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.