Invariant prolongation of the Killing tensor equation

TL;DR

This paper develops a projectively invariant prolongation of the Killing tensor equation on manifolds with affine connections, using tractor calculus to reveal new invariant structures and establish a correspondence between solutions and parallel sections.

Contribution

It introduces a novel prolongation method for the Killing tensor equation that is projectively invariant and utilizes tractor calculus to uncover additional invariant structures.

Findings

Prolongation to a linear connection with parallel sections corresponding to solutions.

The prolongation is projectively invariant.

Reveals new invariant structures related to the Killing tensor equation.

Abstract

The Killing tensor equation is a first order differential equation on symmetric covariant tensors that generalises to higher rank the usual Killing vector equation on Riemannian manifolds. We view this more generally as an equation on any manifold equipped with an affine connection, and in this setting derive its prolongation to a linear connection. This connection has the property that parallel sections are in 1-1 correspondence with solutions of the Killing equation. Moreover this connection is projectively invariant and is derived entirely using the projectively invariant tractor calculus which reveals also further invariant structures linked to the prolongation.