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**Lie group structures on groups of diffeomorphisms and applications to CR manifolds.**
*(English)*
Zbl 1062.22046

The authors obtain some sufficient geometric conditions on a CR manifold \(M\) to guarantee that the group of all its smooth CR automorphisms has the structure of a (finite dimensional) Lie group compatible with its natural topology. First, they establish general theorems on Lie group structures for subgroups of diffeomorphisms of a given smooth real-analytic manifold, then they apply these results together with recent work concerning jet parametrization and complete systems for CR automorphisms. The authors define the notion of a complete system for a set of diffeomorphisms of a manifold and show that it is equivalent to the notion of jet parametrization. Similar results are obtained for subsets of germs of diffeomorphisms fixing a point. Some partial converse results are obtained. Next, the authors present the basic definitions and properties of abstract and embedded CR manifolds that guarantee the existence of a Lie group structure on the group of CR automorphisms.

Reviewer: Vasile Oproiu (Iaşi)

### MSC:

22F50 | Groups as automorphisms of other structures |

57S25 | Groups acting on specific manifolds |

58D05 | Groups of diffeomorphisms and homeomorphisms as manifolds |

32V40 | Real submanifolds in complex manifolds |

22E15 | General properties and structure of real Lie groups |

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\textit{M. S. Baouendi} et al., Ann. Inst. Fourier 54, No. 5, 1279--1303 (2004; Zbl 1062.22046)

### References:

[1] | 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 · Zbl 0449.32008 |

[2] | M.S. Baouendi, P. Ebenfelt & L.P. Rothschild, Parametrization of local biholomorphisms of real analytic hypersurfaces, Asian J. Math1 (1997) p. 1-16 · Zbl 0943.32021 |

[3] | M.S. Baouendi, P. Ebenfelt & L.P. Rothschild, Rational dependence of smooth and analytic CR mappings on their jets, Math. Ann315 (1999) p. 205-249 · Zbl 0942.32027 |

[4] | M.S. Baouendi, P. Ebenfelt & L.P. Rothschild, Real submanifolds in complex space and their mappings., Princeton Mathematical Series 47, Princeton University Press, 1999 · Zbl 0944.32040 |

[5] | M.S. Baouendi, P. Ebenfelt & L.P. Rothschild, Convergence and finite determination of formal CR mappings, J. Amer. Math. Soc13 (2000) p. 697-723 · Zbl 0958.32033 |

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

[7] | M.S. Baouendi, N. Mir & L.P. Rothschild, Reflection ideals and mappings between generic submanifolds in complex space, J. Geom. Anal12 (2002) no. p. 543-580 · Zbl 1039.32021 |

[8] | M.S. Baouendi, L.P. Rothschild & D. Zaitsev, Deformation of generic submanifolds in complex space (in preparation), · Zbl 1129.32019 |

[9] | S. Bochner & D. Montgomery, Locally compact groups of differentiable transformations, Ann. of Math. (2)47 (1946) p. 639-653 · Zbl 0061.04407 |

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

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

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

[13] | S.S. Chern & J.K. Moser, Real hypersurfaces in complex manifolds, Acta Math133 (1974) p. 219-271 · Zbl 0302.32015 |

[14] | D. Dummit & R. Foote, Abstract algebra, Prentice Hall, Inc., 1991 · Zbl 0751.00001 |

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

[16] | 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. p. 546-573 · Zbl 1032.32025 |

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

[18] | C.-K. Han, Complete differential system for the mappings of CR manifolds of nondegenerate Levi forms, Math. Ann309 (1997) p. 401-409 · Zbl 0892.32015 |

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

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

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

[22] | R.T. Kowalski, Rational jet dependence of formal equivalences between real-analytic hypersurfaces in \(\mathbb{C}^2,\) e-print. To appear, Pacific J. Math, http://arXiv.org/abs/math.CV/0108165, 2001 · Zbl 1106.32025 |

[23] | N. Tanaka, On generalized graded Lie algebras and geometric structures I, J. Math. Soc. Japan19 (1967) p. 215-254 · Zbl 0165.56002 |

[24] | 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 · Zbl 0692.58005 |

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

[26] | D. Zaitsev, On the automorphism groups of algebraic bounded domains, Math. Ann302 (1995) p. 105-129 · Zbl 0823.14005 |

[27] | D. Zaitsev, Germs of local automorphisms of real analytic CR structures and analytic dependence on the \(k\)-jets, Math. Res. Lett4 (1997) p. 823-842 · Zbl 0898.32006 |

[28] | 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 · Zbl 0692.58005 |

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