Hodge star operator

Template:Short description In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the Hodge dual of the element. This map was introduced by W. V. D. Hodge.

For example, in an oriented 3-dimensional Euclidean space, an oriented plane can be represented by the exterior product of two basis vectors, and its Hodge dual is the normal vector given by their cross product; conversely, any vector is dual to the oriented plane perpendicular to it, endowed with a suitable bivector. Generalizing this to an Template:Math-dimensional vector space, the Hodge star is a one-to-one mapping of Template:Math-vectors to Template:Math-vectors; the dimensions of these spaces are the binomial coefficients <math>\tbinom nk = \tbinom{n}{n - k}</math>.

The naturalness of the star operator means it can play a role in differential geometry when applied to the cotangent bundle of a pseudo-Riemannian manifold, and hence to [[Differential form|differential Template:Math-forms]]. This allows the definition of the codifferential as the Hodge adjoint of the exterior derivative, leading to the Laplace–de Rham operator. This generalizes the case of 3-dimensional Euclidean space, in which divergence of a vector field may be realized as the codifferential opposite to the gradient operator, and the Laplace operator on a function is the divergence of its gradient. An important application is the Hodge decomposition of differential forms on a closed Riemannian manifold.

Formal definition for k-vectors

Let Template:Math be an [[Dimension (vector space)|Template:Math-dimensional]] oriented vector space with a nondegenerate symmetric bilinear form <math>\langle \cdot,\cdot \rangle</math>, referred to here as a scalar product. (In more general contexts such as pseudo-Riemannian manifolds and Minkowski space, the bilinear form may not be positive-definite.) This induces a scalar product on [[Multivector|Template:Math-vectors]] Template:Nowrap for <math>0 \le k \le n</math>, by defining it on simple Template:Math-vectors <math>\alpha = \alpha_1 \wedge \cdots \wedge \alpha_k</math> and <math>\beta = \beta_1 \wedge \cdots \wedge \beta_k</math> to equal the Gram determinant<ref name="HF">Harley Flanders (1963) Differential Forms with Applications to the Physical Sciences, Academic Press</ref>Template:Rp

<math> \langle \alpha, \beta \rangle = \det \left( \left\langle \alpha_i, \beta_j \right\rangle _{i,j=1}^k\right)</math>

extended to <math display="inline">\bigwedge^{\!k}V</math> through linearity.

The unit Template:Math-vector <math>\omega\in{\textstyle\bigwedge}^{\!n}V</math> is defined in terms of an oriented orthonormal basis <math>\{e_1,\ldots,e_n\}</math> of Template:Math as:

<math>\omega := e_1\wedge\cdots\wedge e_n.</math>

(Note: In the general pseudo-Riemannian case, orthonormality means <math> \langle e_i,e_j\rangle \in\{\delta_{ij},-\delta_{ij}\}</math> for all pairs of basis vectors.) The Hodge star operator is a linear operator on the exterior algebra of Template:Math, mapping Template:Math-vectors to (Template:Math)-vectors, for <math>0 \le k \le n</math>. It has the following property, which defines it completely:<ref name="HF" />Template:Rp

<math>\alpha \wedge ({\star} \beta) = \langle \alpha,\beta \rangle \,\omega </math> for all Template:Math-vectors <math>\alpha,\beta\in {\textstyle\bigwedge}^{\!k}V .</math>

Dually, in the space <math>{\textstyle\bigwedge}^{\!n}V^*</math> of Template:Math-forms (alternating Template:Math-multilinear functions on <math>V^n</math>), the dual to <math>\omega</math> is the volume form <math>\det</math>, the function whose value on <math>v_1\wedge\cdots\wedge v_n</math> is the determinant of the <math>n\times n</math> matrix assembled from the column vectors of <math>v_j</math> in <math>e_i</math>-coordinates. Applying <math>\det</math> to the above equation, we obtain the dual definition:

<math>\det(\alpha \wedge {\star} \beta) = \langle \alpha,\beta \rangle </math> for all Template:Math-vectors <math>\alpha,\beta\in {\textstyle\bigwedge}^{\!k}V .</math>

Equivalently, taking <math>\alpha = \alpha_1 \wedge \cdots \wedge \alpha_k</math>, <math>\beta = \beta_1 \wedge \cdots \wedge \beta_k</math>, and <math>{\star}\beta = \beta_1^\star \wedge \cdots \wedge \beta_{n-k}^\star</math>:

<math>

 \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).

</math>

This means that, writing an orthonormal basis of Template:Math-vectors as <math>e_I \ = \ e_{i_1}\wedge\cdots\wedge e_{i_k}</math> over all subsets <math>I = \{i_1<\cdots<i_k\}</math> of <math>[n]=\{1,\ldots,n\}</math>, the Hodge dual is the (Template:Math)-vector corresponding to the complementary set <math>\bar{I} = [n] \smallsetminus I = \left\{\bar i_1 < \cdots < \bar i_{n-k}\right\}</math>:

<math>{\star} e_I = s\cdot t\cdot e_\bar{I} ,</math>

where <math>s\in\{1,-1\}</math> is the sign of the permutation <math>i_1 \cdots i_k \bar i_1 \cdots \bar i_{n-k}</math> and <math>t\in\{1,-1\}</math> is the product <math>\langle e_{i_1},e_{i_1}\rangle\cdots \langle e_{i_k},e_{i_k}\rangle</math>. In the Riemannian case, <math>t=1</math>.

Since Hodge star takes an orthonormal basis to an orthonormal basis, it is an isometry on the exterior algebra <math display="inline">\bigwedge V</math>.

Geometric explanation

The Hodge star is motivated by the correspondence between a subspace Template:Math of Template:Math 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 Template:Math-vector <math>w_1\wedge\cdots\wedge w_k\in \textstyle\bigwedge^{\!k} V</math> corresponds by the Plücker embedding to the subspace <math>W</math> with oriented basis <math>w_1,\ldots,w_k</math>, endowed with a scaling factor equal to the Template:Math-dimensional volume of the parallelepiped spanned by this basis (equal to the Gramian, the determinant of the matrix of scalar products <math>\langle w_i, w_j \rangle</math>). The Hodge star acting on a decomposable vector can be written as a decomposable (Template:Math)-vector:

<math>{\star}(w_1\wedge\cdots\wedge w_k) \,=\, u_1\wedge\cdots\wedge u_{n-k},</math>

where <math>u_1,\ldots,u_{n-k}</math> form an oriented basis of the orthogonal space <math>U = W^\perp\!</math>. Furthermore, the (Template:Math)-volume of the <math>u_i</math>-parallelepiped must equal the Template:Math-volume of the <math>w_i</math>-parallelepiped, and <math>w_1,\ldots,w_k,u_1,\ldots,u_{n-k}</math> must form an oriented basis of <math>V</math>.

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

Examples

Two dimensions

In two dimensions with the normalized Euclidean metric and orientation given by the ordering Template:Math, the Hodge star on Template:Math-forms is given by <math display="block">\begin{align}

             {\star} \, 1 &= dx \wedge dy \\
            {\star} \, dx &= dy \\
            {\star} \, dy &= -dx \\
 {\star} ( dx \wedge dy ) &= 1 .

\end{align}</math>

Three dimensions

A common example of the Hodge star operator is the case Template:Math, when it can be taken as the correspondence between vectors and bivectors. Specifically, for Euclidean R³ with the basis <math>dx, dy, dz</math> of one-forms often used in vector calculus, one finds that <math display="block">\begin{align}

 {\star} \,dx &= dy \wedge dz \\
 {\star} \,dy &= dz \wedge dx \\
 {\star} \,dz &= dx \wedge dy.

\end{align}</math>

The Hodge star relates the exterior and cross product in three dimensions:<ref name="Lounesto" /> <math display="block">{\star} (\mathbf{u} \wedge \mathbf{v}) = \mathbf{u} \times \mathbf{v} \qquad {\star} (\mathbf{u} \times \mathbf {v}) = \mathbf{u} \wedge \mathbf{v} .</math> Applied to three dimensions, the Hodge star provides an isomorphism between axial vectors and bivectors, so each axial vector Template:Math is associated with a bivector Template:Math and vice versa, that is:<ref name="Lounesto">Template:Cite book</ref> <math>\mathbf{A} = {\star} \mathbf{a}, \ \ \mathbf{a} = {\star} \mathbf{A}</math>.

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 <math>V</math> gives an isomorphism <math>V\cong V^*\!</math> identifying <math>V</math> with its dual space, and the vector space <math>L(V,V)</math> is naturally isomorphic to the tensor product <math>V^*\!\!\otimes V\cong V\otimes V</math>. Thus for <math>V = \mathbb{R}^3</math>, the star mapping <math display="inline">\textstyle {\star} : V\to\bigwedge^{\!2}\! V \subset V\otimes V</math> takes each vector <math>\mathbf{v}</math> to a bivector <math>{\star} \mathbf{v} \in V\otimes V</math>, which corresponds to a linear operator <math>L_{\mathbf{v}} : V\to V</math>. Specifically, <math>L_{\mathbf{v}}</math> is a skew-symmetric operator, which corresponds to an infinitesimal rotation: that is, the macroscopic rotations around the axis <math>\mathbb{v}</math> are given by the matrix exponential <math>\exp(t L_{\mathbf{v}})</math>. With respect to the basis <math>dx, dy, dz</math> of <math>\R^3</math>, the tensor <math>dx\otimes dy</math> corresponds to a coordinate matrix with 1 in the <math>dx</math> row and <math>dy</math> column, etc., and the wedge <math>dx\wedge dy \,=\, dx\otimes dy - dy\otimes dx</math> is the skew-symmetric matrix <math>\scriptscriptstyle\left[\begin{array}{rrr} \,0\!\! & \!\!1 & \!\!\!\!0\!\!\!\!\!\! \\[-.5em] \,\!-1\!\!&\!\!0\!\!&\!\!\!\!0\!\!\!\!\!\! \\[-.5em] \,0\!\! & \!\!0\!\! & \!\!\!\!0\!\!\!\!\!\! \end{array}\!\!\!\right]</math>, etc. That is, we may interpret the star operator as: <math display="block"> \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].</math> Under this correspondence, cross product of vectors corresponds to the commutator Lie bracket of linear operators: <math>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]</math>.

Four dimensions

In case <math>n=4</math>, the Hodge star acts as an endomorphism of the second exterior power (i.e. it maps 2-forms to 2-forms, since Template:Math). 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 Template:Section link 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 <math>\pm 1</math> (or <math>\pm i</math>, depending on the signature).

For concreteness, we discuss the Hodge star operator in Minkowski spacetime where <math>n=4</math> with metric signature Template:Math and coordinates <math>(t,x,y,z)</math>. The volume form is oriented as <math>\varepsilon_{0123} = 1</math>. For one-forms, <math display="block">\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}</math> while for 2-forms, <math display="block">\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}</math>

These are summarized in the index notation as <math display="block">\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}</math>

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

Note that the combinations <math display="block"> (dx^\mu \wedge dx^\nu)^{\pm} := \frac{1}{2} \big( dx^\mu \wedge dx^\nu \mp i {\star} (dx^\mu \wedge dx^\nu) \big)</math> take <math>\pm i</math> as the eigenvalue for Hodge star operator, i.e., <math display="block"> {\star} (dx^\mu \wedge dx^\nu)^{\pm} = \pm i (dx^\mu \wedge dx^\nu)^{\pm} , </math> 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 Template:Math-forms on a Template:Math-dimensional vector space <math> V </math>, i.e. if <math> g </math> is a metric on <math> V </math> and <math> \lambda > 0 </math>, then the induced Hodge stars <math display="block"> {\star}_g, {\star}_{\lambda g} : \Lambda^n V \to \Lambda^n V</math> are the same.

Example: Derivatives in three dimensions

The combination of the <math>{\star}</math> operator and the exterior derivative Template:Math generates the classical operators Template:Math, Template:Math, and Template:Math on vector fields in three-dimensional Euclidean space. This works out as follows: Template:Math 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 <math>f = f(x,y,z)</math>, the first case written out in components gives: <math display="block">df = \frac{\partial f}{\partial x} \, dx + \frac{\partial f}{\partial y} \, dy + \frac{\partial f}{\partial z} \, dz.</math>

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

In the second case, a vector field <math>\mathbf F = (A,B,C)</math> corresponds to the 1-form <math>\varphi = A\,dx + B\,dy + C\,dz</math>, which has exterior derivative: <math display="block">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.</math>

Applying the Hodge star gives the 1-form: <math display="block">{\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,</math> which becomes the vector field <math display="inline">\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)</math>.

In the third case, <math>\mathbf F = (A,B,C)</math> again corresponds to <math>\varphi = A\,dx + B\,dy + C\,dz</math>. Applying Hodge star, exterior derivative, and Hodge star again: <math display="block">\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}</math>

One advantage of this expression is that the identity Template:Math, which is true in all cases, has as special cases two other identities: (1) Template:Math, and (2) Template:Math. 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 <math>{\star}d{\star}</math> (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 Template:Math in terms of the above operations: <math display="block"> \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}.</math>

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

Duality

Applying the Hodge star twice leaves a Template:Math-vector unchanged up to a sign: for <math>\eta\in {\textstyle\bigwedge}^k V</math> in an Template:Mvar-dimensional space Template:Math, one has

<math>{\star} {\star} \eta = (-1)^{k(n-k)} s\, \eta ,</math>

where Template:Mvar is the parity of the signature of the scalar product on Template:Math, that is, the sign of the determinant of the matrix of the scalar product with respect to any basis. For example, if Template:Math and the signature of the scalar product is either Template:Math or Template:Math then Template:Math. For Riemannian manifolds (including Euclidean spaces), we always have Template:Math.

The above identity implies that the inverse of <math>{\star}</math> can be given as

<math>\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}</math>

If Template:Mvar is odd then Template:Math is even for any Template:Mvar, whereas if Template:Mvar is even then Template:Math has the parity of Template:Mvar. Therefore:

<math>{\star}^{-1} = \begin{cases} s\, {\star} & n \text{ is odd} \\ (-1)^k s\, {\star} & n \text{ is even} \end{cases}</math>

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

More generally, if <math>M</math> 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 <math display="inline">\left\{\frac{\partial}{\partial x_1}, \ldots, \frac{\partial}{\partial x_n}\right\}</math> in a tangent space <math>V = T_p M</math> and its dual basis <math>\{dx_1,\ldots,dx_n\}</math> in <math>V^* = T^*_p M</math>, having the metric matrix <math display="inline">(g_{ij}) = \left(\left\langle \frac{\partial}{\partial x_i}, \frac{\partial}{\partial x_j}\right\rangle\right)</math> and its inverse matrix <math>(g^{ij}) = (\langle dx^i, dx^j\rangle)</math>. The Hodge dual of a decomposable k-form is: <math display="block"> {\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}. </math>

Here <math>\varepsilon_{j_1 \dots j_n}</math> is the Levi-Civita symbol with <math>\varepsilon_{1 \dots n} = 1</math>, and we implicitly take the sum over all values of the repeated indices <math> j_1,\ldots,j_n</math>. The factorial <math>(n-k)!</math> accounts for double counting, and is not present if the summation indices are restricted so that <math>j_{k+1} < \dots < j_n</math>. 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: <math display="block"> \alpha \ =\ \frac{1}{k!}\alpha_{i_1, \dots, i_k} dx^{i_1}\wedge \dots \wedge dx^{i_k} \ =\ \sum_{i_1 < \dots < i_k} \alpha_{i_1, \dots, i_k} dx^{i_1}\wedge \dots \wedge dx^{i_k}. </math>

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

Using the above expression for the Hodge dual of <math>dx^{i_1} \wedge \dots \wedge dx^{i_k}</math>, we find:<ref>Template:Cite book</ref> <math display="block"> ({\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}\, . </math>

Although one can apply this expression to any tensor <math>\alpha</math>, 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 <math display="inline">\omega = {\star} 1\in \bigwedge^n V^*</math> is given by: <math display="block">\omega = \sqrt{ \left| \det [g_{ij}] \right| }\;dx^1 \wedge \cdots \wedge dx^n .</math>

Codifferential

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

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: <math display="block"> \langle\!\langle\eta,\delta \zeta\rangle\!\rangle \ =\ \langle\!\langle d\eta,\zeta\rangle\!\rangle, </math> where <math> \zeta </math> is a <math>k</math>-form and <math> \eta </math> a <math>(k\!-\!1)</math>-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: <math display="block"> 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, </math> provided <math>M</math> has empty boundary, or <math> \eta </math> or <math>{\star}\zeta</math> 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 <math>\zeta_i \to \zeta</math> (as <math>i \to \infty</math>) to be interchanged with the combined differential and integral operations, so that <math>\langle\!\langle\eta,\delta \zeta_i\rangle\!\rangle \to \langle\!\langle\eta,\delta \zeta\rangle\!\rangle</math> and likewise for sequences converging to <math>\eta</math>.)

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

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

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 Template:Mvar-forms, and so the Hodge star induces an isomorphism of cohomology groups <math display="block">{\star} : H^k_\Delta (M) \to H^{n-k}_\Delta(M),</math> which in turn gives canonical identifications via Poincaré duality of Template:Math with its dual space.

In coordinates, with notation as above, the codifferential of the form <math>\alpha</math> may be written as <math display="block">\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}},</math> where here <math>\Gamma^{j}_{ml}</math> denotes the Christoffel symbols of <math display="inline">\left\{\frac{\partial}{\partial x_1}, \ldots, \frac{\partial}{\partial x_n}\right\}</math>.

Poincare lemma for codifferential

In analogy to the Poincare lemma for exterior derivative, one can define its version for codifferential, which reads<ref name=":0">Template:Cite journal</ref>

If <math>\delta\omega=0</math> for <math>\omega \in \Lambda^{k}(U)</math>, where <math>U</math> is a star domain on a manifold, then there is <math>\alpha \in \Lambda^{k+1}(U)</math> such that <math>\omega=\delta\alpha</math>.

A practical way of finding <math>\alpha</math> is to use cohomotopy operator <math>h</math>, that is a local inverse of <math>\delta</math>. One has to define a homotopy operator<ref name=":0" />

<math>H\beta = \int_{0}^{1} \mathcal{K}\lrcorner\beta|_{F(t,x)}t^{k}dt,</math>

where <math>F(t,x)=x_{0}+t(x-x_{0})</math> is the linear homotopy between its center <math>x_{0}\in U</math> and a point <math>x \in U</math>, and the (Euler) vector <math>\mathcal{K}=\sum_{i=1}^{n}(x-x_{0})^{i}\partial_{x^{i}}</math> for <math>n=\dim(U)</math> is inserted into the form <math>\beta \in \Lambda^{*}(U)</math>. We can then define cohomotopy operator as<ref name=":0" />

<math>h:\Lambda(U)\rightarrow \Lambda(U), \quad h:=\eta {\star}^{-1}H\star</math>,

where <math>\eta \beta = (-1)^{k}\beta</math> for <math>\beta \in \Lambda^{k}(U)</math>.

The cohomotopy operator fulfills (co)homotopy invariance formula<ref name=":0" />

<math>\delta h + h\delta = I - S_{x_{0}} ,</math>

where <math>S_{x_{0}}={\star}^{-1}s_{x_{0}}^{*}{\star}</math>, and <math>s_{x_{0}}^{*}</math> is the pullback along the constant map <math>s_{x_{0}}:x \rightarrow x_{0}</math>.

Therefore, if we want to solve the equation <math>\delta \omega =0</math>, applying cohomotopy invariance formula we get

<math> \omega= \delta h\omega + S_{x_{0}}\omega,</math> where <math>h\omega\in \Lambda^{k+1}(U)</math> is a differential form we are looking for, and "constant of integration" <math>S_{x_{0}}\omega</math> vanishes unless <math>\omega</math> is a top form.

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

<math>\Lambda(U)=\mathcal{C}(U)\oplus \mathcal{Y}(U)</math>.

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:<ref name=":0" /> <math>(h\delta)^{2}=h\delta, \quad (\delta h)^{2}=\delta h</math>.

These results are extension of similar results for exterior derivative.<ref>Template:Cite book</ref>

References

• David Bleecker (1981) Gauge Theory and Variational Principles. Addison-Wesley Publishing. Template:Isbn. Chpt. 0 contains a condensed review of non-Riemannian differential geometry.
• Template:Cite book
• Charles W. Misner, Kip S. Thorne, John Archibald Wheeler (1970) Gravitation. W.H. Freeman. Template:Isbn. 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. Template:Isbn. 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.