Asymptotically optimal cubature formulas on manifolds for prefixed weights

Given a Riemannian manifold X with Riemannian measure μ and positive weights {ω_j}, j=1,...,N, we study the conditions under which there exist points {x_j}, j=1,...,N in X so that a cubature formula of the form ∫P dμ=∑ω_jP(x_j) holds for all polynomials P of order less than or equal to L. The problem is studied for diffusion polynomials (linear combinations of eigenfunctions of the Laplace–Beltrami operator) in the context of abstract Riemannian manifolds and for algebraic polynomials in the context of algebraic manifolds in R^n.