= Hodge Laplacian
{c}
{title2=$\Delta=d\delta+\delta d$}
The <Hodge Laplacian> acts on <differential forms> by
$$
\Delta_H=d\delta+\delta d,
$$
where $d$ is the <exterior derivative> and $\delta$ its <codifferential>. On a compact oriented <Riemannian manifold> without boundary, <Hodge integration by parts in arbitrary degree> gives
$$
\langle\Delta_H\alpha,\alpha\rangle_{L^2}=\|d\alpha\|_{L^2}^2+\|\delta\alpha\|_{L^2}^2\geq0.
$$
Consequently a <harmonic differential form> is both a <closed differential form> and a <coclosed differential form> on such a manifold. On a manifold with boundary, the same calculation has boundary terms unless compatible boundary conditions remove them.
On scalar <functions>, this nonnegative sign convention gives $\Delta_H=-\Delta_g$ when the <Laplace-Beltrami operator> is defined as $\Delta_g=\operatorname{div}_g\nabla_g$. Stating the convention prevents a sign ambiguity when comparing geometric and analytical formulas.
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