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Second Order Symmetries of the Conformal Laplacian

Jean-Philippe Michel^a, Fabian Radoux^a, Josef Šilhan<sup>b

a</sup> Department of Mathematics of the University of Liège, Grande Traverse 12, 4000 Liège, Belgium
^b Department of Algebra and Geometry of the Masaryk University in Brno, Janàčkovo nàm. 2a, 662 95 Brno, Czech Republic

Abstract: Let $(M,{\rm g})$ be an arbitrary pseudo-Riemannian manifold of dimension at least $3$. We determine the form of all the conformal symmetries of the conformal (or Yamabe) Laplacian on $(M,{\rm g})$, which are given by differential operators of second order. They are constructed from conformal Killing $2$-tensors satisfying a natural and conformally invariant condition. As a consequence, we get also the classification of the second order symmetries of the conformal Laplacian. Our results generalize the ones of Eastwood and Carter, which hold on conformally flat and Einstein manifolds respectively. We illustrate our results on two families of examples in dimension three.

Keywords: Laplacian; quantization; conformal geometry; separation of variables.

MSC: 58J10; 53A30; 70S10; 53D20; 53D55

Received: October 25, 2013; in final form February 5, 2014; Published online February 14, 2014

Language: English

DOI: 10.3842/SIGMA.2014.016

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ArXiv: 1308.1046