= Dimension bound for harmonic one-forms under nonnegative Ricci curvature
{title2=$\dim\mathcal H^1(M)\le\dim M$}
On a compact connected <Riemannian manifold> without boundary with nonnegative <Ricci curvature>, the <Bochner formula for one-forms> makes every <harmonic one-form> parallel. Evaluation at any point is then injective: a parallel form that is zero there stays zero along every path. Hence $\dim\mathcal H^1(M)\le\dim M$. Flat tori attain equality. Positive-definite <Ricci curvature> at even one point instead forces $\mathcal H^1=0$, because the Bochner integral makes its Ricci contraction vanish everywhere.
Back to article page
