Given a finite dimensional (pseudo)-Riemannian manifold $(X,g)$, the Hodge star operator “completes” a $k$-differential form to the volume form of $(X,g)$.
Let $(X,g)$ be an oriented $n$-dimensional smooth manifold $X$ endowed with a (pseudo)-Riemannian metric $g$. For $0 \leq k \leq n$, write $\Omega^k(X)$ for the vector space of $k$-forms on $X$.
The metric $g$ naturally induces a nondegenerate symmetric bilinear form
If $X$ is compact then the integral of this against the volume form $vol_g$ exists. This is the Hodge inner product
The Hodge star operator is the unique linear function
defined by the identity
where $vol_g \in \Omega^n X$ is the volume form induced by $g$.
Therefore in terms of the Hodge operator the Hodge inner product reads
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
where $\epsilon_{i_1,\dots,i_n}$ 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.
Let $(X,g)$ be a Riemannian manifold of dimension $n$ and let $\omega,\lambda \in \Omega^k(X)$. Then
$\star(\star\omega) = (-1)^{k(n+1)} \omega = (-1)^{k(n-k)} \omega$;
$\langle\star\omega , \star\lambda\rangle = \langle\omega | \lambda\rangle$;
$\star 1 = vol$.
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.)
Some useful basic formulas are listed in