# zbMATH — the first resource for mathematics

Collapsing Riemannian manifolds to ones of lower dimensions. (English) Zbl 0606.53027
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.

##### MSC:
 53C20 Global Riemannian geometry, including pinching
Full Text: