Harmonic one-form (${\mathcal{H}}^1(M)$)

A harmonic one-form is a differential one-form $\alpha$ satisfying $\Delta \alpha =0$ for the Hodge Laplacian. On a closed manifold, the Hodge decomposition theorem identifies these forms with degree-one de Rham cohomology. The Bochner-Weitzenbock formula for one-forms implies that harmonic one-forms are parallel under nonnegative Ricci curvature.