Three-manifolds with positive Ricci curvature.

*(English)*Zbl 0504.53034In 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.

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 |

##### Keywords:

Ricci-curvature; equation of evolution; Nash-Moser inverse function theorem; method of a priori estimation; constant positive curvature##### References:

[1] | J. P. Bourguignon, Ricci curvature and Einstein metrics, Lecture Notes in Math., Vol. 838, Springer, Berlin, p. 298. · Zbl 0437.53029 · doi:10.1007/BFb0088841 |

[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 · doi:10.2307/2373037 · links.jstor.org |

[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 · doi:10.1090/S0273-0979-1982-15004-2 |

[6] | J. A. Wolf, Spaces of constant curvature, McGraw-Hill, New York, 1967, p. 408. · Zbl 0162.53304 |

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