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Three-manifolds with positive Ricci curvature. (English) Zbl 0504.53034

In this paper the following important theorem is proved: Let $X$ be a compact 3-manifold which admits a Riemannian metric with strictly positive Ricci curvature. Then $X$ also admits a metric of constant positive curvature.

As all manifolds of constant curvature have been completely classified by J. A. Wolf in his book [Spaces of constant curvature. New York etc.: McGraw-Hill (1967; Zbl 0162.53304)], by the preceding theorem these are the only compact three-manifolds, which can carry metrics of strictly positive Ricci curvature. Accordingly, a conjecture of J.-P. Bourguignon [Global differential geometry and global analysis, Proc. Colloq., Berlin 1979, Lect. Notes Math. 838, 249–250 (1981; Zbl 0437.53011)] is answered affirmatively. As pointed out by the author, this theorem is closely related to the famous Poincaré’s conjecture on the compact, simply-connected 3-manifolds and the Smith’s conjecture related to the group of covering transformations. If both conjectures were true, the main theorem mentioned above would follow as a corollary.

In the proof of this theorem the author studied in advance the equation of evolution $\frac{\partial {g}_{ij}}{\partial t}=\frac{2}{n}r{g}_{ij}-2{R}_{ij}$, where $r$ is the average of the scalar curvature $R$, namely $r=\int R\phantom{\rule{0.166667em}{0ex}}d\mu /\int \phantom{\rule{0.166667em}{0ex}}d\mu$. He proved that if for a compact 3-manifold the initial metric has strictly positive Ricci curvature, then it continues like that for ever, and converges to a metric of constant positive curvature, as $t$ tends to $\infty$. Then he made use of the fact peculiar to three dimensions, that the full Riemannian curvature tensor ${R}_{hijk}$ can be calculated from the Ricci tensor ${R}_{ij}$ (Weyl’s conformal curvature tensor is vanishing). In view of this crucial step, the method used here cannot be generalized to the case $dim\left(M\right)\ge 3$, unless it is extensively modified.

Reviewer: C.-c. Hwang

##### MSC:
 53C21 Methods of Riemannian geometry, including PDE methods; curvature restrictions (global) 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 35K55 Nonlinear parabolic equations