Harmonic one-forms are parallel under nonnegative Ricci curvature
= Harmonic one-forms are parallel under nonnegative Ricci curvature
On a closed <Riemannian manifold> with nonnegative <Ricci curvature>, integration of the <Bochner-Weitzenbock formula for one-forms> for a <harmonic one-form> gives
$$
0=\|\nabla\alpha\|_{L^2}^2+\int_M\operatorname{Ric}(\alpha^{\sharp},\alpha^{\sharp})\,d\mathrm{vol}_g.
$$
Both terms are nonnegative, so $\nabla\alpha=0$. Riemannian volume density makes the argument valid even without orientation.