This paper studies the statistical query (SQ) complexity of estimating d-dimensional submanifolds in ℝn. We propose a purely geometric algorithm called Manifold Propagation, that reduces the problem to three natural geometric routines: projection, tangent space estimation, and point detection. We then provide constructions of these geometric routines in the SQ framework. Given an adversarial STAT(τ) oracle and a target Hausdorff distance precision ε = Ω(τ2/(d+1)), the resulting SQ manifold reconstruction algorithm has query complexity O(n polylog(n) ε−d/2), which is proved to be nearly optimal. In the process, we establish low-rank matrix completion results for SQ’s and lower bounds for randomized SQ estimators in general metric spaces.