= Bochner-Weitzenbock formula for one-forms
{c}
{title2=$\Delta\alpha=\nabla^*\nabla\alpha+\operatorname{Ric}\cdot\alpha$}
For a real one-form $\alpha$, the <Hodge Laplacian> differs from the <rough Laplacian> by the action of <Ricci curvature>: $(\operatorname{Ric}\cdot\alpha)(X)=\alpha(\operatorname{Ric}^{\sharp}X)$, where $g(\operatorname{Ric}^{\sharp}X,Y)=\operatorname{Ric}(X,Y)$. Equivalently, with the nonnegative scalar Laplacian convention,
$$
\tfrac12\Delta|\alpha|^2=\langle\Delta\alpha,\alpha\rangle-|\nabla\alpha|^2-\operatorname{Ric}(\alpha^{\sharp},\alpha^{\sharp}).
$$
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