Weyl collineations that are not curvature collineations

TL;DR

This paper investigates the relationship between Weyl symmetries and curvature symmetries, finding that Weyl symmetries can be strictly larger, but not smaller, than curvature symmetries, suggesting a fundamental asymmetry.

Contribution

It demonstrates that Weyl collineations can form a larger symmetry algebra than curvature collineations, revealing an asymmetry in their Lie algebra structures.

Findings

Weyl symmetries can properly contain curvature symmetries

No example found where curvature symmetries contain Weyl symmetries

Speculation on a fundamental reason for the asymmetry

Abstract

Though the Weyl tensor is a linear combination of the curvature tensor, Ricci tensor and Ricci scalar, it does not have all and only the Lie symmetries of these tensors since it is possible, in principle, that "asymmetries cancel". Here we investigate if, when and how the symmetries can be different. It is found that we can obtain a metric with a finite dimensional Lie algebra of Weyl symmetries that properly contains the Lie algebra of curvature symmetries. There is no example found for the converse requirement. It is speculated that there may be a fundamental reason for this lack of "duality".