The Jordan algebras of Riemann, Weyl and curvature compatible tensors
Carlo Alberto Mantica, Luca Guido Molinari
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.
Topics & Concepts
MathematicsRiemann curvature tensorWeyl tensorRicci decompositionPure mathematicsCurvatureTensor (intrinsic definition)Tensor productAlgebra over a fieldMathematical physicsGeometryAdvanced Differential Geometry ResearchAdvanced Neuroimaging Techniques and ApplicationsGeometric Analysis and Curvature Flows