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Homotopy types of positively curved manifolds with large volume. (English) Zbl 0662.53034
Consider the class of closed Riemannian manifolds M of dimension n and of sectional curvature \(\geq 1\). Let \(V_ 0\) be \(1/2\) of the volume of the round n-sphere of radius 1. The authors show that there is an \(\epsilon =\epsilon (n)>0\) such that any closed n-manifold M as above with \(Vol(M)\geq V_ 0-\epsilon\) has the homotopy type if either \(S^ n\) or \({\mathbb{R}}P^ n\). The proof uses critical point theory of distance functions.
Reviewer: W.Ballmann

MSC:
53C20 Global Riemannian geometry, including pinching
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