Four-manifolds with positive curvature operator. (English) Zbl 0628.53042
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.
Reviewer: W.Ballmann
MSC:
53C20 | Global Riemannian geometry, including pinching |
58J60 | Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) |
53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
57R99 | Differential topology |
58J35 | Heat and other parabolic equation methods for PDEs on manifolds |