Hodge star commutes with the Hodge Laplacian
= Hodge star commutes with the Hodge Laplacian
{c}
{title2=$\Delta*=*\Delta$}
On an oriented <Riemannian manifold>, the <codifferential> convention $\delta=(-1)^{n(p+1)+1}*d*$ and $*^2=(-1)^{p(n-p)}$ give $d*\alpha=(-1)^p*\delta\alpha$ and $\delta*\alpha=(-1)^{p+1}*d\alpha$. Applying these twice proves the displayed identity. Thus the <Hodge star operator> preserves <harmonic differential forms> even without compactness.