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On Symmetric CR Geometries of Hypersurface Type


We study non-degenerate CR geometries of hypersurface type that are symmetric in the sense that, at each point, there is a CR transformation reversing the CR distribution at that point. We show that such geometries are either flat or homogeneous. We show that non-flat non-degenerate symmetric CR geometries of hypersurface type are covered by CR geometries with a compatible pseudo-Riemannian metric preserved by all symmetries. We construct examples of simply connected flat non-degenerate symmetric CR geometries of hypersurface type that do not carry a pseudo-Riemannian metric compatible with the symmetries.

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Correspondence to Jan Gregorovič.

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First author supported by the project P29468 of the Austrian Science Fund (FWF). Second author supported by the Grant 17-01171S of the Czech Science Foundation (GAČR)

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Gregorovič, J., Zalabová, L. On Symmetric CR Geometries of Hypersurface Type. J Geom Anal 29, 3135–3159 (2019).

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  • CR geometry
  • Homogeneous manifold
  • Webster metric

Mathematics Subject Classification

  • 32V05
  • 32V30
  • 53C30