Hodge star operator - Knowlpedia

Let $(M,g)$ be an oriented $n$ -dimensional Riemannian manifold. The metric and orientation determine an inner product on $k$ -forms and a volume form $\operatorname{vol}_g$ . The Hodge star operator is the unique linear isomorphism

*:\Omega^k(M)\longrightarrow\Omega^{n-k}(M)

such that, for all real $k$ -forms $\alpha,\beta$ ,

\alpha\wedge *\beta=\langle\alpha,\beta\rangle_g\,\operatorname{vol}_g.

It is defined pointwise and varies smoothly with $g$ . Thus the star combines the wedge product, the metric, and orientation into an operator complementary in degree.

Characterizing identities

On real $k$ -forms in the Riemannian convention,

*(*\alpha)=(-1)^{k(n-k)}\alpha.

The star is a pointwise isometry, and $*1=\operatorname{vol}_g$ , while $*\operatorname{vol}_g=1$ . These identities follow directly in an oriented orthonormal coframe and characterize the signs used here.

The star also converts the exterior derivative into its formal adjoint, up to a degree- and dimension-dependent sign. This is the basic mechanism behind the Hodge Laplacian and harmonic-form theory.

Computation in an orthonormal coframe

If $e^1,\ldots,e^n$ is a positively oriented orthonormal coframe and $I=(i_1<\cdots<i_k)$ , then $*e^I$ is the uniquely signed complementary wedge $e^{I^c}$ for which

e^I\wedge *e^I=e^1\wedge\cdots\wedge e^n.

For the standard orientation of $\mathbb R^3$ , for example, $*dx=dy\wedge dz$ , $*dy=dz\wedge dx$ , and $*dz=dx\wedge dy$ .

Conventions and scope

References

Shigeyuki Morita, Geometry of Differential Forms, Translations of Mathematical Monographs 201, American Mathematical Society, 2001. Publisher record. Relevant: “Laplacian and harmonic forms.”
Raymond O. Wells Jr., Differential Analysis on Complex Manifolds, 3rd ed., Graduate Texts in Mathematics 65, Springer, 2008. Publisher record. Relevant: Chapter III, “Differential Geometry,” Hermitian exterior algebra and the Hodge star.