= Solvability condition for the Hodge Poisson equation
{title2=$\beta\perp\mathcal H^k(M)$}
The <Poisson equation for differential forms> on a closed oriented <Riemannian manifold> is solvable exactly when its source is $L^2$-orthogonal to every <harmonic differential form> of the same degree. Necessity follows from the <formal adjoint> identity $\langle\Delta\alpha,h\rangle=\langle\alpha,\Delta h\rangle=0$. For sufficiency, the <Green operator of the Hodge Laplacian> gives $\alpha=G\beta$, since $\Delta G=I-P_{\mathcal H}$.
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