# 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

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topBaouendi, 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

top- 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
- 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
- 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
- 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
- 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
- M.S. Baouendi, H. Jacobowitz, F. Trèves, On the analyticity of CR mappings, Ann. of Math. (2) 122 (1985), 365-400 Zbl0583.32021MR808223
- 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
- M.S. Baouendi, L.P. Rothschild, D. Zaitsev, Deformation of generic submanifolds in complex space (in preparation) Zbl1129.32019
- S. Bochner, D. Montgomery, Locally compact groups of differentiable transformations, Ann. of Math. (2) 47 (1946), 639-653 Zbl0061.04407MR18187
- A. Boggess, CR manifolds and the tangential Cauchy-Riemann complex, (1991), CRC Press, Boca Raton, FL Zbl0760.32001MR1211412
- 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
- E. Cartan, Sur la géométrie pseudo-conforme des hypersurfaces de deux variables complexes I, II, Œuvres II-2 (1932), 1217-1238 Zbl58.1256.03
- S.S. Chern, J.K. Moser, Real hypersurfaces in complex manifolds, Acta Math 133 (1974), 219-271 Zbl0302.32015MR425155
- D. Dummit, R. Foote, Abstract algebra, (1991), Prentice Hall, Inc., Englewood Cliffs, NJ Zbl0751.00001MR1138725
- P. Ebenfelt, Finite jet determination of holomorphic mappings at the boundary., Asian J. Math. 5 (2001), 637-662 Zbl1015.32031MR1913814
- 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
- M. Golubitsky, V. Guillemin, Stable mappings and their singularities, Vol. 14 (1973), Springer-Verlag, New York-Heidelberg Zbl0294.58004MR341518
- C.-K. Han, Complete differential system for the mappings of CR manifolds of nondegenerate Levi forms, Math. Ann 309 (1997), 401-409 Zbl0892.32015MR1474199
- S.-Y. Kim, D. Zaitsev, Equivalence and embedding problems for CR-structures of any codimension, (2002) Zbl1079.32022MR2122216
- S.-Y. Kim, D. Zaitsev, Remarks on the rigidity of CR-manifolds (in preparation) Zbl1101.32018
- S. Kobayashi, Transformation groups in differential geometry., Band 70 (1972), Springer-Verlag, New York-Heidelberg Zbl0246.53031MR355886
- R.T. Kowalski, Rational jet dependence of formal equivalences between real-analytic hypersurfaces in ${\u2102}^{2}$, (2001) Zbl1106.32025
- N. Tanaka, On generalized graded Lie algebras and geometric structures I, J. Math. Soc. Japan 19 (1967), 215-254 Zbl0165.56002MR221418
- 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
- V.S. Varadarajan, Lie groups, Lie algebras, and their representations, (1974), Prentice-Hall, Inc., Englewood Cliffs, N.J Zbl0371.22001MR376938
- D. Zaitsev, On the automorphism groups of algebraic bounded domains, Math. Ann 302 (1995), 105-129 Zbl0823.14005MR1329449
- 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
- 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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