Lie group structures on groups of diffeomorphisms and applications to CR manifolds

Baouendi, M. Salah; Preiss Rothschild, Linda; Winkelmann, Jörg; Zaitsev, Dimitri

doi:10.5802/aif.2050

Lie group structures on groups of diffeomorphisms and applications to CR manifolds
[Sous-groupes de difféomorphismes, leur structure de groupe de Lie, et applications aux variétés CR]

Baouendi, M. Salah ¹ ; Preiss Rothschild, Linda ; Winkelmann, Jörg ; Zaitsev, Dimitri

¹ University of California, Department of Mathematics, 0112, San Diego, La Jolla, CA 92093-0112 (USA), Université Henri Poincaré Nancy 1, Institut Élie Cartan, B.P. 239, 54506 Vand{\oe}uvre-lès-Nancy Cedex (France), Trinity College, School of Mathematics, Dublin 2 (Irland)

Annales de l'Institut Fourier, Tome 54 (2004) no. 5, pp. 1279-1303.

Résumé
Abstract

Nous donnons des conditions suffisantes pour qu'un sous-groupe donné du groupe des difféomorphismes d'une variété indéfiniment differentiable ou réelle analytique ait une structure compatible de groupe de Lie. En utilisant ces résultats, ainsi que des travaux récents concernant la paramétrisation des automorphismes CR par leur jets en un point et leur systèmes complets, nous donnons des conditions sous lesquelles le groupe des automorphismes CR globaux d'une variété CR est un groupe de Lie relativement à une topologie appropriée.

We give general sufficient conditions to guarantee that a given subgroup of the group of diffeomorphisms of a smooth or real-analytic manifold has a compatible Lie group structure. These results, together with recent work concerning jet parametrization and complete systems for CR automorphisms, are then applied to determine when the global CR automorphism group of a CR manifold is a Lie group in an appropriate topology.

MR Zbl

DOI : 10.5802/aif.2050

Classification : 22E15, 22F50, 32V20, 32V40, 57S25, 58D05

Affiliations des auteurs :

Baouendi, M. Salah ¹ ; Preiss Rothschild, Linda ; Winkelmann, Jörg ; Zaitsev, Dimitri

@article{AIF_2004__54_5_1279_0,
     author = {Baouendi, M. Salah and Preiss Rothschild, Linda and Winkelmann, J\"org and Zaitsev, Dimitri},
     title = {Lie group structures on groups of diffeomorphisms and applications to {CR} manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {1279--1303},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {54},
     number = {5},
     year = {2004},
     doi = {10.5802/aif.2050},
     mrnumber = {2127849},
     zbl = {1062.22046},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2050/}
}

TY  - JOUR
AU  - Baouendi, M. Salah
AU  - Preiss Rothschild, Linda
AU  - Winkelmann, Jörg
AU  - Zaitsev, Dimitri
TI  - Lie group structures on groups of diffeomorphisms and applications to CR manifolds
JO  - Annales de l'Institut Fourier
PY  - 2004
SP  - 1279
EP  - 1303
VL  - 54
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2050/
DO  - 10.5802/aif.2050
LA  - en
ID  - AIF_2004__54_5_1279_0
ER  -

%0 Journal Article
%A Baouendi, M. Salah
%A Preiss Rothschild, Linda
%A Winkelmann, Jörg
%A Zaitsev, Dimitri
%T Lie group structures on groups of diffeomorphisms and applications to CR manifolds
%J Annales de l'Institut Fourier
%D 2004
%P 1279-1303
%V 54
%N 5
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2050/
%R 10.5802/aif.2050
%G en
%F AIF_2004__54_5_1279_0

Baouendi, M. Salah; Preiss Rothschild, Linda; Winkelmann, Jörg; Zaitsev, Dimitri. Lie group structures on groups of diffeomorphisms and applications to CR manifolds. Annales de l'Institut Fourier, Tome 54 (2004) no. 5, pp. 1279-1303. doi : 10.5802/aif.2050. http://www.numdam.org/articles/10.5802/aif.2050/

Bibliographie
Cité par

[AF79] A. Andreotti & G.A. Fredricks, Embeddability of real analytic Cauchy-Riemann manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 6 (1979) p. 285-304 | Numdam | MR | Zbl

[BER97] M.S. Baouendi, P. Ebenfelt & L.P. Rothschild, Parametrization of local biholomorphisms of real analytic hypersurfaces, Asian J. Math 1 (1997) p. 1-16 | MR | Zbl

[BER99a] M.S. Baouendi, P. Ebenfelt & L.P. Rothschild, Rational dependence of smooth and analytic CR mappings on their jets, Math. Ann 315 (1999) p. 205-249 | MR | Zbl

[BER99b] M.S. Baouendi, P. Ebenfelt & L.P. Rothschild, Real submanifolds in complex space and their mappings., Princeton Mathematical Series 47, Princeton University Press, 1999 | MR | Zbl

[BER00] M.S. Baouendi, P. Ebenfelt & L.P. Rothschild, Convergence and finite determination of formal CR mappings, J. Amer. Math. Soc 13 (2000) p. 697-723 | MR | Zbl

[BJT85] M.S. Baouendi, H. Jacobowitz & F. Trèves, On the analyticity of CR mappings, Ann. of Math. (2) 122 (1985) p. 365-400 | MR | Zbl

[BMR02] M.S. Baouendi, N. Mir & L.P. Rothschild, Reflection ideals and mappings between generic submanifolds in complex space, J. Geom. Anal 12 (2002) no.4 p. 543-580 | MR | Zbl

[BRZ04] M.S. Baouendi, L.P. Rothschild & D. Zaitsev, Deformation of generic submanifolds in complex space (in preparation),

[BM46] S. Bochner & D. Montgomery, Locally compact groups of differentiable transformations, Ann. of Math. (2) 47 (1946) p. 639-653 | MR | Zbl

[Bo91] A. Boggess, CR manifolds and the tangential Cauchy-Riemann complex, Studies in Advanced Mathematics, CRC Press, 1991 | MR | Zbl

[BS77] D. Burns Jr. & S. Shnider, Real hypersurfaces in complex manifolds, XXX, Amer. Math. Soc., 1977, p. 141-168 | Zbl

[C32] E. Cartan, Sur la géométrie pseudo-conforme des hypersurfaces de deux variables complexes I, II, Œuvres II-2 (1932) p. 1217-1238

[CM74] S.S. Chern & J.K. Moser, Real hypersurfaces in complex manifolds, Acta Math 133 (1974) p. 219-271 | MR | Zbl

[DF91] D. Dummit & R. Foote, Abstract algebra, Prentice Hall, Inc., 1991 | MR | Zbl

[E01] P. Ebenfelt, Finite jet determination of holomorphic mappings at the boundary., Asian J. Math. 5 (2001) p. 637-662 | MR | Zbl

[ELZ03] P. Ebenfelt, B. Lamel & D. Zaitsev, Finite jet determination of local analytic CR automorphisms and their parametrization by $2$ -jets in the finite type case, Geom. Funct. Anal. 13 (2003) no.3 p. 546-573 | MR | Zbl

[GG73] M. Golubitsky & V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics Vol. 14, Springer-Verlag, 1973 | MR | Zbl

[H97] C.-K. Han, Complete differential system for the mappings of CR manifolds of nondegenerate Levi forms, Math. Ann 309 (1997) p. 401-409 | MR | Zbl

[KZ02] S.-Y. Kim & D. Zaitsev, Equivalence and embedding problems for CR-structures of any codimension, Preprint, 2002 | MR | Zbl

[KZ03] S.-Y. Kim & D. Zaitsev, Remarks on the rigidity of CR-manifolds (in preparation), | Zbl

[Ko72] S. Kobayashi, Transformation groups in differential geometry., Ergebnisse der Mathematik und ihrer Grenzgebiete Band 70, Springer-Verlag, 1972 | MR | Zbl

[Kw01] R.T. Kowalski, Rational jet dependence of formal equivalences between real-analytic hypersurfaces in $ℂ^{2}$ , e-print. To appear, Pacific J. Math, http://arXiv.org/abs/math.CV/0108165, 2001 | Zbl

[T67] N. Tanaka, On generalized graded Lie algebras and geometric structures I, J. Math. Soc. Japan 19 (1967) p. 215-254 | MR | Zbl

[Tu88] A.E. Tumanov, Extension of CR-functions into a wedge from a manifold of finite type (Russian), Mat. Sb. (N.S.) 136(178) (1988) p. 128-139 | MR | Zbl

[V74] V.S. Varadarajan, Lie groups, Lie algebras, and their representations, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., 1974 | MR | Zbl

[Z95] D. Zaitsev, On the automorphism groups of algebraic bounded domains, Math. Ann 302 (1995) p. 105-129 | MR | Zbl

[Z97] D. Zaitsev, Germs of local automorphisms of real analytic CR structures and analytic dependence on the $k$ -jets, Math. Res. Lett 4 (1997) p. 823-842 | MR | Zbl

[<L>Lu88</L] A.E. Tumanov, Extension of CR-functions into a wedge from a manifold of finite type, Math. USSR-Sb. (translation) 64 (1989) p. 129-140 | MR | Zbl

Cité par Sources :