= Parallel differential form
{title2=$\nabla\alpha=0$}
A <differential form> $\alpha$ on a <Riemannian manifold> is parallel when its <covariant derivative> under the <Levi-Civita connection> vanishes:
$$
\nabla\alpha=0.
$$
Equivalently, <parallel transport> preserves the <differential form>. The <exterior derivative> is the alternating part of this <covariant derivative>, and the <codifferential> is a contraction of it, so $d\alpha=0$ and $\delta\alpha=0$. Consequently the <Hodge Laplacian> satisfies $\Delta_H\alpha=(d\delta+\delta d)\alpha=0$.
This is a pointwise harmonicity statement, requiring no compactness. On a manifold with boundary it does not by itself impose a chosen boundary condition. Constant <functions> and the <Riemannian volume form> on an oriented <Riemannian manifold> are examples.
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