Differentiable sphere theorem by curvature pinching (Q1179723)

A Riemannian manifold \(M\) is said to be \(\delta\)-pinched if the sectional curvature \(s\) of \(M\) satisfies \(\delta\leqq s\leqq 1\) everywhere. The problem whether \(M\) is diffeomorphic to the ordinary sphere, if \(M\) is \(\delta\)-pinched for a certain positive number \(\delta\), is known to be the differentiable pinching problem and evaluations for \(\delta\) are made by several authors, whose approaches may be classified into two main types. The author improves the pinching theorem to the pinching up to \(\delta\leqq 0.681\) by combining these two types of approaches.