Dégénerescence locale des transformations conformes pseudo-riemanniennes

Frances, Charles

Annales de l'Institut Fourier, Tome 62 (2012), p. 1627-1669 / Harvested from Numdam

Résumé

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.

Abstract

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.

Détail

Publié le : 2012-01-01

MR 3025150 | Zbl 1261.53052 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.2732
Classification:  53A30,  53C50
Mots clés: transformations conformes, structures pseudo-riemanniennes.

Bibliographie

[1] Alekseevski, D. Self-similar Lorentzian manifolds, Ann. Global Anal. Geom., Tome 3 (1985) no. 1, pp. 59-84 | Article | MR 812313 | Zbl 0538.53060

[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 2436232 | Zbl 1154.53047

[3] Besse, A Einstein manifolds, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Tome 10 (1987) | Zbl 1147.53001

[4] Čap, A.; Slovák, J.; Žádník, V. On distinguished curves in parabolic geometries, Transform. Groups, Tome 9 (2004) no. 2, pp. 143-166 | Article | MR 2056534 | Zbl 1070.53021

[5] Ferrand, J. Les géodésiques des structures conformes, C. R. Acad. Sci. Paris Sér. I Math., Tome 294 (1982) no. 18, pp. 629-632 | MR 664297 | Zbl 0487.53011

[6] Ferrand, J. Convergence and degeneracy of quasiconformal maps of Riemannian manifolds, J. Anal. Math., Tome 69 (1996), pp. 1-24 | Article | MR 1428092 | Zbl 0871.53031

[7] Ferrand, J. The action of conformal transformations on a Riemannian manifold, Math. Ann., Tome 304 (1996) no. 2, pp. 277-291 | Article | MR 1371767 | Zbl 0866.53027

[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), Tome 40 (2007) no. 5, pp. 741-764 (English version : eprint arXiv :math/0608.537v1) | Numdam | MR 2382860 | Zbl 1135.53016

[10] Gehring, F. W. The Carathéodory convergence theorem for quasiconformal mappings in space, Ann. Acad. Sci. Fenn. Ser. A I No., Tome 336/11 (1963), pp. 21 | MR 160898 | Zbl 0136.38102

[11] Kobayashi, S. Transformation groups in differential geometry. Reprint of the 1972 edition., Springer-Verlag, Berlin, Classics in Mathematics (1995) | MR 1336823 | Zbl 0829.53023

[12] Kuehnel, W.; Rademacher, H.B. Liouville’s theorem in conformal geometry, Pures et Appl. (9) (to appear) (Classics in Mathematics) | MR 2355458 | Zbl 1127.53014

[13] Obata, M. The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geometry, Tome 6 (1971/72), pp. 247-258 | MR 303464 | Zbl 0236.53042

[14] Schoen, R. On the conformal and CR automorphism groups, Geom. Funct. Anal., Tome 5 (1995) no. 2, pp. 464-481 | Article | MR 1334876 | Zbl 0835.53015

[15] Schottenloher, M. A Mathematical Introduction to Conformal Field Theory, Springer-Verlag, Berlin (1997) | MR 1473464 | Zbl 0874.17031

[16] Sharpe, R.W. Differential Geometry : Cartan’s generalization of Klein’s Erlangen Program, Springer, New York (1997) | MR 1453120 | Zbl 0876.53001

[17] Vässälä, J. Lectures on $n$-dimensional quasiconformal mappings, Springer-Verlag, Berlin-New York, Lecture Notes in Mathematics, Tome 229 (1971) | MR 454009 | Zbl 0221.30031

[18] Zeghib, A. Sur les actions affines des groupes discrets, Ann. Inst. Fourier, Tome 47 (1997), pp. 641-685 | Article | Numdam | MR 1450429 | Zbl 0865.57038

[19] Zeghib, A. Isometry groups and geodesic foliations of Lorentz manifolds. I. Foundations of Lorentz dynamics, Geom. Funct. Anal., Tome 9 (1999) no. 4, pp. 775-822 | Article | MR 1719606 | Zbl 0946.53035

[20] Zeghib, A. Isometry groups and geodesic foliationsof Lorentz manifolds. II. Geometry of analytic Lorentz manifolds with large isometry groups, Geom. Funct. Anal., Tome 9 (1999) no. 4, pp. 823-854 | Article | MR 1719610 | Zbl 0946.53036