Local degeneracy of pseudo-Riemannian conformal transformations

Charles Frances[1]

  • [1] Université Paris-Sud Laboratoire de Mathématiques, 91405 ORSAY Cedex.

Annales de l’institut Fourier (2012)

  • Volume: 62, Issue: 5, page 1627-1669
  • ISSN: 0373-0956

Abstract

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We study the set Conf ( M , N ) of conformal immersions between two pseudo-Riemannian manifolds ( M , g ) and ( N , h ) . We characterize the closure of Conf ( M , N ) in the space of continuous maps from M to N , and we investigate the geometric properties of ( M , g ) whenever this closure is nontrivial.

How to cite

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Frances, Charles. "Dégénerescence locale des transformations conformes pseudo-riemanniennes." Annales de l’institut Fourier 62.5 (2012): 1627-1669. <http://eudml.org/doc/251095>.

@article{Frances2012,
abstract = {Nous étudions l’ensemble Conf$(M,N)$ des immersions conformes entre deux variétés pseudo-riemanniennes $(M,g)$ et $(N,h)$. Nous caractérisons notamment l’adhérence de Conf$(M,N)$ dans l’espace des applications continues $\{\mathcal\{C\}\}^\{0\}(M,N)$, et décrivons quelques propriétés géométriques de $(M,g)$ lorsque cette adhérence est non triviale.},
affiliation = {Université Paris-Sud Laboratoire de Mathématiques, 91405 ORSAY Cedex.},
author = {Frances, Charles},
journal = {Annales de l’institut Fourier},
keywords = {conformal maps; pseudo-Riemannian structures},
language = {fre},
number = {5},
pages = {1627-1669},
publisher = {Association des Annales de l’institut Fourier},
title = {Dégénerescence locale des transformations conformes pseudo-riemanniennes},
url = {http://eudml.org/doc/251095},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Frances, Charles
TI - Dégénerescence locale des transformations conformes pseudo-riemanniennes
JO - Annales de l’institut Fourier
PY - 2012
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 5
SP - 1627
EP - 1669
AB - Nous étudions l’ensemble Conf$(M,N)$ des immersions conformes entre deux variétés pseudo-riemanniennes $(M,g)$ et $(N,h)$. Nous caractérisons notamment l’adhérence de Conf$(M,N)$ dans l’espace des applications continues ${\mathcal{C}}^{0}(M,N)$, et décrivons quelques propriétés géométriques de $(M,g)$ lorsque cette adhérence est non triviale.
LA - fre
KW - conformal maps; pseudo-Riemannian structures
UR - http://eudml.org/doc/251095
ER -

References

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  1. D. Alekseevski, Self-similar Lorentzian manifolds, Ann. Global Anal. Geom. 3 (1985), 59-84 Zbl0538.53060MR812313
  2. T. Barbot, V. Charette, T. Drumm, W.M. Goldman, K. Melnick, A primer on the (2+1) Einstein universe., Recent Developments in Pseudo-Riemannian Geometry (2008), ESI Lectures in Mathematics and Physics Zbl1154.53047MR2436232
  3. A Besse, Einstein manifolds, 10 (1987), Springer-Verlag, Berlin Zbl1147.53001
  4. A. Čap, J. Slovák, V. Žádník, On distinguished curves in parabolic geometries, Transform. Groups 9 (2004), 143-166 Zbl1070.53021MR2056534
  5. J. Ferrand, Les géodésiques des structures conformes, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), 629-632 Zbl0487.53011MR664297
  6. J. Ferrand, Convergence and degeneracy of quasiconformal maps of Riemannian manifolds, J. Anal. Math. 69 (1996), 1-24 Zbl0871.53031MR1428092
  7. J. Ferrand, The action of conformal transformations on a Riemannian manifold, Math. Ann. 304 (1996), 277-291 Zbl0866.53027MR1371767
  8. C. Frances, Géométrie et dynamique lorentziennes conformes  
  9. C. Frances, Sur le groupe d’automorphismes des géométries paraboliques de rang 1, Ann. Sci. École Norm. Sup. (4) 40 (2007), 741-764 Zbl1135.53016MR2382860
  10. F. W. Gehring, The Carathéodory convergence theorem for quasiconformal mappings in space, Ann. Acad. Sci. Fenn. Ser. A I No. 336/11 (1963) Zbl0136.38102MR160898
  11. S. Kobayashi, Transformation groups in differential geometry. Reprint of the 1972 edition., (1995), Springer-Verlag, Berlin Zbl0829.53023MR1336823
  12. W. Kuehnel, H.B. Rademacher, Liouville’s theorem in conformal geometry, (9) (to appear) Zbl1127.53014
  13. M. Obata, The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geometry 6 (1971/72), 247-258 Zbl0236.53042MR303464
  14. R. Schoen, On the conformal and CR automorphism groups, Geom. Funct. Anal. 5 (1995), 464-481 Zbl0835.53015MR1334876
  15. M. Schottenloher, A Mathematical Introduction to Conformal Field Theory, (1997), Springer-Verlag, Berlin Zbl0874.17031MR1473464
  16. R.W. Sharpe, Differential Geometry : Cartan’s generalization of Klein’s Erlangen Program, (1997), Springer, New York Zbl0876.53001MR1453120
  17. J. Vässälä, Lectures on n -dimensional quasiconformal mappings, 229 (1971), Springer-Verlag, Berlin-New York MR454009
  18. A. Zeghib, Sur les actions affines des groupes discrets, Ann. Inst. Fourier 47 (1997), 641-685 Zbl0865.57038MR1450429
  19. A. Zeghib, Isometry groups and geodesic foliations of Lorentz manifolds. I. Foundations of Lorentz dynamics, Geom. Funct. Anal. 9 (1999), 775-822 Zbl0946.53035MR1719606
  20. A. Zeghib, Isometry groups and geodesic foliationsof Lorentz manifolds. II. Geometry of analytic Lorentz manifolds with large isometry groups, Geom. Funct. Anal. 9 (1999), 823-854 Zbl0946.53036MR1719610

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