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


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
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