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
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.
Keywords
rigidity, Hopf conjecture, definite, Bochner technique,
harmonic Weyl, Einstein
