\star} \zeta\right) \ =
\langle!\langle d\eta,\zeta\rangle!\rangle - \langle!\langle\eta,\delta\zeta\rangle!\rangle, provided M has empty boundary, or \eta or {\star}\zeta has zero boundary values. (The proper definition of the above requires specifying a topological vector space that is closed and complete on the space of smooth forms. The Sobolev space is conventionally used; it allows the convergent sequence of forms \zeta_i \to \zeta (as i \to \infty) to be interchanged with the combined differential and integral operations, so that \langle!\langle\eta,\delta \zeta_i\rangle!\rangle \to \langle!\langle\eta,\delta \zeta\rangle!\rangle and likewise for sequences converging to \eta.)

Since the differential satisfies d^2 = 0, the codifferential has the corresponding property \delta^2 = (-1)^n s^2 {\star} d {\star} {\star} d {\star} = (-1)^{nk+k+1} s^3 {\star} d^2 {\star} = 0.

The Laplace–deRham operator is given by \Delta = (\delta + d)^2 = \delta d + d\delta and lies at the heart of Hodge theory. It is symmetric: \langle!\langle\Delta \zeta,\eta\rangle!\rangle = \langle!\langle\zeta,\Delta \eta\rangle!\rangle and non-negative: \langle!\langle\Delta\eta,\eta\rangle!\rangle \ge 0.

The Hodge star sends harmonic forms to harmonic forms. As a consequence of Hodge theory, the de Rham cohomology is naturally isomorphic to the space of harmonic k-forms, and so the Hodge star induces an isomorphism of cohomology groups {\star} : H^k_\Delta (M) \to H^{n-k}_\Delta(M), which in turn gives canonical identifications via Poincaré duality of H k(M) with its dual space.

In coordinates, with notation as above, the codifferential of the form \alpha may be written as \delta \alpha=\ -\frac{1}{k!}g^{ml}\left(\frac{\partial}{\partial x_l} \alpha_{m,i_1, \dots, i_{k-1}} - \Gamma^j_{ml} \alpha_{j,i_1, \dots, i_{k-1}} \right) dx^{i_1} \wedge \dots \wedge dx^{i_{k-1}}, where here \Gamma^{j}_{ml} denotes the Christoffel symbols of \left{\frac{\partial}{\partial x_1}, \ldots, \frac{\partial}{\partial x_n}\right}.

Poincare lemma for codifferential

In analogy to the Poincare lemma for exterior derivative, one can define its version for codifferential, which reads : If \delta\omega=0 for \omega \in \Lambda^{k}(U), where * U is a star domain on a manifold, then there is \alpha \in \Lambda^{k+1}(U) such that \omega=\delta\alpha.*

A practical way of finding \alpha is to use cohomotopy operator h, that is a local inverse of \delta. One has to define a homotopy operator : H\beta = \int_{0}^{1} \mathcal{K}\lrcorner\beta|{F(t,x)}t^{k}dt, where F(t,x)=x{0}+t(x-x_{0}) is the linear homotopy between its center x_{0}\in U and a point x \in U, and the (Euler) vector \mathcal{K}=\sum_{i=1}^{n}(x-x_{0})^{i}\partial_{x^{i}} for n=\dim(U) is inserted into the form \beta \in \Lambda^{*}(U). We can then define cohomotopy operator as : h:\Lambda(U)\rightarrow \Lambda(U), \quad h:=\eta {\star}^{-1}H\star, where \eta \beta = (-1)^{k}\beta for \beta \in \Lambda^{k}(U).

The cohomotopy operator fulfills (co)homotopy invariance formula : \delta h + h\delta = I - S_{x_{0}} , where S_{x_{0}}={\star}^{-1}s_{x_{0}}^{}{\star}, and s_{x_{0}}^{} is the pullback along the constant map s_{x_{0}}:x \rightarrow x_{0}.

Therefore, if we want to solve the equation \delta \omega =0, applying cohomotopy invariance formula we get : \omega= \delta h\omega + S_{x_{0}}\omega, where h\omega\in \Lambda^{k+1}(U) is a differential form we are looking for, and "constant of integration" S_{x_{0}}\omega vanishes unless \omega is a top form.

Cohomotopy operator fulfills the following properties: h^{2}=0, \quad \delta h \delta =\delta, \quad h\delta h =h. They make it possible to use it to define anticoexact forms on U by \mathcal{Y}(U)={ \omega\in\Lambda(U)| \omega = h\delta \omega }, which together with exact forms \mathcal{C}(U) ={ \omega\in\Lambda(U)|\omega = \delta h\omega } make a direct sum decomposition : \Lambda(U)=\mathcal{C}(U)\oplus \mathcal{Y}(U).

This direct sum is another way of saying that the cohomotopy invariance formula is a decomposition of unity, and the projector operators on the summands fulfills idempotence formulas: (h\delta)^{2}=h\delta, \quad (\delta h)^{2}=\delta h.

These results are extension of similar results for exterior derivative.

Citations

References

• David Bleecker (1981) Gauge Theory and Variational Principles. Addison-Wesley Publishing. . Chpt. 0 contains a condensed review of non-Riemannian differential geometry.
• Charles W. Misner, Kip S. Thorne, John Archibald Wheeler (1970) Gravitation. W.H. Freeman. . A basic review of differential geometry in the special case of four-dimensional spacetime.
• Steven Rosenberg (1997) The Laplacian on a Riemannian manifold. Cambridge University Press. . An introduction to the heat equation and the Atiyah–Singer theorem.
• Tevian Dray (1999) The Hodge Dual Operator. A thorough overview of the definition and properties of the Hodge star operator.

References

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8. Kycia, Radosław Antoni. (2022-07-29). "The Poincare Lemma for Codifferential, Anticoexact Forms, and Applications to Physics". Results in Mathematics.
9. Edelen, Dominic G. B.. (2005). "Applied exterior calculus".