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
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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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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DOI: https://doi.org/10.1007/s10711-016-0212-y