Dégénerescence locale des transformations conformes pseudo-riemanniennes
[Local degeneracy of pseudo-Riemannian conformal transformations]
Annales de l'Institut Fourier, Volume 62 (2012) no. 5, pp. 1627-1669.

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.

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 𝒞 0 (M,N), et décrivons quelques propriétés géométriques de (M,g) lorsque cette adhérence est non triviale.

DOI: 10.5802/aif.2732
Classification: 53A30, 53C50
Mot clés : transformations conformes, structures pseudo-riemanniennes.
Keywords: conformal maps, pseudo-Riemannian structures.
Frances, Charles 1

1 Université Paris-Sud Laboratoire de Mathématiques, 91405 ORSAY Cedex.
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 Frances, Charles. Dégénerescence locale des transformations conformes pseudo-riemanniennes. Annales de l'Institut Fourier, Volume 62 (2012) no. 5, pp. 1627-1669. doi : 10.5802/aif.2732. http://www.numdam.org/articles/10.5802/aif.2732/

[1] Alekseevski, D. Self-similar Lorentzian manifolds, Ann. Global Anal. Geom., Volume 3 (1985) no. 1, pp. 59-84 | DOI | MR | Zbl

[2] Barbot, T.; Charette, V.; Drumm, T.; Goldman, W.M.; Melnick, K. A primer on the (2+1) Einstein universe., Recent Developments in Pseudo-Riemannian Geometry, ESI Lectures in Mathematics and Physics, 2008 | MR | Zbl

[3] Besse, A Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 10, Springer-Verlag, Berlin, 1987 | Zbl

[4] Čap, A.; Slovák, J.; Žádník, V. On distinguished curves in parabolic geometries, Transform. Groups, Volume 9 (2004) no. 2, pp. 143-166 | DOI | MR | Zbl

[5] Ferrand, J. Les géodésiques des structures conformes, C. R. Acad. Sci. Paris Sér. I Math., Volume 294 (1982) no. 18, pp. 629-632 | MR | Zbl

[6] Ferrand, J. Convergence and degeneracy of quasiconformal maps of Riemannian manifolds, J. Anal. Math., Volume 69 (1996), pp. 1-24 | DOI | MR | Zbl

[7] Ferrand, J. The action of conformal transformations on a Riemannian manifold, Math. Ann., Volume 304 (1996) no. 2, pp. 277-291 | DOI | MR | Zbl

[8] Frances, C. Géométrie et dynamique lorentziennes conformes (Thèse, ENS Lyon, 2002, available at http ://mahery.math.u-psud.fr/frances/)

[9] Frances, C. Sur le groupe d’automorphismes des géométries paraboliques de rang 1, Ann. Sci. École Norm. Sup. (4), Volume 40 (2007) no. 5, pp. 741-764 (English version : eprint arXiv :math/0608.537v1) | Numdam | MR | Zbl

[10] Gehring, F. W. The Carathéodory convergence theorem for quasiconformal mappings in space, Ann. Acad. Sci. Fenn. Ser. A I No., Volume 336/11 (1963), pp. 21 | MR | Zbl

[11] Kobayashi, S. Transformation groups in differential geometry. Reprint of the 1972 edition., Classics in Mathematics, Springer-Verlag, Berlin, 1995 | MR | Zbl

[12] Kuehnel, W.; Rademacher, H.B. Liouville’s theorem in conformal geometry, Pures et Appl. (9) (to appear) (Classics in Mathematics) | MR | Zbl

[13] Obata, M. The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geometry, Volume 6 (1971/72), pp. 247-258 | MR | Zbl

[14] Schoen, R. On the conformal and CR automorphism groups, Geom. Funct. Anal., Volume 5 (1995) no. 2, pp. 464-481 | DOI | EuDML | MR | Zbl

[15] Schottenloher, M. A Mathematical Introduction to Conformal Field Theory, Springer-Verlag, Berlin, 1997 | MR | Zbl

[16] Sharpe, R.W. Differential Geometry : Cartan’s generalization of Klein’s Erlangen Program, Springer, New York, 1997 | MR | Zbl

[17] Vässälä, J. Lectures on n -dimensional quasiconformal mappings, Lecture Notes in Mathematics, 229, Springer-Verlag, Berlin-New York, 1971 | MR | Zbl

[18] Zeghib, A. Sur les actions affines des groupes discrets, Ann. Inst. Fourier, Volume 47 (1997), pp. 641-685 | DOI | EuDML | Numdam | MR | Zbl

[19] Zeghib, A. Isometry groups and geodesic foliations of Lorentz manifolds. I. Foundations of Lorentz dynamics, Geom. Funct. Anal., Volume 9 (1999) no. 4, pp. 775-822 | DOI | MR | Zbl

[20] Zeghib, A. Isometry groups and geodesic foliationsof Lorentz manifolds. II. Geometry of analytic Lorentz manifolds with large isometry groups, Geom. Funct. Anal., Volume 9 (1999) no. 4, pp. 823-854 | DOI | MR | Zbl

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