The classification of simple Jacobi--Ricci commuting algebraic curvature tensors

Abstract

We classify algebraic curvature tensors such that the Ricci operator is simple (i.e. the Ricci operator is complex diagonalizable and either the complex spectrum consists of a single real eigenvalue or the complex spectrum consists of a pair of eigenvalues which are complex conjugates of each other) and which are Jacobi--Ricci commuting (i.e. the Ricci operator commutes with the Jacobi operator of any vector).