Let $(M^{4},g)$ be a Riemannian $4$-manifold and let $\omega\in\Omega^{1}(M^{4})$ satisfy $\mathscr{D}_{g}\omega=0$, then

(7.116)

$$\Delta_{H}\omega=0,$$

where $\mathscr{D}_{g}\equiv d^{+}+d^{*}$ and $\Delta_{H}$ is the Hodge Laplacian.