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 $d \geq 3$. For, then, the symmetric linear transformation field $Q$ defined by the Ricci tensor satisfies Codazzi’s equation $({\nabla _X}Q)Y = ({\nabla _Y}Q)X$. This observation leads to a pinching theorem on the length of the Ricci tensor.