Hodge star operator

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 -dimensional vector space, the Hodge star is a one-to-one mapping of -vectors to -vectors; the dimensions of these spaces are the binomial coefficients formula_1.
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 -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 be an -dimensional vector space with a nondegenerate symmetric bilinear form formula_2, referred to here as an inner product. This induces an inner product on "k"-vectors for formula_3, by defining it on decomposable -vectors formula_4 and formula_5 to equal the Gram determinant
extended to formula_7 through linearity.
The unit -vector formula_8 is defined in terms of an oriented orthonormal basis formula_9 of as:
The Hodge star operator is a linear operator on the exterior algebra of , mapping -vectors to ()-vectors, for formula_3. It has the following property, which defines it completely:
Dually, in the space formula_14of -forms (alternating -multilinear functions on formula_15), the dual to formula_16 is the volume form formula_17, the function whose value on formula_18 is the determinant of the formula_19 matrix assembled from the column vectors of formula_20 in formula_21-coordinates.
Applying formula_17 to the above equation, we obtain the dual definition:
or equivalently, taking formula_4, formula_5, and formula_26:
This means that, writing an orthonormal basis of -vectors as formula_28 over all subsets formula_29 of formula_30, the Hodge dual is the ()-vector corresponding to the complementary set formula_31:
where formula_33 is the sign of the permutation formula_34.
Since Hodge star takes an orthonormal basis to an orthonormal basis, it is an isometry on the exterior algebra formula_35.
Geometric explanation.
The Hodge star is motivated by the correspondence between a subspace of and its orthogonal subspace (with respect to the inner product), where each space is endowed with an orientation and a numerical scaling factor. Specifically, a non-zero decomposable -vector formula_36 corresponds by the Plücker embedding to the subspace formula_37 with oriented basis formula_38, endowed with a scaling factor equal to the -dimensional volume of the parallelepiped spanned by this basis (equal to the Gramian, the determinant of the matrix of inner products formula_39). The Hodge star acting on a decomposable vector can be written as a decomposable ()-vector:
where formula_41 form an oriented basis of the orthogonal space formula_42. Furthermore, the ()-volume of the formula_43-parallelepiped must equal the -volume of the formula_44-parallelepiped, and formula_45 must form an oriented basis of .
A general -vector is a linear combination of decomposable -vectors, and the definition of the Hodge star is extended to general -vectors by defining it as being linear.
Examples.
Two dimensions.
In two dimensions with the normalized Euclidean metric and orientation given by the ordering , the Hodge star on -forms is given by
On the complex plane regarded as a real vector space with the standard sesquilinear form as the metric, the Hodge star has the remarkable property that it is invariant under holomorphic changes of coordinate. If is a holomorphic function of , then by the Cauchy–Riemann equations we have that and . In the new coordinates
so that
proving the claimed invariance.
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 formula_49 of one-forms often used in vector calculus, one finds that
The Hodge star relates the exterior and cross product in three dimensions:formula_51Applied to three dimensions, the Hodge star provides an isomorphism between axial vectors and bivectors, so each axial vector is associated with a bivector and vice versa, that is: formula_52.
The Hodge star can also be interpreted as a form of the geometric correspondence between an axis and an infinitesimal rotation around the axis, with speed equal to the length of the axis vector. An inner product on a vector space formula_53 gives an isomorphism formula_54 identifying formula_53 with its dual space, and the space of all linear operators formula_56 is naturally isomorphic to the tensor product formula_57. Thus for formula_58, the star mapping formula_59 takes each vector formula_60 to a bivector formula_61, which corresponds to a linear operator formula_62. Specifically, formula_63 is a skew-symmetric operator, which corresponds to an infinitesimal rotation: that is, the macroscopic rotations around the axis formula_64 are given by the matrix exponential formula_65. With respect to the basis formula_49 of formula_67, the tensor formula_68 corresponds to a coordinate matrix with 1 in the formula_69 row and formula_70 column, etc., and the wedge formula_71 is the skew-symmetric matrix formula_72, etc. That is, we may interpret the star operator as:formula_73Under this correspondence, cross product of vectors corresponds to the commutator Lie bracket of linear operators: formula_74.
Four dimensions.
In case , 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, then application twice will return the argument up to a sign – see "" below. For example, in Minkowski spacetime where with metric signature and coordinates where (using formula_75):
for one-forms while
for 2-forms. Because their determinants are the same in both and , the signs of the Minkowski space 2-form duals depend only on the chosen orientation.
An easy rule to remember for the above Hodge operations is that given a form formula_78, its Hodge dual formula_79 may be obtained by writing the components not involved in formula_78 in an order such that formula_81. An extra minus sign will enter only if formula_78 does not contain formula_83. (The latter convention stems from the choice for the metric signature. For , one puts in a minus sign only if formula_78 involves formula_83.)
Conformal invariance.
The Hodge star is conformally invariant on n forms on a 2n dimensional vector space V, i.e. if formula_86 is a metric on formula_87 and formula_88, then the induced Hodge stars 
are the same. 
Example: Derivatives in three dimensions.
The combination of the formula_90 operator and the exterior derivative generates the classical operators , , and on vector fields in three-dimensional Euclidean space. This works out as follows: 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 formula_91, the first case written out in components gives:
The inner product identifies 1-forms with vector fields as formula_93, etc., so that formula_94 becomes formula_95.
In the second case, a vector field formula_96 corresponds to the 1-form formula_97, which has exterior derivative:
Applying the Hodge star gives the 1-form:
which becomes the vector field formula_100.
In the third case, formula_96 again corresponds to formula_97. Applying Hodge star, exterior derivative, and Hodge star again:
One advantage of this expression is that the identity , which is true in all cases, sums up two others, namely that and . 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 formula_104 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:
The Laplacian can also be seen as a special case of the more general Laplace–deRham operator formula_106 where formula_107 is the codifferential for formula_108-forms. Any function formula_109 is a 0-form, and formula_110 and so this reduces to the ordinary Laplacian. For the 1-form formula_111 above, the codifferential is formula_112 and after some plug and chug, one obtains the Laplacian acting on formula_111.
Duality.
Applying the Hodge star twice leaves a -vector unchanged except for its sign: for formula_114 in an -dimensional space , one has
where is the parity of the signature of the inner product on , that is, the sign of the determinant of the matrix of the inner product with respect to any basis. For example, if and the signature of the inner product is either or then . For Riemannian manifolds (including Euclidean spaces), we always have .
The above identity implies that the inverse of formula_90 can be given as
If is odd then is even for any , whereas if is even then has the parity of . Therefore:
where 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 formula_119 and its exterior powers formula_120, and hence to the differential "k"-forms formula_121, the global sections of the bundle formula_122. The Riemannian metric induces an inner product on formula_120 at each point formula_124. We define the Hodge dual of a "k"-form formula_125, defining formula_126 as the unique ("n" – "k")-form satisfying
for every "k"-form formula_128, where formula_129 is a real-valued function on formula_130, and the volume form formula_131 is induced by the Riemannian metric. Integrating this equation over formula_130, the right side becomes the formula_133 (square-integrable) inner product on "k"-forms, and we obtain:
More generally, if formula_130 is non-oriented, 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 formula_136 in a tangent space formula_137 and its dual basis formula_138 in formula_139, having the metric matrix formula_140 and its inverse matrix formula_141. The Hodge dual of a decomposable "k"-form is:
Here formula_143 is the Levi-Civita symbol with formula_144, and we implicitly take the sum over all values of the repeated indices formula_145. The factorial formula_146 accounts for double counting, and is not present if the summation indices are restricted so that formula_147. 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:
The factorial formula_149 is again included to account for double counting when we allow non-increasing indices. We would like to define the dual of the component formula_150 so that the Hodge dual of the form is given by
Using the above expression for the Hodge dual of formula_152, we find:
Although one can apply this expression to any tensor formula_78, 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 formula_155 is given by:
Codifferential.
The most important application of the Hodge star on manifolds is to define the codifferential formula_157 on "k"-forms. Let
where formula_159 is the exterior derivative or differential, and formula_160 for Riemannian manifolds. Then
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 with respect to the square-integrable inner product:
where formula_125 is a -form and formula_128 a -form. This identity follows from Stokes' theorem for smooth forms:
provided "M" has empty boundary, or formula_128 or formula_168 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 of a sequence of forms formula_169 (as formula_170) to be interchanged with the combined differential and integral operations, so that formula_171 and likewise for sequences converging to formula_172.)
Since the differential satisfies formula_173, 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 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 -forms, and so the Hodge star induces an isomorphism of cohomology groups
which in turn gives canonical identifications via Poincaré duality of with its dual space.