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.


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