Abstract

We study closed four-dimensional manifolds. In particular, we show that under various pinching curvature conditions (for example, the sectional curvature is no more than $5∕6$ of the smallest Ricci eigenvalue), the manifold is definite. If restricting to a metric with harmonic Weyl tensor, then it must be self-dual or anti-self-dual under the same conditions. Similarly, if restricting to an Einstein metric, then it must be either the complex projective space with its Fubini–Study metric, the round sphere, or the quotient of one of these. Furthermore, we also classify Einstein manifolds with positive intersection form and an upper bound on the sectional curvature.

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