Let be a smooth compact oriented manifold without boundary, imbedded in a Euclidean space , and let be a smooth map of into a Riemannian manifold Λ. An unknown state is observed via X = , where > 0 is a small parameter and is a white Gaussian noise. For a given smooth prior on and smooth estimators of the map we derive a secondorder asymptotic expansion for the related Bayesian risk. The calculation involves the geometry of
the underlying spaces and , in particular, the integration-by-parts formula. Using this result, a second-order minimax estimator of is found based on the modern theory of harmonic maps and hypo-elliptic differential operators.