Topological Description Of Riemannian Foliations With Dense Leaves

A foliation is called Riemannian if its holonomy pseudogroup consists of local isometries for some Riemannian metric. By combining the work on Hilbert's fifth problem for local groups with our work on equicontinuous foliated spaces, we prove that, if a foliated space is strongly equicontinuous, locally connected and of finite dimension, has a dense leaf, and has holonomy pseudogroup whose closure is quasianalytic, then it is a Riemannian foliation.