Conformal Symmetries of the Yamabe and Paneitz Operators

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.