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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.

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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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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