Hodge star operator

Let {{math|V}} be an Dimension (vector space) oriented vector space with a nondegenerate symmetric bilinear form $\backslash langle\; \backslash cdot,\backslash cdot\; \backslash rangle$, 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 {{nowrap|$\backslash alpha,\; \backslash beta\backslash in\; \backslash bigwedge^\{\backslash !k\}V$,}} for $0\; \backslash le\; k\; \backslash le\; n$, by defining it on simple {{math|k}}-vectors $\backslash alpha\; =\; \backslash alpha\_1\; \backslash wedge\; \backslash cdots\; \backslash wedge\; \backslash alpha\_k$ and $\backslash beta\; =\; \backslash beta\_1\; \backslash wedge\; \backslash cdots\; \backslash wedge\; \backslash beta\_k$ to equal the Gram determinantHarley Flanders (1963) Differential Forms with Applications to the Physical Sciences, Academic Press{{rp|14}}

: $\backslash langle\; \backslash alpha,\; \backslash beta\; \backslash rangle\; =\; \backslash det\; \backslash left(\; \backslash left\backslash langle\; \backslash alpha\_i,\; \backslash beta\_j\; \backslash right\backslash rangle\; \_\{i,j=1\}^k\backslash right)$

extended to $\backslash bigwedge^\{\backslash !k\}V$ through linearity.

The unit {{math|n}}-vector $\backslash omega\backslash in\{\backslash textstyle\backslash bigwedge\}^\{\backslash !n\}V$ is defined in terms of an oriented orthonormal basis $\backslash \{e\_1,\backslash ldots,e\_n\backslash \}$ of {{math|V}} as:

: $\backslash omega\; :=\; e\_1\backslash wedge\backslash cdots\backslash wedge\; e\_n.$

(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.)

The Hodge star operator is a linear operator on the exterior algebra of {{math|V}}, mapping {{math|k}}-vectors to ({{math|n – k}})-vectors, for $0\; \backslash le\; k\; \backslash le\; n$. It has the following property, which defines it completely:{{rp|15}}

: $\backslash alpha\; \backslash wedge\; (\{\backslash star\}\; \backslash beta)\; =\; \backslash langle\; \backslash alpha,\backslash beta\; \backslash rangle\; \backslash ,\backslash omega$ for all {{math|k}}-vectors $\backslash alpha,\backslash beta\backslash in\; \{\backslash textstyle\backslash bigwedge\}^\{\backslash !k\}V\; .$

Dually, in the space $\{\backslash textstyle\backslash bigwedge\}^\{\backslash !n\}V^*$ of {{math|n}}-forms (alternating {{math|n}}-multilinear functions on $V^n$), the dual to $\backslash omega$ is the volume form $\backslash det$, the function whose value on $v\_1\backslash wedge\backslash cdots\backslash wedge\; v\_n$ is the determinant of the $n\backslash times\; n$ matrix assembled from the column vectors of $v\_j$ in $e\_i$-coordinates. Applying $\backslash det$ to the above equation, we obtain the dual definition:

: $\backslash det(\backslash alpha\; \backslash wedge\; \{\backslash star\}\; \backslash beta)\; =\; \backslash langle\; \backslash alpha,\backslash beta\; \backslash rangle$ for all {{math|k}}-vectors $\backslash alpha,\backslash beta\backslash in\; \{\backslash textstyle\backslash bigwedge\}^\{\backslash !k\}V\; .$

Equivalently, taking $\backslash alpha\; =\; \backslash alpha\_1\; \backslash wedge\; \backslash cdots\; \backslash wedge\; \backslash alpha\_k$, $\backslash beta\; =\; \backslash beta\_1\; \backslash wedge\; \backslash cdots\; \backslash wedge\; \backslash beta\_k$, and $\{\backslash star\}\backslash beta\; =\; \backslash beta\_1^\backslash star\; \backslash wedge\; \backslash cdots\; \backslash wedge\; \backslash beta\_\{n-k\}^\backslash star$:

: $$

\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 {{math|k}}-vectors as $e\_I\; \backslash \; =\; \backslash \; e\_\{i\_1\}\backslash wedge\backslash cdots\backslash wedge\; e\_\{i\_k\}$ over all subsets n – k}})-vector corresponding to the complementary set :

: $\{\backslash star\}\; e\_I\; =\; s\backslash cdot\; t\backslash cdot\; e\_\backslash bar\{I\}\; ,$

where $s\backslash in\backslash \{1,-1\backslash \}$ is the sign of the permutation $i\_1\; \backslash cdots\; i\_k\; \backslash bar\; i\_1\; \backslash cdots\; \backslash bar\; i\_\{n-k\}$

and $t\backslash in\backslash \{1,-1\backslash \}$ is the product

$\backslash langle\; e\_\{i\_1\},e\_\{i\_1\}\backslash rangle\backslash cdots\; \backslash langle\; e\_\{i\_k\},e\_\{i\_k\}\backslash rangle$. In the Riemannian case, $t=1$.

Since Hodge star takes an orthonormal basis to an orthonormal basis, it is an isometry on the exterior algebra $\backslash bigwedge\; V$.

The Hodge star is motivated by the correspondence between a subspace {{math|W}} of {{math|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 {{math|k}}-vector $w\_1\backslash wedge\backslash cdots\backslash wedge\; w\_k\backslash in\; \backslash textstyle\backslash bigwedge^\{\backslash !k\}\; V$ corresponds by the Plücker embedding to the subspace $W$ with oriented basis $w\_1,\backslash ldots,w\_k$, endowed with a scaling factor equal to the {{math|k}}-dimensional volume of the parallelepiped spanned by this basis (equal to the Gramian, the determinant of the matrix of scalar products $\backslash langle\; w\_i,\; w\_j\; \backslash rangle$). The Hodge star acting on a decomposable vector can be written as a decomposable ({{math|n − k}})-vector:

: $\{\backslash star\}(w\_1\backslash wedge\backslash cdots\backslash wedge\; w\_k)\; \backslash ,=\backslash ,\; u\_1\backslash wedge\backslash cdots\backslash wedge\; u\_\{n-k\},$

where $u\_1,\backslash ldots,u\_\{n-k\}$ form an oriented basis of the orthogonal space $U\; =\; W^\backslash perp\backslash !$. Furthermore, the ({{math|n − k}})-volume of the $u\_i$-parallelepiped must equal the {{math|k}}-volume of the $w\_i$-parallelepiped, and $w\_1,\backslash ldots,w\_k,u\_1,\backslash ldots,u\_\{n-k\}$ must form an oriented basis of $V$.

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

In two dimensions with the normalized Euclidean metric and orientation given by the ordering {{math|(x, y)}}, the Hodge star on {{math|k}}-forms is given by

$$\backslash begin\{align\}$$

{\star} \, 1 &= dx \wedge dy \\

{\star} \, dx &= dy \\

{\star} \, dy &= -dx \\

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

\end{align}

A common example of the Hodge star operator is the case {{math|1=n = 3}}, 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

$$\backslash begin\{align\}$$

{\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: $$\{\backslash star\}\; (\backslash mathbf\{u\}\; \backslash wedge\; \backslash mathbf\{v\})\; =\; \backslash mathbf\{u\}\; \backslash times\; \backslash mathbf\{v\}\; \backslash qquad\; \{\backslash star\}\; (\backslash mathbf\{u\}\; \backslash times\; \backslash mathbf\; \{v\})\; =\; \backslash mathbf\{u\}\; \backslash wedge\; \backslash mathbf\{v\}\; .$$ Applied to three dimensions, the Hodge star provides an isomorphism between axial vectors and bivectors, so each axial vector {{math|a}} is associated with a bivector {{math|A}} and vice versa, that is:{{cite book |title=Clifford Algebras and Spinors, Volume 286 of London Mathematical Society Lecture Note Series |author=Pertti Lounesto |chapter-url = https://books.google.com/books?id=E_xvJuA4M7QC&pg=PA39 |page=39 |chapter=§3.6 The Hodge dual |isbn=0-521-00551-5 |year=2001 |edition=2nd |publisher=Cambridge University Press}} $\backslash mathbf\{A\}\; =\; \{\backslash star\}\; \backslash mathbf\{a\},\; \backslash \; \backslash \; \backslash mathbf\{a\}\; =\; \{\backslash star\}\; \backslash mathbf\{A\}$.

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 $V\backslash cong\; V^*\backslash !$ identifying $V$ with its dual space, and the vector space $L(V,V)$ is naturally isomorphic to the tensor product $V^*\backslash !\backslash !\backslash otimes\; V\backslash cong\; V\backslash otimes\; V$. Thus for $V\; =\; \backslash mathbb\{R\}^3$, the star mapping $\backslash textstyle\; \{\backslash star\}\; :\; V\backslash to\backslash bigwedge^\{\backslash !2\}\backslash !\; V\; \backslash subset\; V\backslash otimes\; V$ takes each vector $\backslash mathbf\{v\}$ to a bivector $\{\backslash star\}\; \backslash mathbf\{v\}\; \backslash in\; V\backslash otimes\; V$, which corresponds to a linear operator $L\_\{\backslash mathbf\{v\}\}\; :\; V\backslash to\; V$. Specifically, $L\_\{\backslash mathbf\{v\}\}$ is a skew-symmetric operator, which corresponds to an infinitesimal rotation: that is, the macroscopic rotations around the axis $\backslash mathbb\{v\}$ are given by the matrix exponential $\backslash exp(t\; L\_\{\backslash mathbf\{v\}\})$. With respect to the basis $dx,\; dy,\; dz$ of $\backslash R^3$, the tensor $dx\backslash otimes\; dy$ corresponds to a coordinate matrix with 1 in the $dx$ row and $dy$ column, etc., and the wedge $dx\backslash wedge\; dy\; \backslash ,=\backslash ,\; dx\backslash otimes\; dy\; -\; dy\backslash otimes\; dx$ is the skew-symmetric matrix $\backslash scriptscriptstyle\backslash left[\backslash begin\{array\}\{rrr\}$

\,0\!\! & \!\!1 & \!\!\!\!0\!\!\!\!\!\!

\\[-.5em]

\,\!-1\!\!&\!\!0\!\!&\!\!\!\!0\!\!\!\!\!\!

\\[-.5em]

\,0\!\! & \!\!0\!\! & \!\!\!\!0\!\!\!\!\!\!

\end{array}\!\!\!\right], etc. That is, we may interpret the star operator as: $$\backslash mathbf\{v\}\; =\; a\backslash ,dx\; +\; b\backslash ,dy\; +\; c\backslash ,dz$$

\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: $L\_\{\backslash mathbf\{u\}\backslash times\backslash mathbf\{v\}\}\; =\; L\_\{\backslash mathbf\{v\}\}\; L\_\{\backslash mathbf\{u\}\}\; -\; L\_\{\backslash mathbf\{u\}\}\; L\_\{\backslash mathbf\{v\}\}=-\backslash left[L\_\{\backslash mathbf\{u\}\},\; L\_\{\backslash mathbf\{v\}\}\backslash right]$.

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 {{math|1=4 − 2 = 2}}). 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 {{section link|#Duality}} 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 $\backslash pm\; 1$ (or $\backslash pm\; i$, depending on the signature).

For concreteness, we discuss the Hodge star operator in Minkowski spacetime where $n=4$ with metric signature {{math|(− + + +)}} and coordinates $(t,x,y,z)$. The volume form is oriented as $\backslash varepsilon\_\{0123\}\; =\; 1$. For one-forms,

$$\backslash begin\{align\}$$

{\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,

$$\backslash begin\{align\}$$

{\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

$$\backslash begin\{align\}$$

{\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, $\{\backslash star\}^2=1$ for odd-rank forms and $\{\backslash star\}^2=-1$ for even-rank forms. An easy rule to remember for these Hodge operations is that given a form $\backslash alpha$, its Hodge dual $\{\backslash star\}\backslash alpha$ may be obtained by writing the components not involved in $\backslash alpha$ in an order such that $\backslash alpha\; \backslash wedge\; (\{\backslash star\}\; \backslash alpha)\; =\; dt\; \backslash wedge\; dx\; \backslash wedge\; dy\; \backslash wedge\; dz$.{{verify source|date=September 2019}} An extra minus sign will enter only if $\backslash alpha$ contains $dt$. (For {{math|(+ − − −)}}, one puts in a minus sign only if $\backslash alpha$ involves an odd number of the space-associated forms $dx$, $dy$ and $dz$.)

Note that the combinations

$$(dx^\backslash mu\; \backslash wedge\; dx^\backslash nu)^\{\backslash pm\}\; :=\; \backslash frac\{1\}\{2\}\; \backslash big(\; dx^\backslash mu\; \backslash wedge\; dx^\backslash nu\; \backslash mp\; i\; \{\backslash star\}\; (dx^\backslash mu\; \backslash wedge\; dx^\backslash nu)\; \backslash big)$$

take $\backslash pm\; i$ as the eigenvalue for Hodge star operator, i.e.,

$$\{\backslash star\}\; (dx^\backslash mu\; \backslash wedge\; dx^\backslash nu)^\{\backslash pm\}\; =\; \backslash pm\; i\; (dx^\backslash mu\; \backslash wedge\; dx^\backslash nu)^\{\backslash pm\}\; ,$$

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.

The Hodge star is conformally invariant on {{math|n}}-forms on a {{math|2n}}-dimensional vector space $V$, i.e. if $g$ is a metric on $V$ and $\backslash lambda\; >\; 0$, then the induced Hodge stars

$$\{\backslash star\}\_g,\; \{\backslash star\}\_\{\backslash lambda\; g\}\; :\; \backslash Lambda^n\; V\; \backslash to\; \backslash Lambda^n\; V$$

are the same.

The combination of the $\{\backslash star\}$ operator and the exterior derivative {{math|d}} generates the classical operators {{math|grad}}, {{math|curl}}, and {{math|div}} on vector fields in three-dimensional Euclidean space. This works out as follows: {{math|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:

$$df\; =\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; x\}\; \backslash ,\; dx\; +\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; y\}\; \backslash ,\; dy\; +\; \backslash frac\{\backslash partial\; f\}\{\backslash partial\; z\}\; \backslash ,\; dz.$$

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

In the second case, a vector field $\backslash mathbf\; F\; =\; (A,B,C)$ corresponds to the 1-form $\backslash varphi\; =\; A\backslash ,dx\; +\; B\backslash ,dy\; +\; C\backslash ,dz$, which has exterior derivative:

$$d\backslash varphi\; =$$

\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:

$$\{\backslash star\}\; d\backslash varphi\; =\; \backslash left(\{\backslash partial\; C\; \backslash over\; \backslash partial\; y\}\; -\; \{\backslash partial\; B\; \backslash over\; \backslash partial\; z\}\; \backslash right)\; \backslash ,\; dx\; -\; \backslash left(\{\backslash partial\; C\; \backslash over\; \backslash partial\; x\}\; -\; \{\backslash partial\; A\; \backslash over\; \backslash partial\; z\}\; \backslash right)\; \backslash ,\; dy\; +\; \backslash left(\{\backslash partial\; B\; \backslash over\; \backslash partial\; x\}\; -\; \{\backslash partial\; A\; \backslash over\; \backslash partial\; y\}\backslash right)\; \backslash ,\; dz,$$

which becomes the vector field $\backslash operatorname\{curl\}\backslash mathbf\{F\}\; =\; \backslash left($

\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, $\backslash mathbf\; F\; =\; (A,B,C)$ again corresponds to $\backslash varphi\; =\; A\backslash ,dx\; +\; B\backslash ,dy\; +\; C\backslash ,dz$. Applying Hodge star, exterior derivative, and Hodge star again:

$$\backslash begin\{align\}$$

{\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 {{math|1=d{{i sup|2}} = 0}}, which is true in all cases, has as special cases two other identities: (1) {{math|1=curl grad f = 0}}, and (2) {{math|1=div curl F = 0}}. 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 $\{\backslash star\}d\{\backslash star\}$ (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 {{math|1=Δf = div grad f}} in terms of the above operations:

$$\backslash Delta\; f\; =\; \{\backslash star\}d\{\backslash star\}d\; f\; =\; \backslash frac\{\backslash partial^2\; f\}\{\backslash partial\; x^2\}\; +\; \backslash frac\{\backslash partial^2\; f\}\{\backslash partial\; y^2\}\; +\; \backslash frac\{\backslash partial^2\; f\}\{\backslash partial\; z^2\}.$$

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

Applying the Hodge star twice leaves a {{math|k}}-vector unchanged up to a sign: for $\backslash eta\backslash in\; \{\backslash textstyle\backslash bigwedge\}^k\; V$ in an {{mvar|n}}-dimensional space {{math|V}}, one has

: $\{\backslash star\}\; \{\backslash star\}\; \backslash eta\; =\; (-1)^\{k(n-k)\}\; s\backslash ,\; \backslash eta\; ,$

where {{mvar|s}} is the parity of the signature of the scalar product on {{math|V}}, that is, the sign of the determinant of the matrix of the scalar product with respect to any basis. For example, if {{math|n {{=}} 4}} and the signature of the scalar product is either {{math|(+ − − −)}} or {{math|(− + + +)}} then {{math|s {{=}} −1}}. For Riemannian manifolds (including Euclidean spaces), we always have {{math|s {{=}} 1}}.

The above identity implies that the inverse of $\{\backslash star\}$ can be given as

: $\backslash begin\{align\}$

{\star}^{-1}: ~ {\textstyle\bigwedge}^{\!k} V &\to {\textstyle\bigwedge}^{\!n-k} V \\

\eta &\mapsto (-1)^{k(n-k)} \!s\, {\star} \eta

\end{align}

If {{mvar|n}} is odd then {{math|k(n − k)}} is even for any {{mvar|k}}, whereas if {{mvar|n}} is even then {{math|k(n − k)}} has the parity of {{mvar|k}}. Therefore:

:

where {{mvar|k}} is the degree of the element operated on.

For an n-dimensional oriented pseudo-Riemannian manifold M, we apply the construction above to each cotangent space $\backslash text\{T\}^*\_p\; M$ and its exterior powers $\backslash bigwedge^k\backslash text\{T\}^*\_p\; M$, and hence to the differential k-forms $\backslash zeta\backslash in\backslash Omega^k(M)\; =\; \backslash Gamma\backslash left(\backslash bigwedge^k\backslash text\{T\}^*\backslash !M\backslash right)$, the global sections of the bundle $\backslash bigwedge^k\; \backslash mathrm\{T\}^*\backslash !\; M\backslash to\; M$. The Riemannian metric induces a scalar product on $\backslash bigwedge^k\; \backslash text\{T\}^*\_p\; M$ at each point $p\backslash in\; M$. We define the Hodge dual of a k-form $\backslash zeta$, defining $\{\backslash star\}\; \backslash zeta$ as the unique (n – k)-form satisfying

$$\backslash eta\backslash wedge\; \{\backslash star\}\; \backslash zeta\; \backslash \; =\backslash \; \backslash langle\; \backslash eta,\; \backslash zeta\; \backslash rangle\; \backslash ,\; \backslash omega$$

for every k-form $\backslash eta$, where $\backslash langle\backslash eta,\backslash zeta\backslash rangle$ is a real-valued function on $M$, and the volume form $\backslash omega$ 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:

$$\backslash int\_M\; \backslash eta\backslash wedge\; \{\backslash star\}\; \backslash zeta$$

\ =\ \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.

We compute in terms of tensor index notation with respect to a (not necessarily orthonormal) basis $\backslash left\backslash \{\backslash frac\{\backslash partial\}\{\backslash partial\; x\_1\},\; \backslash ldots,\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\_n\}\backslash right\backslash \}$ in a tangent space $V\; =\; T\_p\; M$ and its dual basis $\backslash \{dx\_1,\backslash ldots,dx\_n\backslash \}$ in $V^*\; =\; T^*\_p\; M$, having the metric matrix $(g\_\{ij\})\; =\; \backslash left(\backslash left\backslash langle\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\_i\},\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\_j\}\backslash right\backslash rangle\backslash right)$ and its inverse matrix $(g^\{ij\})\; =\; (\backslash langle\; dx^i,\; dx^j\backslash rangle)$. 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 $\backslash varepsilon\_\{j\_1\; \backslash dots\; j\_n\}$ is the Levi-Civita symbol with $\backslash varepsilon\_\{1\; \backslash dots\; n\}\; =\; 1$, and we implicitly take the sum over all values of the repeated indices $j\_1,\backslash ldots,j\_n$. The factorial $(n-k)!$ accounts for double counting, and is not present if the summation indices are restricted so that . 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 < \dots < i_k} \alpha_{i_1, \dots, i_k} dx^{i_1}\wedge \dots \wedge dx^{i_k}.

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 $\backslash alpha\_\{i\_1,\; \backslash dots,\; i\_k\}$ 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 $dx^\{i\_1\}\; \backslash wedge\; \backslash dots\; \backslash wedge\; dx^\{i\_k\}$, we find:{{cite book| last=Frankel|first=T.| title=The Geometry of Physics| publisher=Cambridge University Press| year=2012| isbn=978-1-107-60260-1| edition=3rd}}

({\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 $\backslash alpha$, 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 $\backslash omega\; =\; \{\backslash star\}\; 1\backslash in\; \backslash bigwedge^n\; V^*$ is given by:

$$\backslash omega\; =\; \backslash sqrt\{\; \backslash left|\; \backslash det\; [g\_\{ij\}]\; \backslash right|\; \}\backslash ;dx^1\; \backslash wedge\; \backslash cdots\; \backslash wedge\; dx^n\; .$$

The most important application of the Hodge star on manifolds is to define the codifferential $\backslash delta$ on $k$-forms. Let

$$\backslash delta\; =\; (-1)^\{n(k\; +\; 1)\; +\; 1\}\; s\backslash \; \{\backslash star\}\; d\; \{\backslash star\}\; =\; (-1)^\{k\}\backslash ,\; \{\backslash star\}^\{-1\}\; d\; \{\backslash star\}$$

where $d$ is the exterior derivative or differential, and $s\; =\; 1$ for Riemannian manifolds. Then

$$d:\backslash Omega^k(M)\backslash to\; \backslash Omega^\{k+1\}(M)$$

while

$$\backslash delta:\backslash Omega^k(M)\backslash to\; \backslash Omega^\{k-1\}(M).$$

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:

$$\backslash langle\backslash !\backslash langle\backslash eta,\backslash delta\; \backslash zeta\backslash rangle\backslash !\backslash rangle\; \backslash \; =\backslash \; \backslash langle\backslash !\backslash langle\; d\backslash eta,\backslash zeta\backslash rangle\backslash !\backslash rangle,$$

where $\backslash zeta$ is a $k$-form and $\backslash eta$ a $(k\backslash !-\backslash !1)$-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 $\backslash eta$ or $\{\backslash star\}\backslash zeta$ 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 $\backslash zeta\_i\; \backslash to\; \backslash zeta$ (as $i\; \backslash to\; \backslash infty$) to be interchanged with the combined differential and integral operations, so that $\backslash langle\backslash !\backslash langle\backslash eta,\backslash delta\; \backslash zeta\_i\backslash rangle\backslash !\backslash rangle\; \backslash to\; \backslash langle\backslash !\backslash langle\backslash eta,\backslash delta\; \backslash zeta\backslash rangle\backslash !\backslash rangle$ and likewise for sequences converging to $\backslash eta$.)

Since the differential satisfies $d^2\; =\; 0$, the codifferential has the corresponding property

$$\backslash delta^2\; =\; (-1)^n\; s^2\; \{\backslash star\}\; d\; \{\backslash star\}\; \{\backslash star\}\; d\; \{\backslash star\}\; =\; (-1)^\{nk+k+1\}\; s^3\; \{\backslash star\}\; d^2\; \{\backslash star\}\; =\; 0.$$

The Laplace–deRham operator is given by

$$\backslash Delta\; =\; (\backslash delta\; +\; d)^2\; =\; \backslash delta\; d\; +\; d\backslash delta$$

and lies at the heart of Hodge theory. It is symmetric:

$$\backslash langle\backslash !\backslash langle\backslash Delta\; \backslash zeta,\backslash eta\backslash rangle\backslash !\backslash rangle\; =\; \backslash langle\backslash !\backslash langle\backslash zeta,\backslash Delta\; \backslash eta\backslash rangle\backslash !\backslash rangle$$

and non-negative:

$$\backslash langle\backslash !\backslash langle\backslash Delta\backslash eta,\backslash eta\backslash rangle\backslash !\backslash rangle\; \backslash ge\; 0.$$

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 {{mvar|k}}-forms, and so the Hodge star induces an isomorphism of cohomology groups

$$\{\backslash star\}\; :\; H^k\_\backslash Delta\; (M)\; \backslash to\; H^\{n-k\}\_\backslash Delta(M),$$

which in turn gives canonical identifications via Poincaré duality of {{math|H^k(M)}} with its dual space.

In coordinates, with notation as above, the codifferential of the form $\backslash alpha$ may be written as

$$\backslash delta\; \backslash alpha=\backslash \; -\backslash frac\{1\}\{k!\}g^\{ml\}\backslash left(\backslash frac\{\backslash partial\}\{\backslash partial\; x\_l\}\; \backslash alpha\_\{m,i\_1,\; \backslash dots,\; i\_\{k-1\}\}\; -\; \backslash Gamma^j\_\{ml\}\; \backslash alpha\_\{j,i\_1,\; \backslash dots,\; i\_\{k-1\}\}\; \backslash right)\; dx^\{i\_1\}\; \backslash wedge\; \backslash dots\; \backslash wedge\; dx^\{i\_\{k-1\}\},$$

where here $\backslash Gamma^\{j\}\_\{ml\}$ denotes the Christoffel symbols of $\backslash left\backslash \{\backslash frac\{\backslash partial\}\{\backslash partial\; x\_1\},\; \backslash ldots,\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\_n\}\backslash right\backslash \}$.

In analogy to the Poincare lemma for exterior derivative, one can define its version for codifferential, which reads{{Cite journal |last=Kycia |first=Radosław Antoni |date=2022-07-29 |title=The Poincare Lemma for Codifferential, Anticoexact Forms, and Applications to Physics |url=https://doi.org/10.1007/s00025-022-01646-z |journal=Results in Mathematics |language=en |volume=77 |issue=5 |pages=182 |doi=10.1007/s00025-022-01646-z |issn=1420-9012|arxiv=2009.08542 |s2cid=221802588 }}

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

A practical way of finding $\backslash alpha$ is to use cohomotopy operator $h$, that is a local inverse of $\backslash delta$. One has to define a homotopy operator

: $H\backslash beta\; =\; \backslash int\_\{0\}^\{1\}\; \backslash mathcal\{K\}\backslash lrcorner\backslash beta|\_\{F(t,x)\}t^\{k\}dt,$

where $F(t,x)=x\_\{0\}+t(x-x\_\{0\})$ is the linear homotopy between its center $x\_\{0\}\backslash in\; U$ and a point $x\; \backslash in\; U$, and the (Euler) vector $\backslash mathcal\{K\}=\backslash sum\_\{i=1\}^\{n\}(x-x\_\{0\})^\{i\}\backslash partial\_\{x^\{i\}\}$ for $n=\backslash dim(U)$ is inserted into the form $\backslash beta\; \backslash in\; \backslash Lambda^\{*\}(U)$. We can then define cohomotopy operator as

: $h:\backslash Lambda(U)\backslash rightarrow\; \backslash Lambda(U),\; \backslash quad\; h:=\backslash eta\; \{\backslash star\}^\{-1\}H\backslash star$,

where $\backslash eta\; \backslash beta\; =\; (-1)^\{k\}\backslash beta$ for $\backslash beta\; \backslash in\; \backslash Lambda^\{k\}(U)$.

The cohomotopy operator fulfills (co)homotopy invariance formula

: $\backslash delta\; h\; +\; h\backslash delta\; =\; I\; -\; S\_\{x\_\{0\}\}\; ,$

where $S\_\{x\_\{0\}\}=\{\backslash star\}^\{-1\}s\_\{x\_\{0\}\}^\{*\}\{\backslash star\}$, and $s\_\{x\_\{0\}\}^\{*\}$ is the pullback along the constant map $s\_\{x\_\{0\}\}:x\; \backslash rightarrow\; x\_\{0\}$.

Therefore, if we want to solve the equation $\backslash delta\; \backslash omega\; =0$, applying cohomotopy invariance formula we get

: $\backslash omega=\; \backslash delta\; h\backslash omega\; +\; S\_\{x\_\{0\}\}\backslash omega,$ where $h\backslash omega\backslash in\; \backslash Lambda^\{k+1\}(U)$ is a differential form we are looking for, and "constant of integration" $S\_\{x\_\{0\}\}\backslash omega$ vanishes unless $\backslash omega$ is a top form.

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

: $\backslash Lambda(U)=\backslash mathcal\{C\}(U)\backslash oplus\; \backslash mathcal\{Y\}(U)$.

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: $(h\backslash delta)^\{2\}=h\backslash delta,\; \backslash quad\; (\backslash delta\; h)^\{2\}=\backslash delta\; h$.

These results are extension of similar results for exterior derivative.{{Cite book |last=Edelen |first=Dominic G. B. |url=https://www.worldcat.org/oclc/56347718 |title=Applied exterior calculus |date=2005 |isbn=978-0-486-43871-9 |edition=Revised |location=Mineola, N.Y. |oclc=56347718}}

{{reflist}}

{{refbegin}}

• David Bleecker (1981) Gauge Theory and Variational Principles. Addison-Wesley Publishing. {{isbn|0-201-10096-7}}. Chpt. 0 contains a condensed review of non-Riemannian differential geometry.
• {{cite book|author-last=Jost|author-first=Jürgen|author-link=Jürgen Jost|title=Riemannian Geometry and Geometric Analysis|date=2002|publisher=Springer-Verlag|isbn=3-540-42627-2}}
• Charles W. Misner, Kip S. Thorne, John Archibald Wheeler (1970) Gravitation. W.H. Freeman. {{isbn|0-7167-0344-0}}. 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. {{isbn|0-521-46831-0}}. An introduction to the heat equation and the Atiyah–Singer theorem.
• [https://sites.science.oregonstate.edu/~tevian/onid/Courses/MTH434/2009/dual.pdf Tevian Dray (1999) The Hodge Dual Operator]. A thorough overview of the definition and properties of the Hodge star operator.

{{refend}}

{{Tensors}}

{{DEFAULTSORT:Hodge Dual}}

Category:Differential forms

Category:Riemannian geometry

Category:Duality theories

Category:Differential operators