A new version of differentiable sphere theorem (Q1263818)

The classical differentiable sphere theorem states that a simply connected complete Riemannian manifold satisfying some strong curvature pinching is diffeomorphic to the standard sphere \(S^ n\). In this paper the upper curvature bound is replaced by a lower bound on the volume. More precisely the authors show that there exists a positive number \(\epsilon =\epsilon (n)\) such that every connected complete Riemannian manifold M of dimension n with \(K_ M\geq 1\) and \(Vol(M)\geq Vol(S^ n)- \epsilon\) is diffeomorphic to \(S^ n\). They also present an analogue for the real projective space involving an additional upper bound on the diameter.