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Hodge star operator

Idea

Given a finite dimensional (pseudo)-Riemannian manifold $(X,g)$, the Hodge star operator “completes” a $k$-differential form to the volume form of $(X,g)$.

Definition

Let $(X,g)$ be an oriented $n$-dimensional smooth manifold $X$ endowed with a (pseudo)-Riemannian metric $g$. For $0\le k\le n$, write ${\Omega }^k(X)$ for the vector space of $k$-forms on $X$.

Hodge inner product

The metric $g$ naturally induces a nondegenerate symmetric bilinear form

$$(-\mid -):{\Omega }^k(X)\otimes {\Omega }^k(X)\to {\Omega }^0(X).$$(-\mid-) : \Omega^k(X) \otimes \Omega^k(X) \to \Omega^0(X) \,.

If $X$ is compact then the integral of this against the volume form ${\mathrm{vol}}_g$ exists. This is the Hodge inner product

$$⟨-,-⟩:{\Omega }^k(X)\otimes {\Omega }^k(X)\to ℝ$$\langle - , - \rangle : \Omega^k(X)\otimes \Omega^k(X) \to \mathbb{R}

$$⟨\alpha ,\beta ⟩:={\int }_X(\alpha \mid \beta )\mathrm{vol}.$$\langle \alpha, \beta \rangle := \int_X (\alpha\mid \beta) vol \,.

Hodge star operator

The Hodge star operator is the unique linear function

$$\star :{\Omega }^k(X)\to {\Omega }^{n-k}(X)$${\star}\colon \Omega^k (X) \to \Omega^{n-k} (X)

defined by the identity

$$\alpha \wedge \star \beta =(\alpha \mid \beta ){\mathrm{vol}}_g,\forall \alpha ,\beta \in \stackrel{k}{\bigwedge }X,$$\alpha \wedge \star\beta = (\alpha \mid \beta) vol_g, \qquad \forall \alpha,\beta \in \bigwedge^k X \,,

where ${\mathrm{vol}}_g\in {\Omega }^{n}X$ is the volume form induced by $g$.

Therefore in terms of the Hodge operator the Hodge inner product reads

$$⟨\alpha ,\beta ⟩={\int }_X\alpha \wedge \star \beta .$$\langle \alpha , \beta\rangle = \int_X \alpha \wedge \star \beta \,.

Properties

Component formulas

If $e_1,\dots ,e_n$ is a local basis on $X$ and $e^1,\dots ,e^n$ is the dual basis, so that $\alpha =\frac{1}{k!}{\alpha }_{i_1,\dots ,i_k}{e}^{i_1}\wedge \cdots \wedge e^{i_k}$, then

$$\star \alpha =\frac{1}{k!(n-k)!}{ϵ}_{i_1,\dots ,i_n}\sqrt{\mid \det (g)\mid }{\alpha }_{j_1,\dots ,j_k}{g}^{i_1,j_1}\cdots g^{i_k,j_k}{e}^{i_{k+1}}\wedge \cdots \wedge e^{i_n},$$\star \alpha = \frac{1}{k!(n-k)!} \epsilon_{i_1,\dots,i_n} \sqrt{|det(g)|} \alpha_{j_1,\dots,j_k} g^{i_1,j_1} \cdots g^{i_k,j_k} e^{i_{k+1}} \wedge \cdots \wedge e^{i_n},

where ${ϵ}_{i_1,\dots ,i_n}$ is the sign of the permutation $(1,2,\dots ,n)\mapsto (i_1,i_2,\dots ,i_n)$ and $\det (g)$ is the determinant of $g$ in the local basis.

Basis-independent formulas

Let $(X,g)$ be a Riemannian manifold of dimension $n$ and let $\omega ,\lambda \in {\Omega }^k(X)$. Then

• $\star (\star \omega )=(-1{)}^{k(n+1)}\omega =(-1{)}^{k(n-k)}\omega$;

• $⟨\star \omega ,\star \lambda ⟩=⟨\omega \mid \lambda ⟩$;

• $\star 1=\mathrm{vol}$.

Generalisations

The metric $g$ is used in two places in the specification of the Hodge operator: in the inner product on forms and in the volume form. If $X$ is equipped only with a volume form (not necessarily coming from a metric), then the Hodge operator still takes $k$-forms to $(n-k)$-vector fields. If the manifold is not oriented, then the metric only gives a volume pseudoform, but the Hodge operator still takes $k$-forms to $(n-k)$-pseudoforms. Finally, if $X$ is equipped with only a volume pseudoform (which is equivalent to an absolutely continuous Radon measure on $X$), then the Hodge operator takes $k$-forms to $(n-k)$-pseudovector fields. (Of course, in every case, one might apply the operator to pseudoforms or multivector fields to begin with.)

References

Some useful basic formulas are listed in

• Hodge theory on Riemannian manifolds , lecture notes (pdf)