= Compact harmonic-spinor rigidity under nonnegative scalar curvature
{title2=$\not\nabla\psi=0\Longrightarrow\nabla\psi=0$}
On a compact boundaryless Riemannian <spin manifold> with $R\ge0$, integrating the <Lichnerowicz spinor-square formula> against a harmonic spinor gives $\int(|\nabla\psi|^2+R|\psi|^2/4)=0$. Positivity forces the spinor to be parallel. Its norm is constant on each connected component, so a nontrivial solution is nowhere zero there. Strictly positive <scalar curvature> at any point on such a component excludes nonzero harmonic spinors.
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