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Essential conformal actions of \({{\mathrm{PSL}}}(2,\mathbf {R})\) on real-analytic compact Lorentz manifolds

Abstract

The main result of this paper is the conformal flatness of real-analytic compact Lorentz manifolds of dimension at least three admitting a conformal essential action of a Lie group locally isomorphic to \({{\mathrm{PSL}}}(2,\mathbf {R})\). It is established by using a general result on local isometries of real-analytic rigid geometric structures. As corollary, we deduce the same conclusion for conformal essential actions of connected semi-simple Lie groups on real-analytic compact Lorentz manifolds. This work is a contribution to the understanding of the Lorentzian version of a question asked by A. Lichnerowicz.

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Notes

  1. 1.

    The Jordan decomposition is valid in algebraic groups. Here, \(P \subset {{\mathrm{PO}}}(2,n)\) is not algebraic. Nevertheless, it is the quotient of an algebraic subgroup of O(2, n) by \(\{\pm {{\mathrm{id}}}\}\). So, when we speak of algebraic properties of elements or subgroups of P, we deal with the lifts of these elements or subgroups to O(2, n).

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Acknowledgements

This work has been done during my PhD and I would like to deeply thank my advisor, Charles Frances, for his constant support.

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Correspondence to Vincent Pecastaing.

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This work has been finalized while the author was supported by a DAAD grant.

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Pecastaing, V. Essential conformal actions of \({{\mathrm{PSL}}}(2,\mathbf {R})\) on real-analytic compact Lorentz manifolds. Geom Dedicata 188, 171–194 (2017). https://doi.org/10.1007/s10711-016-0212-y

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Keywords

  • Conformal geometry
  • Lorentzian geometry
  • Rigid geometric structures
  • Actions of Lie groups

Mathematics Subject Classification (2010)

  • 22F50
  • 53A30
  • 53B30