Conformal Killing tensors and their Killing scales
A. Rod Gover, Jonathan Kress, Thomas Leistner

TL;DR
This paper develops a conformally invariant framework using tractor calculus to characterize when conformal Killing tensors are derived from Einstein metrics, providing algebraic and differential criteria for Einstein Killing scales.
Contribution
It introduces a new invariant characterization of Einstein Killing scales via tractor calculus and partial prolongation, extending the understanding of conformal Killing tensors in conformal geometry.
Findings
Invariant characterization of Einstein Killing scales
Full prolongation of conformal Killing equation on conformally flat manifolds
Algebraic description of compatible conformal Killing tensors
Abstract
We address the problem of how to characterise when a rank-two conformal Killing tensor is the trace-free part of a Killing tensor for a metric in the conformal class. We call such a metric a Killing scale. Our approach is via differential prolongation using conformally invariant tractor calculus. First, we show that there is a useful partial prolongation of the conformal Killing equation to a simplified equation for sections of some tractor bundle. We then use this partial prolongation to provide such an invariant characterisation in terms of the scale tractor and this partial prolongation. This captures invariantly the relevant Bertrand--Darboux equation. We show that Einstein Killing scales have a special place in the theory. On conformally flat manifolds, we give the full prolongation of the conformally Killing equation to a conformally invariant connection on a tractor bundle. Using…
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Taxonomy
TopicsBlack Holes and Theoretical Physics · Quantum Mechanics and Non-Hermitian Physics · Spectral Theory in Mathematical Physics
