Conformal Symmetries of the Yamabe and Paneitz Operators

TL;DR

This paper investigates the conformal symmetries of the Yamabe and Paneitz operators, revealing how conformal Killing vectors relate to symmetries and correcting previous formulas in related literature.

Contribution

It establishes the relationship between conformal Killing vectors and symmetries of the Yamabe and Paneitz operators on Einstein spaces and general manifolds.

Findings

First order symmetries induce second order symmetries for Yamabe operator.

Conformal Killing vectors induce symmetries of the Paneitz operator on Einstein spaces.

Corrections are provided for earlier formulas in the literature.

Abstract

We study first and second order conformal symmetries of the Yamabe Laplacian on a general pseudo-Riemannian manifold and of the Paneitz operator on Einstein spaces. We show that first order conformal symmetries of the Yamabe operator induce second order conformal symmetries. We show that on an Einstein space every conformal Killing vector field induces a conformal symmetry of the Paneitz operator and, conversely, every first order conformal symmetry of the Paneitz operator arises from a conformal Killing vector field. Possible extensions of our main theorems are illustrated by examples. We correct formulas in a paper of B. Carter and a paper of Kamran and McLenaghan.