= Riemannian parallel-spinor Ricci-flatness
{title2=$\nabla\psi=0,\ \psi\ne0\Longrightarrow\operatorname{Ric}=0$}
A parallel spinor has zero <spin> <curvature>. Contracting the spinorial <Ricci identity> with a <gamma matrix> gives $R_{\mu\nu}\gamma^\nu\psi=0$. For each fixed $\mu$, squaring <Clifford multiplication> by that Ricci row gives $|R_{\mu\cdot}|^2\psi=0$. Riemannian positivity and a nonzero spinor force every row to vanish. This implication is local and does not itself require compactness; the compact hypothesis is used to obtain parallelism from harmonicity.
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