Four-manifolds with positive curvature operator
The main result of this paper states that a compact Riemannian \(M^ 4\) with nonnegative curvature operator is diffeomorphic to a quotient of \(S^ 4\) or \(\mathbb CP^ 2\) or \(S^ 3\times \mathbb R^ 1\) or \(S^ 2\times \mathbb R^ 2\) or \(\mathbb R^ 4\) by a group of isometries (in the standard metric). A corresponding result in dimension 3, with curvature operator replaced by Ricci curvature, is also obtained. In the proof of these results, the parabolic equation \(\partial g/\partial t=2/nrg-2 Ric\) is considered, where \(Ric\) is the Ricci-tensor and \(r\) the mean of the Ricci curvature. Under the assumption of the theorems, it is shown that the solution either converges, as \(t\to \infty\), to a metric of constant sectional curvature or the geometry can be already identified.
scientific article; zbMATH DE number 4022085