Definition:Hodge Star Operator

Definition

Let $\struct {M,g}$ be an oriented $n$-dimensional Riemannian manifold.

Let $T^* M$ be the cotangent bundle of $M$.

Let $T^k T^*M$ be the space of contravariant $k$-tensors on $T^* M$.

Let $\Lambda^k T^* M$ the subbundle of $T^k T^*M$ consisting of alternating tensors.

Let $\omega, \eta$ be smooth $k$-forms.

Let $\innerprod \cdot \cdot_g$ be the inner product on $k$-forms.

Let $\rd V_g$ be the Riemannian volume form.

Let $\wedge$ denote the wedge product.

Hodge star operator is a smooth bundle homomorphism $* : \Lambda^k T^* M \to \Lambda^{n - k} T^*M$ such that:

$\omega \wedge *\eta = \innerprod \omega \eta_g \rd V_g$

Source of Name

This entry was named for William Vallance Douglas Hodge.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Problems