The Jordan algebras of Riemann, Weyl and curvature compatible tensors

TL;DR

This paper investigates the algebraic structure of tensors compatible with Riemann, Weyl, and generalized curvature tensors, revealing they form a special Jordan algebra with closure under symmetrized products.

Contribution

It demonstrates that K-compatible tensors form a Jordan algebra, extending known properties and establishing new algebraic relations among these tensors.

Findings

K-compatible tensors form a Jordan algebra

Symmetrized product of K-compatible tensors remains K-compatible

New properties of curvature-compatible tensors are established

Abstract

Given the Riemann, or the Weyl, or a generalized curvature tensor K, a symmetric tensor $b_{ij}$ is named `compatible' with the curvature tensor if $b_i{}^m K_{jklm} + b_j{}^m K_{kilm} + b_k{}^m K_{ijlm} = 0$. Amongst showing known and new properties, we prove that they form a special Jordan algebra, i.e. the symmetrized product of K-compatible tensors is K-compatible.