Prolongation of symmetric Killing tensors and commuting symmetries of the Laplace operator

TL;DR

This paper characterizes the algebra of symmetries commuting with the Laplace operator on constant curvature manifolds using tractor calculus, linking it to projective geometry.

Contribution

It introduces a prolongation method for symmetric Killing tensors and explores their algebraic structure and geometric relations.

Findings

Determined the space of commuting symmetries of the Laplace operator.

Constructed a prolongation of the differential system for symmetric Killing tensors.

Connected the symmetry algebra to projective differential geometry.

Abstract

We determine the space of commuting symmetries of the Laplace operator on pseudo-Riemannian manifolds of constant curvature, and derive its algebra structure. Our construction is based on the Riemannian tractor calculus, allowing to construct a prolongation of the differential system for symmetric Killing tensors. We also discuss some aspects of its relation to projective differential geometry.