Conformally Flat Semi-Riemannian Manifolds with Commuting Curvature and Ricci Operators

TLDR: In this article, conformally flat, semi-Riemannian manifolds with curvature tensors and Ricci operators were classified. But the Ricci operator has pure imaginary eigenvalues.

Abstract: We classify the conformally flat, semi-Riemannian manifolds satisfying $R(X,Y) \cdot Q = 0$, where $R$ and $Q$ are the curvature tensor and the Ricci operator, respectively. As the cases which do not occur in the Riemannian manifolds, the Ricci operator $Q$ has pure imaginary eigenvalues or it satisfies $Q^2 = 0$.