Adjoint of the Hodge star on middle-degree forms
= Adjoint of the Hodge star on middle-degree forms
{c}
{title2=$*^\dagger=(-1)^{m^2}*$}
On $m$-forms in an oriented Riemannian manifold of dimension $2m$, the <Hodge star operator> is orthogonal and satisfies $*^2=(-1)^{m^2}$. Hence
$$
*^\dagger=*^{-1}=(-1)^{m^2}*.
$$
It is self-adjoint when $m$ is even and skew-adjoint when $m$ is odd.