Harmonic differential form ($\Delta \alpha =0$)

A differential form is harmonic when it lies in the kernel of the Hodge Laplacian $\Delta =d\delta +\delta d$. On a compact manifold, this is equivalent to $d\alpha =0$ and $\delta \alpha =0$.

On a compact oriented Riemannian manifold,as an $L^2$-orthogonal direct sum. Every de Rham cohomology class consequently has a unique harmonic representative.