Hodge star operator - Imported Knowlpedia Content

Definition

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 construction and its role in the Laplacian are treated in Morita, “Laplacian and harmonic forms”.

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.