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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 PoincarĂ©’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

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