Topological Description of Riemannian Foliations with Dense Leaves

Riemannian foliations occupy an important place in geometry. An excellent survey is A. Haefliger’s Bourbaki seminar [11], and the book of P. Molino [18] is the standard reference for Riemannian foliations. In one of the appendices to this book, E. Ghys proposes the problem of developing a theory of equicontinuous foliated spaces paralleling that of Riemannian foliations; he uses the suggestive term “qualitative Riemannian foliations” for such foliated spaces. In our previous paper [1], we discussed the structure of equicontinuous foliated spaces and, more generally, of equicontinuous pseudogroups of local homeomorphisms of topological spaces. This concept was difficult to develop because of the local nature of pseudogroups and the failure of having an infinitesimal characterization of local isometries, as one does have in the Riemannian case. These difficulties give rise to two versions of equicontinuity: a weaker version seems to be more natural, but a stronger version is more useful for generalizing topological properties of Riemannian foliations. Another relevant property for this purpose is that of quasi-effectiveness, which is a generalization to pseudogroups of effectiveness for group actions. In the case of locally connected foliated spaces, quasi-effectiveness is equivalent to the quasi-analyticity introduced by Haefliger [9]. For instance, the following well-known topological properties of Riemannian foliations were generalized to strongly equicontinuous quasi-effective compact foliated spaces [1] (remark that we also assume that all foliated spaces are locally compact and Polish): • Leaves without holonomy are quasi-isometric to one another (our original motivation for that study).