Collapsing Riemannian manifolds while keeping their curvature bounded. I.

*(English)*Zbl 0606.53028This is a very interesting paper of fundamental importance. It is the first of two papers devoted to the study of Riemannian manifolds with bounded curvature and (uniformly) ”small” injectivity radius. Here it is shown that if a smooth manifold M admits a certain topological structure called an F-structure of positive rank (to be thought of as compatible partial actions by tori), then M also admits a family of Riemannian metrics, \(g_{\delta}\), with uniformly bounded curvature, such that as \(\delta\to 0\), the injectivity radius, \(i_ p\), converges uniformly to zero at all points, \(p\in M\). The paper is enhanced by a number of illuminating examples. We are looking forward to the second part in which a sort of strengthened converse of the result indicated above is proved.

Reviewer: K.Grove

##### MSC:

53C20 | Global Riemannian geometry, including pinching |