= Bochner identity for Killing vector fields
{c}
{title2=$\int_M|\nabla X|^2\,dV=\int_M\operatorname{Ric}(X,X)\,dV$}
= Integrated Bochner identity for Killing fields
{synonym}
On a closed <Riemannian manifold>, a <Killing vector field> $X$ satisfies $\nabla^*\nabla X=\operatorname{Ric}^{\sharp}X$ for the nonnegative <rough Laplacian>. This follows by differentiating $\nabla_iX_j+\nabla_jX_i=0$, contracting and commuting <covariant derivatives>; the contraction also gives $\operatorname{div}X=0$. Taking the <inner product> with $X$ and using <integration by parts> gives the displayed identity. If <Ricci curvature> is nonpositive, the nonnegative left side must vanish, so $X$ is a <parallel vector field>. In particular every <Killing vector field> on a closed <flat Riemannian manifold> is a <parallel vector field>.
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