Abstract
It is shown that if ann dimensional Riemannian or pseudo-Riemannian manifold admits a proper conformal scalar, every (local) conformal group is conformally isometric, and that if it admits a proper conformal gradient every (local) conformal group is conformally homothetic. In the Riemannian case there is always a conformal scalar unless the metric is conformally Euclidean. In the case of a Lorentzian 4-manifold it is proved that the only metrics with no conformal scalars (and hence the only ones admitting a (local) conformal group not conformally isometric) are either conformal to the plane wave metric with parallel rays or conformally Minkowskian.
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Communicated by J. Ehlers
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Defrise-Carter, L. Conformal groups and conformally equivalent isometry groups. Commun.Math. Phys. 40, 273–282 (1975). https://doi.org/10.1007/BF01610003
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DOI: https://doi.org/10.1007/BF01610003