Lie group structures on groups of diffeomorphisms and applications to CR manifolds
Annales de l'Institut Fourier, Volume 54 (2004) no. 5, p. 1279-1303
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.
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.
DOI : https://doi.org/10.5802/aif.2050
Classification:  22E15,  22F50,  32V20,  32V40,  57S25,  58D05
@article{AIF_2004__54_5_1279_0,
     author = {Baouendi, Mohamed 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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {54},
     number = {5},
     year = {2004},
     pages = {1279-1303},
     doi = {10.5802/aif.2050},
     mrnumber = {2127849},
     zbl = {1062.22046},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/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, Volume 54 (2004) no. 5, pp. 1279-1303. doi : 10.5802/aif.2050. https://aif.centre-mersenne.org/item/AIF_2004__54_5_1279_0/

[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 541450 | Zbl 0449.32008

[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 1480988 | Zbl 0943.32021

[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 1721797 | Zbl 0942.32027

[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 1668103 | Zbl 0944.32040

[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 1775734 | Zbl 0958.32033

[BJT85] M.S. Baouendi, H. Jacobowitz & F. Trèves, On the analyticity of CR mappings, Ann. of Math. (2) 122 (1985) p. 365-400 | MR 808223 | Zbl 0583.32021

[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 1916859 | Zbl 1039.32021

[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 18187 | Zbl 0061.04407

[Bo91] A. Boggess, CR manifolds and the tangential Cauchy-Riemann complex, Studies in Advanced Mathematics, CRC Press, 1991 | MR 1211412 | Zbl 0760.32001

[BS77] D. Burns Jr. & S. Shnider, Real hypersurfaces in complex manifolds, XXX, Amer. Math. Soc., 1977, p. 141-168 | Zbl 0422.32016

[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 425155 | Zbl 0302.32015

[DF91] D. Dummit & R. Foote, Abstract algebra, Prentice Hall, Inc., 1991 | MR 1138725 | Zbl 0751.00001

[E01] P. Ebenfelt, Finite jet determination of holomorphic mappings at the boundary., Asian J. Math. 5 (2001) p. 637-662 | MR 1913814 | Zbl 1015.32031

[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 1995799 | Zbl 1032.32025

[GG73] M. Golubitsky & V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics Vol. 14, Springer-Verlag, 1973 | MR 341518 | Zbl 0294.58004

[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 1474199 | Zbl 0892.32015

[KZ02] S.-Y. Kim & D. Zaitsev, Equivalence and embedding problems for CR-structures of any codimension, Preprint, 2002 | MR 2122216 | Zbl 1079.32022

[KZ03] S.-Y. Kim & D. Zaitsev, Remarks on the rigidity of CR-manifolds (in preparation), | Zbl 1101.32018

[Ko72] S. Kobayashi, Transformation groups in differential geometry., Ergebnisse der Mathematik und ihrer Grenzgebiete Band 70, Springer-Verlag, 1972 | MR 355886 | Zbl 0246.53031

[Kw01] R.T. Kowalski, Rational jet dependence of formal equivalences between real-analytic hypersurfaces in 2 , e-print. To appear, Pacific J. Math, http://arXiv.org/abs/math.CV/0108165, 2001 | Zbl 1106.32025

[T67] N. Tanaka, On generalized graded Lie algebras and geometric structures I, J. Math. Soc. Japan 19 (1967) p. 215-254 | MR 221418 | Zbl 0165.56002

[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 945904 | Zbl 0692.58005

[V74] V.S. Varadarajan, Lie groups, Lie algebras, and their representations, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., 1974 | MR 376938 | Zbl 0371.22001

[Z95] D. Zaitsev, On the automorphism groups of algebraic bounded domains, Math. Ann 302 (1995) p. 105-129 | MR 1329449 | Zbl 0823.14005

[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 1492123 | Zbl 0898.32006

[Lu88A.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 945904 | Zbl 0692.58005