For numerical integration on compact Riemannian manifolds one may use cubature rules weighting function values at a suitable point set to approximate the integral with respect to the Riemannian volume. Building on our earlier work on bounded convex domains, we present a characterization of point sets with asymptotically optimal worst-case error in Sobolev spaces on such manifolds. Applied to random points, our result closes (except for boundary cases) a logarithmic gap left open by Ehler, Gräf and Oates [Stat. Comput. 29, 2019]. Using a well-known relation on the sphere connected to Stolarsky's principle, we apply the Hilbert space case to weighted spherical L2-discrepancy. This talk is based on joint work with D. Krieg.