Conformally flat manifolds and a pinching problem on the Ricci tensor

Abstract

There is a formal similarity between the theory of hypersurfaces and conformally flat d-dimensional spaces of constant scalar curvature provided d > 3. For, then, the symmetric linear transformation field Q defined by the Ricci tensor satisfies Codazzi's equation (Vx Q)Y = (Vy Q)X. This observation leads to a pinching theorem on the length of the Ricci tensor. 1. Statement of results. Recently, one of the authors [1] obtained THEOREM G. Let M be a d-dimensional compact conformallyflat manifold with definite Ricci curvature. If the scalar curvature r is constant and if the square of the length of the Ricci tensor is not greater than r2/(d 1), d > 3, then M is a space of constant curvature. Note that the square length of the Ricci tensor is greater than or equal to r2ld, so the Ricci tensor has been pinched. In the present paper the following two theorems are proved, the first of which generalizes Theorem G. THEOREM 1. Let M be a d-dimensional compact conformally flat manifold with constant scalar curvature r. If the length of the Ricci tensor is less than r/ d -1, d > 3, then M is a space of constant curvature. THEOREM 2. In a d-dimensional compact conformally flat manifold M, if the length of the Ricci tensor is constant and less than r/ d -1, then M is a space of constant curvature. 2. Conformally flat manifolds. Let M be a Riemannian manifold of dimension d > 3. We cover M by a system of local coordinate neighborhoods (U, xh), and denote by gji, V>, Rk_h, and R1i the Riemannian metric, the operator of covariant differentiation in terms of the Riemannian connection, the curvature tensor and the Ricci tensor, respectively. We say that M is conformally flat if its Riemannian metric is conformally related to a locally Euclidean metric. In a conformally flat manifold, Received by the editors June 9, 1975 and, in revised form, October 15, 1975. AMS (MOS) subject classifications (1970). Primary 53A30, 53B20, 53C20.