Asymptotic quantization for probability measures on Riemannian manifolds

In this paper we study the quantization problem for probability measures on Riemannian manifolds. Under a suitable assumption on the growth at infinity of the measure we find asymptotic estimates for the quantization error, generalizing the results on ${\mathbb{R}}^d\mathrm{.}$ Our growth assumption depends on the curvature of the manifold and reduces, in the flat case, to a moment condition. We also build an example showing that our hypothesis is sharp.