= Betti-number obstruction to nonnegative Ricci curvature
{c}
{title2=$b_1(M)>\dim M\implies\text{no metric with }\operatorname{Ric}\ge0$}
On a closed connected <Riemannian manifold> with nonnegative <Ricci curvature>, <harmonic one-forms are parallel under nonnegative Ricci curvature>. Evaluation at one point injects their space into the <cotangent space>. The <Hodge decomposition theorem> identifies its dimension with the first <Betti number>, giving $b_1(M)\le\dim M$. For example, a genus-two closed orientable surface times a <circle> has dimension three and first <Betti number> five, so no <Riemannian metric> on that product has nonnegative <Ricci curvature>.
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