A sharp characterization of the smooth 4-sphere in curvature terms
The weak pinching of a Riemannian manifold is given by the norm of its tracefree curvature operator, appropriately normalized by the scalar curvature to obtain a scale invariant. Using the Ricci flow on the space of metrics, the author gives a simplified version of the original proof outlined in [\textit{C. Margerin}, C.R. Acad.\ Sci., Paris, Sér.~I 303, 877-880 (1986; Zbl 0607.53027)] of the theorem: Any compact 4-manifold admitting a metric of positive scalar curvature which is 1/6-weakly pinched is diffeomorphic either to the 4-sphere \(S^4\), the real projective 4-space, the complex projective plane or the product \(S^1\times S^3\). If it is strictly 1/6-weakly pinched, then we have diffeomorphism to \(S^4\) or the the real projective 4-space.
scientific article; zbMATH DE number 1187449