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

TL;DR

This paper classifies algebraic curvature tensors with simple Ricci operators that commute with their Jacobi operators, providing a detailed understanding of their spectral properties and algebraic structure.

Contribution

It offers a complete classification of algebraic curvature tensors with simple Ricci operators that are Jacobi--Ricci commuting, focusing on their spectral and algebraic characteristics.

Findings

Classification of algebraic curvature tensors with simple Ricci operators

Characterization of Jacobi--Ricci commuting tensors

Spectral properties of the Ricci and Jacobi operators

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).