= Harmonic top-degree forms on a closed oriented manifold
{title2=$\mathcal H^n(M)=\mathbb R\,\operatorname{vol}_g$}
On a connected compact oriented <Riemannian manifold> without boundary, a top-degree form is $f\operatorname{vol}_g$. Its exterior derivative is automatically zero, and the Hodge codifferential vanishes exactly when $df=0$. Thus it is harmonic exactly when $f$ is constant. Equivalently the <Hodge star operator> identifies this space with harmonic functions. Without connectedness the dimension is the number of connected components.
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