The Jordan algebras of Riemann, Weyl and curvature compatible tensors
Volume 167 / 2022
Colloquium Mathematicum 167 (2022), 63-72
MSC: Primary 53B20; Secondary 17C90.
DOI: 10.4064/cm8067-10-2020
Published online: 12 April 2021
Abstract
Given the Riemann, or the Weyl, or a generalized curvature tensor $K$, a symmetric tensor $b_{ij}$ is called \emph {compatible} with the curvature tensor if $b_i{}^m K_{jklm} + b_j{}^m K_{kilm} + b_k{}^m K_{ijlm}=0$. In addition to establishing some known and some new properties of such tensors, we prove that they form a special Jordan algebra, i.e. the symmetrized product of $K$-compatible tensors is $K$-compatible.
