# 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.

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 *k*-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.

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

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

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 *z* = *x* + *iy* is a holomorphic function of *w* = *u* + *iv*, then by the Cauchy–Riemann equations we have that ∂*x*/∂*u* =
∂*y*/∂*v* and
∂*y*/∂*u* = −
∂*x*/∂*v*. In the new coordinates

The Hodge star relates the exterior and cross product in three dimensions:^{[2]}

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.

One can also obtain the Laplacian Δ *f* = div grad *f* in terms of the above operations:

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

If n is odd then *k*(*n* − *k*) is even for any k, whereas if n is even then *k*(*n* − *k*) has the parity of k. Therefore:

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:

The Hodge star sends harmonic forms to harmonic forms. As a consequence of Hodge theory, the de Rham cohomology is naturally isomorphic to the space of harmonic k-forms, and so the Hodge star induces an isomorphism of cohomology groups

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

*M*) with its dual space.