Abstract
A curvature-type tensor invariant called para contact (pc) conformal curvature is defined on a paracontact manifold. It is shown that a paracontact manifold is locally paracontact conformal to the hyperbolic Heisenberg group or to a hyperquadric of neutral signature iff the pc conformal curvature vanishes. In the three dimensional case the corresponding result is achieved through employing a certain symmetric (0,2) tensor. The well known result of Cartan–Chern–Moser giving necessary and sufficient condition a CR-structure to be CR equivalent to a hyperquadric in \({\mathbb{C}^{n+1}}\) is presented in-line with the paracontact case. An explicit formula for the regular part of a solution to the sub-ultrahyperbolic Yamabe equation on the hyperbolic Heisenberg group is shown.
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This project has been funded in part by the National Academy of Sciences under the [Collaboration in Basic Science and Engineering Program 1 Twinning Program] supported by Contract No. INT-0002341 from the National Science Foundation. The contents of this publication do not necessarily reflect the views or policies of the National Academy of Sciences or the National Science Foundation, nor does mention of trade names, commercial products or organizations imply endorsement by the National Academy of Sciences or the National Science Foundation.
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Ivanov, S., Vassilev, D. & Zamkovoy, S. Conformal paracontact curvature and the local flatness theorem. Geom Dedicata 144, 79–100 (2010). https://doi.org/10.1007/s10711-009-9388-8
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DOI: https://doi.org/10.1007/s10711-009-9388-8
Keywords
- Paracontact
- CR structures
- Pseudo conformal flat
- Paracontact conformally invariant curvature
- Cartan–Chern–Moser theorem