 Hodge Laplacian

The Hodge Laplacian acts on differential forms bywhere $d$ is the exterior derivative and $\delta$ its codifferential. On a compact oriented Riemannian manifold without boundary, Hodge integration by parts in arbitrary degree givesConsequently 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 ${\Delta }_g={\mathrm{div}}_g{\nabla }_g$. Stating the convention prevents a sign ambiguity when comparing geometric and analytical formulas.