Homotopy types of positively curved manifolds with large volume. (English) Zbl 0662.53034

Grove, Karsten; Petersen, Peter V

Am. J. Math. 110, No. 6, 1183-1188 (1988).

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

Keywords:

positive curvature; homotopy type; critical point theory; distance functions