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L’équation de la chaleur associée à la courbure de Ricci (d’après R. S. Hamilton). (The heat equation associated with Ricci curvature (according to R. S. Hamilton)). (French) Zbl 0613.53018
Sémin. Bourbaki, 38ème année, Vol. 1985/86, Exp. No. 653, Astérisque 145/146, 45-61 (1987).
[For the entire collection see Zbl 0601.00002.]
Quite recently, R. S. Hamilton [”Four-manifolds with positive curvature operator”, Preprint, U. C. San Diego (1985), an extension of J. Differ. Geom. 17, 255-306 (1982; Zbl 0504.53034)] has proved the following important result: Any compact three-dimensional manifold with positive or zero Ricci curvature is diffeomorphic to a quotient of \(S^ 3\), of \(S^ 2\times {\mathbb{R}}\) or of \({\mathbb{R}}^ 3\) with respect to a group of isometries of their standard metric without fixed point.
The paper gives a clear description of the method of proving this theorem, whose basic idea is to integrate the Ricci curvature conceived as vector field on the space of Riemannian metrics of the manifold. It contains a unified and comprehensive treatment of the problems, among them some of historical interest, connected with this result and references concerning the latest works in this area. Other geometrical applications of the method are also pointed out.
Reviewer: M.Craioveanu
MSC:
53C20 Global Riemannian geometry, including pinching
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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