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

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

  • [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 Vandœuvre-lès-Nancy Cedex (France), Trinity College, School of Mathematics, Dublin 2 (Irland)

Annales de l’institut Fourier (2004)

  • Volume: 54, Issue: 5, page 1279-1303
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Baouendi, M. Salah, et al. "Lie group structures on groups of diffeomorphisms and applications to CR manifolds." Annales de l’institut Fourier 54.5 (2004): 1279-1303. <http://eudml.org/doc/116143>.

@article{Baouendi2004,
abstract = {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.},
affiliation = {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 Vandœuvre-lès-Nancy Cedex (France), Trinity College, School of Mathematics, Dublin 2 (Irland)},
author = {Baouendi, M. Salah, Preiss Rothschild, Linda, Winkelmann, Jörg, Zaitsev, Dimitri},
journal = {Annales de l’institut Fourier},
keywords = {lie group; CR manifold; CR automorphism; jet parametrization; complete system; Lie group; CR manifolds},
language = {eng},
number = {5},
pages = {1279-1303},
publisher = {Association des Annales de l'Institut Fourier},
title = {Lie group structures on groups of diffeomorphisms and applications to CR manifolds},
url = {http://eudml.org/doc/116143},
volume = {54},
year = {2004},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 5
SP - 1279
EP - 1303
AB - 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.
LA - eng
KW - lie group; CR manifold; CR automorphism; jet parametrization; complete system; Lie group; CR manifolds
UR - http://eudml.org/doc/116143
ER -

References

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  1. A. Andreotti, G.A. Fredricks, Embeddability of real analytic Cauchy-Riemann manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 6 (1979), 285-304 Zbl0449.32008MR541450
  2. M.S. Baouendi, P. Ebenfelt, L.P. Rothschild, Parametrization of local biholomorphisms of real analytic hypersurfaces, Asian J. Math 1 (1997), 1-16 Zbl0943.32021MR1480988
  3. M.S. Baouendi, P. Ebenfelt, L.P. Rothschild, Rational dependence of smooth and analytic CR mappings on their jets, Math. Ann 315 (1999), 205-249 Zbl0942.32027MR1721797
  4. M.S. Baouendi, P. Ebenfelt, L.P. Rothschild, Real submanifolds in complex space and their mappings., 47 (1999), Princeton University Press, Princeton, NJ Zbl0944.32040MR1668103
  5. M.S. Baouendi, P. Ebenfelt, L.P. Rothschild, Convergence and finite determination of formal CR mappings, J. Amer. Math. Soc 13 (2000), 697-723 Zbl0958.32033MR1775734
  6. M.S. Baouendi, H. Jacobowitz, F. Trèves, On the analyticity of CR mappings, Ann. of Math. (2) 122 (1985), 365-400 Zbl0583.32021MR808223
  7. M.S. Baouendi, N. Mir, L.P. Rothschild, Reflection ideals and mappings between generic submanifolds in complex space, J. Geom. Anal 12 (2002), 543-580 Zbl1039.32021MR1916859
  8. M.S. Baouendi, L.P. Rothschild, D. Zaitsev, Deformation of generic submanifolds in complex space (in preparation) Zbl1129.32019
  9. S. Bochner, D. Montgomery, Locally compact groups of differentiable transformations, Ann. of Math. (2) 47 (1946), 639-653 Zbl0061.04407MR18187
  10. A. Boggess, CR manifolds and the tangential Cauchy-Riemann complex, (1991), CRC Press, Boca Raton, FL Zbl0760.32001MR1211412
  11. D. Burns Jr., S. Shnider, Real hypersurfaces in complex manifolds, Several complex variables. Proc. Sympos. Pure Math. Part 2. (Williams Coll., Williamstown, Mass, 1975) XXX (1977), 141-168, Amer. Math. Soc., Providence, R.I Zbl0422.32016
  12. E. Cartan, Sur la géométrie pseudo-conforme des hypersurfaces de deux variables complexes I, II, Œuvres II-2 (1932), 1217-1238 Zbl58.1256.03
  13. S.S. Chern, J.K. Moser, Real hypersurfaces in complex manifolds, Acta Math 133 (1974), 219-271 Zbl0302.32015MR425155
  14. D. Dummit, R. Foote, Abstract algebra, (1991), Prentice Hall, Inc., Englewood Cliffs, NJ Zbl0751.00001MR1138725
  15. P. Ebenfelt, Finite jet determination of holomorphic mappings at the boundary., Asian J. Math. 5 (2001), 637-662 Zbl1015.32031MR1913814
  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), 546-573 Zbl1032.32025MR1995799
  17. M. Golubitsky, V. Guillemin, Stable mappings and their singularities, Vol. 14 (1973), Springer-Verlag, New York-Heidelberg Zbl0294.58004MR341518
  18. C.-K. Han, Complete differential system for the mappings of CR manifolds of nondegenerate Levi forms, Math. Ann 309 (1997), 401-409 Zbl0892.32015MR1474199
  19. S.-Y. Kim, D. Zaitsev, Equivalence and embedding problems for CR-structures of any codimension, (2002) Zbl1079.32022MR2122216
  20. S.-Y. Kim, D. Zaitsev, Remarks on the rigidity of CR-manifolds (in preparation) Zbl1101.32018
  21. S. Kobayashi, Transformation groups in differential geometry., Band 70 (1972), Springer-Verlag, New York-Heidelberg Zbl0246.53031MR355886
  22. R.T. Kowalski, Rational jet dependence of formal equivalences between real-analytic hypersurfaces in ℂ 2 , (2001) Zbl1106.32025
  23. N. Tanaka, On generalized graded Lie algebras and geometric structures I, J. Math. Soc. Japan 19 (1967), 215-254 Zbl0165.56002MR221418
  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), 128-139 Zbl0692.58005MR945904
  25. V.S. Varadarajan, Lie groups, Lie algebras, and their representations, (1974), Prentice-Hall, Inc., Englewood Cliffs, N.J Zbl0371.22001MR376938
  26. D. Zaitsev, On the automorphism groups of algebraic bounded domains, Math. Ann 302 (1995), 105-129 Zbl0823.14005MR1329449
  27. D. Zaitsev, Germs of local automorphisms of real analytic CR structures and analytic dependence on the k -jets, Math. Res. Lett 4 (1997), 823-842 Zbl0898.32006MR1492123
  28. A.E. Tumanov, Extension of CR-functions into a wedge from a manifold of finite type, Math. USSR-Sb. (translation) 64 (1989), 129-140 Zbl0692.58005MR945904

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