The differentiable pinching problem and the diffeotopy theorem (Q1174410)

The author improves results on the differentiable pinching problems obtained by Sugimoto-Shiohama, Im Hof-Ruh, down to the pinching number of 0.681, that is: Theorem. A 0.681-pinched Riemannian manifold is diffeomorphic to the standard sphere. The result is obtained by a modification of Sugimoto-Shiohama's method. The author shows that the equatorial diffeomorphism \(f\) of \(S^{n-1}\) onto \(S^{n-1}\) deduced by the pinching property is diffeotopic to the identity map constructing an approximation map in \(SO(n,\mathbb{R})\) of the differential of \(f\).