  We study manifolds $M$ equipped with a quadrature rule $$ \int_{M}{\phi(x)
dx} \simeq \sum_{i=1}^{n}{a_i \phi(x_i)}.$$ We show that $n-$point quadrature
rules with nonnegative weights on a compact $d-$dimensional manifold cannot
integrate more than at most the first $c_{d}n + o(n)$ Laplacian eigenfunctions
exactly. The constants $c_d$ are explicitly computed and $c_2 = 4$. The result
is new even on $\mathbb{S}^2$ where it generalizes results on spherical
designs.
