Conformally flat 3-manifolds with constant scalar curvature. (English) Zbl 0949.53023

Cheng, Qing-Ming; Ishikawa, Susumu; Shiohama, Katsuhiro

J. Math. Soc. Japan 51, No. 1, 209-226 (1999).

The authors consider the problem of the classification of complete, conformally flat manifolds with constant scalar curvature, \(r\). They study it in the 3-dimensional case under the additional condition of the constancy of the squared length of the Ricci curvature tensor, \(S\). Obvious members of this family are 3-dimensional space forms and products of 2-dimensional space forms with either \(S^1\) or the real line. They show that no more examples exist when \(r\) is a non-negative constant. They conjecture that the same happens when \(r\) is a negative constant and prove it provided \(S\) does not lie in the interval (\(r^2/3,r^2/2\)]. In this direction, they also obtain that there are no 3-dimensional compact conformally flat manifolds, with constant \(S\), constant negative \(r\) and with the eigenvalues of the Ricci curvature tensor being different everywhere.

Reviewer: O.J.Garay (Bilbao)

MSC:

53C20

Global Riemannian geometry, including pinching

Keywords:

conformally flat; scalar curvature; Ricci tensor