Collapsing Riemannian manifolds to ones of lower dimensions (Q1084680)

The purpose of this paper is to find a condition on topological types of manifolds which converge to a manifold with lower dimension. More precisely, the following problem is discussed. Let \(M_ i\) be a sequence of Riemannian manifolds with bounded curvature and N be a Riemannian manifold. Assume that \(M_ i\) converges to N with respect to the Hausdorff distance. Then, describe the relation between the topological types of \(M_ i\) and N. In the paper it is proved that there exists a map \(f: M_ i\to N\), such that f is a fibration with infranilmanifolds as fibres. It is also proved that f is similar to a Riemannian submersion.