In this paper the following important theorem is proved: Let be a compact 3-manifold which admits a Riemannian metric with strictly positive Ricci curvature. Then 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 , where is the average of the scalar curvature , namely . 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 tends to . Then he made use of the fact peculiar to three dimensions, that the full Riemannian curvature tensor can be calculated from the Ricci tensor (Weyl’s conformal curvature tensor is vanishing). In view of this crucial step, the method used here cannot be generalized to the case , unless it is extensively modified.