Optimal Cubature Formulas on Compact Homogeneous Manifolds

We find lower bounds for the rate of convergence of optimal cubature formulas on sets of differentiable functions on compact homogeneous manifolds of rank I or two-point homogeneous spaces. It is shown that these lower bounds are sharp in the power scale in the case of $\mathbb{S}^{2}$, the unit sphere in $\mathbb{R}^{3}$.