Manifolds with $$4\frac{1}{2}$$-Positive Curvature Operator of the Second Kind

Abstract

We show that a closed four-manifold with \(4\frac{1}{2}\)-positive curvature operator of the second kind is diffeomorphic to a spherical space form. The curvature assumption is sharp as both \({\mathbb{CP}\mathbb{}}^2\) and \({\mathbb {S}}^3 \times {\mathbb {S}}^1\) have \(4\frac{1}{2}\)-nonnegative curvature operator of the second kind. In higher dimensions \(n\ge 5\), we show that closed Riemannian manifolds with \(4\frac{1}{2}\)-positive curvature operator of the second kind are homeomorphic to spherical space forms. These results are proved by showing that \(4\frac{1}{2}\)-positive curvature operator of the second kind implies both positive isotropic curvature and positive Ricci curvature. Rigidity results for \(4\frac{1}{2}\)-nonnegative curvature operator of the second kind are also obtained.