Geometry and marginals

It is known that observations of a real-valued random variable defined over a smooth manifold M can be used to make inferences about M, at least when M is a curve or surface. We refine and extend the underlying asymptotic results and remove the condition dim M <= 2. New examples of nonsmoothness in marginals are described in detail for dim M = 3, 4, and methods are given for calculations in general. (C) 2005 Published by Elsevier Inc.