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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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