## 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 $$\frac{\partial g_{ij}}{\partial t}=\frac 2n rg_{ij} - 2R_{ij}$$, where $$r$$ is the average of the scalar curvature $$R$$, namely $$r=\int R\,d\mu/\int\,d\mu$$. 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 $$\dim(M)\geq 3$$, unless it is extensively modified.
Reviewer: C.-c. Hwang
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Usefulness of Nash embedding theorem

### MSC:

 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) 35K55 Nonlinear parabolic equations

### Citations:

Zbl 0162.53304; Zbl 0437.53011
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### References:

  J. P. Bourguignon, Ricci curvature and Einstein metrics, Lecture Notes in Math., Vol. 838, Springer, Berlin, p. 298. · Zbl 0437.53029  J. Cheeger and D. Ebin, Comparison theorems in Riemannian geometry, North-Holland, Amsterdam, 1975, p. 174. · Zbl 0309.53035  J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964) 109-160. JSTOR: · Zbl 0122.40102  R. S. Hamilton, Harmonic maps of manifolds with boundary, Lecture Notes in Math., Vol. 471, Springer, Berlin, 1975, p. 168. · Zbl 0308.35003  R. S. Hamilton, The inverse function theorem of Nash and Moser (new version), preprint, Cornell University, p. 294. · Zbl 0499.58003  J. A. Wolf, Spaces of constant curvature, McGraw-Hill, New York, 1967, p. 408. · Zbl 0162.53304
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