Coclosed differential form ($\delta \alpha =0$)

A differential form is coclosed when its codifferential vanishes. On a closed Riemannian manifold, a form is a harmonic differential form exactly when it is both a closed differential form and coclosed: the identity $⟨\Delta \alpha ,\alpha ⟩=\Vert d\alpha {\Vert }^2+\Vert \delta \alpha {\Vert }^2$ proves the converse as well as necessity.