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Differential operators canonically associated to a conformal structure. (English) Zbl 0596.53009

In this article, the author describes differential operators acting on tensor bundles over a manifold which are naturally associated to a pseudo-Riemannian metric, and change covariantly when the metric is modified conformally. Various non-linear generalizations (analogous to the Yamabe operator for the Laplacian acting on functions) are also contemplated.

As a result, the author is able to construct representations of the group of conformal transformations of the base Lorentz manifold. In the Riemannian setting, numerical invariants attached to conformal classes are also exhibited. The methods are mainly tensorial, and the proofs obtained by inspection.

Reviewer: J.P.Bourguignon

MSC:
53A30Conformal differential geometry
58J60Relations of PDE with special manifold structures
53B30Lorentz metrics, indefinite metrics