Bochner-Weitzenbock formula for one-forms ($\Delta \alpha ={\nabla }^{\ast }\nabla \alpha +\mathrm{Ric}\cdot \alpha$)

For a real one-form $\alpha$, the Hodge Laplacian differs from the rough Laplacian by the action of Ricci curvature: $(\mathrm{Ric}\cdot \alpha )(X)=\alpha ({\mathrm{Ric}}^{♯}X)$, where $g({\mathrm{Ric}}^{♯}X,Y)=\mathrm{Ric}(X,Y)$. Equivalently, with the nonnegative scalar Laplacian convention,

On a closed Riemannian manifold with nonnegative Ricci curvature, integration of the Bochner-Weitzenbock formula for one-forms for a harmonic one-form givesBoth terms are nonnegative, so $\nabla \alpha =0$. Riemannian volume density makes the argument valid even without orientation.