# Hodge dual

In mathematics, the **Hodge star operator** or **Hodge dual** is a significant linear map introduced in general by W. V. D. Hodge. It is defined on the exterior algebra of a finite-dimensional oriented inner product space.

## Dimensionalities and algebra

Suppose that Template:Mvar is the dimensionality of the oriented inner product space and Template:Mvar is an integer such that 0 ≤ *k* ≤ *n*, then the Hodge star operator establishes a one-to-one mapping from the space of [[p-vector|Template:Mvar-vectors]] to the space of (*n* − *k*)-vectors. The image of a Template:Mvar-vector under this mapping is called the *Hodge dual* of the Template:Mvar-vector. The former space, of Template:Mvar-vectors, has dimensionality

while the latter has dimensionality

and by the symmetry of the binomial coefficients, these two dimensionalities are equal. Two vector spaces over the same field with the same dimensionality are always isomorphic; but not necessarily in a natural or canonical way. The Hodge duality, however, in this case exploits the inner product and orientation of the vector space. It singles out a unique isomorphism, that reflects therefore the pattern of the binomial coefficients in algebra. This in turn induces an inner product on the space of Template:Mvar-vectors. The 'natural' definition means that this duality relationship can play a geometrical role in theories.

The first interesting case is on three-dimensional Euclidean space Template:Mvar. In this context the relevant row of Pascal's triangle reads

- 1, 3, 3, 1

and the Hodge dual sets up an isomorphism between the two three-dimensional spaces, which are Template:Mvar itself and the space of wedge products of two vectors from Template:Mvar. See the Examples section for details. In this case the content is just that of the cross product of traditional vector calculus. While the properties of the cross product are special to three dimensions, the Hodge dual applies to all dimensionalities.

## Extensions

Since the space of alternating linear forms in Template:Mvar arguments on a vector space is naturally isomorphic to the dual of the space of Template:Mvar-vectors over that vector space, the Hodge dual can be defined for these spaces as well. As with most constructions from linear algebra, the Hodge dual can then be extended to a vector bundle. Thus a context in which the Hodge dual is very often seen is the exterior algebra of the cotangent bundle (i.e. the space of differential forms on a manifold) where it can be used to construct the **codifferential** from the exterior derivative, and thus the Laplace-de Rham operator, which leads to the Hodge decomposition of differential forms in the case of compact Riemannian manifolds.

## Formal definition of the Hodge star of Template:Mvar-vectors

The **Hodge star operator** on a vector space Template:Mvar with a nondegenerate symmetric bilinear form (herein referred to as the *inner product*) is a linear operator on the exterior algebra of Template:Mvar, mapping Template:Mvar-vectors to (*n* − *k*)-vectors where *n* = dim *V*, for 0 ≤ *k* ≤ *n*. It has the following property, which defines it completely: given two Template:Mvar-vectors *α*, *β*

where denotes the [[Exterior algebra#Inner product|inner product on Template:Mvar-vectors]] and Template:Mvar is the preferred unit Template:Mvar-vector.

The inner product on Template:Mvar-vectors is extended from that on Template:Mvar by requiring that

for any decomposable Template:Mvar-vectors and .

The unit Template:Mvar-vector Template:Mvar is unique up to a sign. The preferred choice of Template:Mvar defines an orientation on Template:Mvar.

## Explanation

Let Template:Mvar be a vector space, with an inner product . The Riesz representation theorem states that for every continuous (every in the finite-dimensional case) linear functional there exists a unique vector Template:Mvar in Template:Mvar such that for all Template:Mvar in Template:Mvar. The map given by is an isomorphism. This holds for *all* vector spaces with an inner product, and can be used to explain the Hodge dual.

Let Template:Mvar be an Template:Mvar-dimensional vector space with basis . For 0 ≤ *k* ≤ *n*, consider the exterior power spaces and . For

we have

There is, up to a scalar, only one Template:Mvar-vector, namely . In other words, must be a scalar multiple of for all and .

Consider a *fixed* . There exists a unique linear function

such that

This is the scalar multiple mentioned in the previous paragraph. If denotes the inner product on (*n* − *k*)-vectors, then there exists a unique (*n* − *k*)-vector, say

such that

This (*n* − *k*)-vector ★*λ* is the Hodge dual of Template:Mvar, and is the image of the under the isomorphism induced by the inner product,

Thus,

## Computation of the Hodge star

Given an orthonormal basis ordered such that , we see that

where is an even permutation of {1, 2, ..., *n*}.

Of these relations, only are independent. The first one in the usual lexicographical order reads

## Index notation for the star operator

Using index notation, the Hodge dual is obtained by contracting the indices of a Template:Mvar-form with the Template:Mvar-dimensional completely antisymmetric **Levi-Civita tensor**. This differs from the Levi-Civita symbol by a factor of |det *g*|^{{{ safesubst:#invoke:Unsubst||$B=1/2}}}, where *g* is an inner product (the metric tensor). The absolute value of the determinant is necessary if *g* is not positive-definite, e.g. for tangent spaces to Lorentzian manifolds.

Thus one writes^{[1]}

where Template:Mvar is an arbitrary antisymmetric tensor in Template:Mvar indices. It is understood that indices are raised and lowered using the same inner product *g* as in the definition of the Levi-Civita tensor. Although one can take the star of any tensor, the result is antisymmetric, since the symmetric components of the tensor completely cancel out when contracted with the completely anti-symmetric Levi-Civita symbol.

## Examples

A common example of the star operator is the case *n* = 3, when it can be taken as the correspondence between the vectors and the skew-symmetric matrices of that size. This is used implicitly in vector calculus, for example to create the cross product vector from the wedge product of two vectors. Specifically, for Euclidean **R**^{3}, one easily finds that

where d*x*, d*y* and d*z* are the standard orthonormal differential one-forms on **R**^{3}. The Hodge dual in this case clearly relates the cross-product to the wedge product in three dimensions. A detailed presentation not restricted to differential geometry is provided next.

### Three-dimensional example

Applied to three dimensions, the Hodge dual provides an isomorphism between axial vectors and bivectors, so each axial vector **a** is associated with a bivector **A** and vice versa, that is:^{[2]}

where ★ indicates the dual operation. These dual relations can be implemented using multiplication by the unit pseudoscalar in *C*ℓ_{3}(R),^{[3]} *i* = **e**_{1}**e**_{2}**e**_{3} (the vectors {**e**_{ℓ}} are an orthonormal basis in three dimensional Euclidean space) according to the relations:^{[4]}

The dual of a vector is obtained by multiplication by Template:Mvar, as established using the properties of the geometric product of the algebra as follows:

and also, in the dual space spanned by {**e**_{ℓ}**e**_{m}}:

In establishing these results, the identities are used:

and:

These relations between the dual ★ and Template:Mvar apply to any vectors. Here they are applied to relate the axial vector created as the cross product **a** = **u** × **v** to the bivector-valued exterior product **A** = **u** ∧ **v** of two polar (that is, not axial) vectors **u** and **v**; the two products can be written as determinants expressed in the same way:

using the notation **e**_{ℓm} = **e**_{ℓ}**e**_{m}. These expressions show these two types of vector are Hodge duals:^{[2]}

as a result of the relations:

with ℓ, *m*, *n* cyclic,

and:

also with ℓ, *m*, *n* cyclic.

Using the implementation of ★ based upon Template:Mvar, the commonly used relations are:^{[5]}

### Four dimensions

In case *n* = 4, the Hodge dual acts as an endomorphism of the second exterior power (i.e. it maps two-forms to two-forms, since 4 − 2 = 2). It is an involution, so it splits it into *self-dual* and *anti-self-dual* subspaces, on which it acts respectively as +1 and −1.

Another useful example is *n* = 4 Minkowski spacetime with metric signature (+ − − −) and coordinates (*t*, *x*, *y*, *z*) where (using )

for one-forms while

for two-forms.

## Inner product of Template:Mvar-vectors

The Hodge dual induces an inner product on the space of Template:Mvar-vectors, that is, on the exterior algebra of Template:Mvar. Given two Template:Mvar-vectors Template:Mvar and Template:Mvar, one has

where Template:Mvar is the normalised Template:Mvar-form (i.e. *ω* ∧ ★*ω* = *ω*). In the calculus of exterior differential forms on a pseudo-Riemannian manifold of dimension Template:Mvar, the normalised Template:Mvar-form is called the volume form and can be written as

where is the matrix of components of the metric tensor on the manifold in the coordinate basis.

If an inner product is given on , then this equation can be regarded as an alternative definition of the Hodge dual.^{[6]} The wedge products of elements of an orthonormal basis in Template:Mvar form an orthonormal basis of the exterior algebra of Template:Mvar.

## Duality

The Hodge star defines a dual in that when it is applied twice, the result is an identity on the exterior algebra, up to sign. Given a Template:Mvar-vector *η* in Λ^{k}(*V*) in an Template:Mvar-dimensional space Template:Mvar, one has

where Template:Mvar is related to the signature of the inner product on Template:Mvar. Specifically, Template:Mvar is the sign of the determinant of the inner product tensor. Thus, for example, if *n* = 4 and the signature of the inner product is either (+ − − −) or (− + + +) then *s* = −1. For ordinary Euclidean spaces, the signature is always positive, and so *s* = 1. When the Hodge star is extended to pseudo-Riemannian manifolds, then the above inner product is understood to be the metric in diagonal form.

Note that the above identity implies that the inverse of ★ can be given as

Note that if Template:Mvar is odd *k*(*n* − *k*) is even for any Template:Mvar whereas if Template:Mvar is even *k*(*n* − *k*) has the parity of Template:Mvar. Therefore:

where Template:Mvar is the degree of the forms operated on.

## Hodge star on manifolds

One can repeat the construction above for each cotangent space of an Template:Mvar-dimensional oriented Riemannian or pseudo-Riemannian manifold, and get the **Hodge dual** (*n* − *k*)-form, of a [[differential form|Template:Mvar-form]]. The Hodge star then induces an *L*^{2}-norm inner product on the differential forms on the manifold. One writes

for the inner product of sections Template:Mvar and Template:Mvar of . (The set of sections is frequently denoted as . Elements of are called exterior Template:Mvar-forms).

More generally, in the non-oriented case, one can define the hodge star of a Template:Mvar-form as a (*n* − *k*)-pseudo differential form; that is, a differential forms with values in the canonical line bundle.

### The codifferential

The most important application of the Hodge dual on manifolds is to define the **codifferential** Template:Mvar. Let

where d is the exterior derivative or differential, and *s* = 1 for Riemannian manifolds.

while

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, in that

where Template:Mvar is a (*k*+1)-form and Template:Mvar a Template:Mvar-form. This identity follows from Stokes' theorem for smooth forms, when

i.e. when Template:Mvar has empty boundary or when Template:Mvar or ★*ζ* has zero boundary values (of course, true adjointness follows after continuous continuation to the appropriate topological vector spaces as closures of the spaces of smooth forms).

Notice that since the differential satisfies d^{2} = 0, the codifferential has the corresponding property

The Laplace-deRham operator is given by

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

and non-negative:

The Hodge dual sends harmonic forms to harmonic forms. As a consequence of the 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

which in turn gives canonical identifications via Poincaré duality of *H ^{ k}*(

*M*) with its dual space.

## Derivatives in three dimensions

The combination of the ★ operator and the exterior derivative d generates the classical operators grad, curl, and div, in three dimensions. This works out as follows: d can take a 0-form (function) to a 1-form, a 1-form to a 2-form, and a 2-form to a 3-form (applied to a 3-form it just gives zero). For a 0-form, , the first case written out in components is identifiable as the grad operator:

The second case followed by ★ is an operator on 1-forms () that in components is the curl operator:

Applying the Hodge star gives:

The final case prefaced and followed by ★, takes a 1-form () to a 0-form (function); written out in components it is the divergence operator:

One advantage of this expression is that the identity d^{2} = 0, which is true in all cases, sums up two others, namely that curl(grad( *f* )) = 0 and 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.

One can also obtain the Laplacian. Using the information above and the fact that Δ *f* = div grad *f* then for a 0-form, :

## Notes

- ↑ The Geometry of Physics (3rd edition), T. Frankel, Cambridge University Press, 2012, ISBN 978-1107-602601
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## References

- 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. - Jurgen Jost (2002)
*Riemannian Geometry and Geometric Analysis*. Springer-Verlag. ISBN 3-540-42627-2. A detailed exposition starting from basic principles; does not treat the pseudo-Riemannian case. - 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. - Tevian Dray (1999)
*The Hodge Dual Operator*. A thorough overview of the definition and properties of the Hodge dual operator.