  The theory of optimal design of experiments has been traditionally developed
on an Euclidean space. In this paper, new theoretical results and an algorithm
for finding the optimal design of an experiment located on a Riemannian
manifold are provided. It is shown that analogously to the results in Euclidean
spaces, D-optimal and G-optimal designs are equivalent on manifolds, and we
provide a lower bound for the maximum prediction variance of the response
evaluated over the manifold. In addition, a converging algorithm that finds the
optimal experimental design on manifold data is proposed. Numerical experiments
demonstrate the importance of considering the manifold structure in a designed
experiment when present, and the superiority of the proposed algorithm.
