Duke Mathematical Journal

Conformal actions of nilpotent groups on pseudo-Riemannian manifolds

Charles Frances and Karin Melnick

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Abstract

We study conformal actions of connected nilpotent Lie groups on compact pseudo-Riemannian manifolds. We prove that if a type-(p,q) compact manifold M supports a conformal action of a connected nilpotent group H, then the degree of nilpotence of H is at most 2p+1, assuming pq; further, if this maximal degree is attained, then M is conformally equivalent to the universal type-(p,q), compact, conformally flat space, up to finite or cyclic covers. The proofs make use of the canonical Cartan geometry associated to a pseudo-Riemannian conformal structure

Article information

Source
Duke Math. J., Volume 153, Number 3 (2010), 511-550.

Dates
First available in Project Euclid: 4 June 2010

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1275671396

Digital Object Identifier
doi:10.1215/00127094-2010-030

Mathematical Reviews number (MathSciNet)
MR2667424

Zentralblatt MATH identifier
1204.53056

Subjects
Primary: 53A30: Conformal differential geometry
Secondary: 53C50: Lorentz manifolds, manifolds with indefinite metrics

Citation

Frances, Charles; Melnick, Karin. Conformal actions of nilpotent groups on pseudo-Riemannian manifolds. Duke Math. J. 153 (2010), no. 3, 511--550. doi:10.1215/00127094-2010-030. https://projecteuclid.org/euclid.dmj/1275671396


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References

  • M. A. Akivis and V. V. Goldberg, Conformal Differential Geometry and Its Generalizations, Pure and Appl. Math. (New York), Wiley-Intersci., New York, 1996.
  • A. Besse, Einstein Manifolds, reprint of the 1987 ed., Classics in Math., Springer, Berlin, 2008.
  • T. Barbot, V. Charette, T. Drumm, W. M. Goldman, and K. Melnick, ``A primer on the ($2+1$) Einstein universe'' in Recent Developments in Pseudo-Riemannian Geometry, ESI Lect. Math. Phys., Eur. Math. Soc., Zürich, 2008.
  • U. Bader, C. Frances, and K. Melnick, An embedding theorem for automorphism groups of Cartan geometries, Geom. Funct. Anal. 19 (2009), 333--355.
  • U. Bader and A. Nevo, Conformal actions of simple Lie groups on compact pseudo-Riemannian manifolds, J. Differential Geom. 60 (2002), 355--387.
  • A. Borel, Linear Algebraic Groups, 2nd ed., Springer, New York, 1991.
  • J. Ferrand, The action of conformal transformations on a Riemannian manifold, Math. Ann. 304 (1996), 277--291.
  • A. Fialkow, Conformal geodesics, Trans. Amer. Math. Soc. 45 (1939), 443--473.
  • C. Frances, Une preuve du théorème de Liouville en géométrie conforme dans le cas analytique, Enseign. Math. (2) 49 (2003), 95--100.
  • —, Sur les variétés lorentziennes dont le groupe conforme est essentiel, Math. Ann. 332 (2005), 103--119.
  • —, Causal conformal vector fields, and singularities of twistor spinors. Ann. Global Anal. Geom. 32 (2007), 277--295.
  • —, Sur le groupe d'automorphismes des géométries paraboliques de rang $1$, Ann. Sci. École Norm. Sup. (4) 40 (2007), 741--764.
  • —, ``Géométrie et dynamique lorentziennes conformes'', Ph.D. dissertation, École Normale Supérieure de Lyon, Lyon, France, 2002.
  • —, Rigidity at the boundary for conformal structures and other Cartan geometries, preprint.
  • C. Frances and A. Zeghib, Some remarks on conformal pseudo-Riemannian actions of simple Lie groups, Math. Res. Lett. 12 (2005), 49--56.
  • H. Friedrich, Conformal geodesics on vacuum space-times, Comm. Math. Phys. 235 (2003), 513--543.
  • H. Friedrich and B. G. Schmidt, Conformal geodesics in general relativity, Proc. Roy. Soc. London Ser. A 414 (1987), 171--195.
  • W. M. Goldman, ``Geometric structures on manifolds and varieties of representations'' in Geometry of Group Representations (Boulder, Colo., 1987), Contemp. Math. 74, Amer. Math. Soc., Providence, 1988, 169--198.
  • S. Kobayashi, Transformation Groups in Differential Geometry, reprint of the 1972 ed., Classics in Math., Springer, Berlin, 1995.
  • D. Witte Morris, Ratner's Theorems on Unipotent Flows, Chicago Lectures in Math., Univ. Chicago Press, Chicago, 2005.
  • —, Introduction to arithmetic groups, preprint.
  • M. Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geom. 6 (1971/72), 247--258.
  • R. W. Sharpe, Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Grad. Texts in Math. 166, Springer, New York, 1997.
  • W. P Thurston, Three-Dimensional Geometry and Topology, Vol. 1, ed. Silvio Levy, Princeton Math. Ser. 35, Princeton Univ. Press, Princeton, 1997.
  • R. J. Zimmer, On the automorphism group of a compact Lorentz manifold and other geometric manifolds, Invent. Math. 83 (1986), 411--424.
  • —, Split rank and semisimple automorphism groups of $G$-structures, J. Differential Geom. 26 1987, 169--173.