## Collapsing and pinching under a lower curvature bound.(English)Zbl 0737.53041

For a positive integer $$n$$ and a real number $$D>0$$, denote by $${\mathcal M}$$ the set of compact Riemannian manifolds $$M$$ with sectional curvatures $$K_ M\geq -1$$ and diameters $$\text{diam}(M)\leq D$$. By a theorem of Gromov, $${\mathcal M}$$ is relatively compact in the set of compact metric spaces with respect to the Hausdorff distance. For a number of reasons (given in the paper) it is natural to ask what collapsing phenomena occur in the limits of convergent sequences in $${\mathcal M}$$ and what types of pinching theorems hold in the class $${\mathcal M}$$. In his main theorem, the author proves that for complete manifolds $$M$$ and $$N$$ of dimension $$n$$ with $$K_ M\geq -1$$, $$| K_ N|\leq 1$$ and injectivity radius of $$N\geq\mu>0$$, there exists a positive number $$\epsilon_ n(\mu)$$, depending only on $$n$$ and $$\mu$$, such that if the Hausdorff distance $$\epsilon$$ between $$M$$ and $$N$$ is less than $$\epsilon_ n(\mu)$$, then $$M$$ fibers over $$N$$ (as an almost Riemannian submersion) with fiber $$F$$, which — in general through a finite covering space — is diffeomorphically related to a torus of dimension equal to the first Betti number of $$F$$. As a corollary he obtains the following Pinching Theorem: There exists a positive number $$\epsilon_ n$$ depending only on $$n$$ such that if the curvature and diameter of a compact Riemannian $$n$$- manifold $$M$$ satisfy $$K_ M\text{diam}(M)^ 2>-\epsilon_ n$$, then a finite covering of $$M$$ fibers over a torus of dimension equal to the first Betti number of $$M$$. In case this Betti number equals $$n$$, the manifold $$M$$ is diffeomorphic to a torus.

### MSC:

 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 53C20 Global Riemannian geometry, including pinching
Full Text: