  We extend to the case of a $d$-dimensional compact connected oriented
Riemannian manifold $\mathcal M$ the theorem of A. Bondarenko, D. Radchenko and
M. Viazovska on the existence of $L$-designs consisting of $N$ nodes, for any
$N\ge C_{\mathcal M} L^d$. For this, we need to prove a version of the
Marcinkiewicz-Zygmund inequality for the gradient of diffusion polynomials.
