Skip to main content
SpringerLink
Log in

Representation of the group of holomorphic symmetries of a real germ in the symmetry group of the model surface of the germ

  • Short Communications
  • Published:
Russian Journal of Mathematical Physics Aims and scope Submit manuscript

Abstract

There is a series of papers devoted to the construction and investigation of local polynomial models of real submanifolds of complex space (see, e.g., [1]). Among the main properties of model surfaces, we can mention the following fact. The dimension of the local group of holomorphic symmetries of a germ does not exceed the dimension of the similar group for the tangent model surface of the germ. In the present note, this assertion is presented in a much stronger form. A faithful representation of the Lie algebra of infinitesimal automorphisms of a germ in the Lie algebra of the tangent model surface is constructed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. K. Beloshapka, “Real Submanifolds of a Complex Space: Their Polynomial Models, Automorphisms, and Classification Problems,” Uspekhi Mat. Nauk 57(1), 3–44 (2002) [Russ. Math. Surv. 57 (1), 1–41 (2002)].

    MathSciNet  Google Scholar 

  2. H. Poincaré, “Les fonctions analytiques de deux variables et la représentation conforme,” Rend. Circ. Mat. Palermo 23, 185–220 (1907).

    Article  MATH  Google Scholar 

  3. V. K. Beloshapka, “A Universal Model for a Real Submanifold,” Mat. Zametki 75(4), 507–522 (2004) [Math. Notes 75 (3–4), 475–488 (2004)].

    MathSciNet  Google Scholar 

  4. M. S. Baouendi, L. P. Rothschild, J. Winkelmann, and D. Zaitsev, “Lie Group Structures of Groups of Diffeomorphisms and Applications to CR Manifolds,” Ann. Inst. Fourier (Grenoble) 54(5), 1279–1303 (2004).

    MATH  MathSciNet  Google Scholar 

  5. A. G. Vitushkin, “Holomorphic Mappings and the Geometry of Surfaces,” in Current Problems in Mathematics. Fundamental Directions, Vol. 7 (Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985), pp. 167–226.

    Google Scholar 

  6. R. V. Gammel’ and I. G. Kossovskii, “The Envelope of Holomorphy of a Model Third-Degree Surface and the Rigidity Phenomenon,” Tr. Mat. Inst. Steklova 253, 30–45 (2006) [Proc. Steklov Inst. Math. 253, 22–36 (2006)].

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanics and Mathematics, Moscow State University, Vorob’evy gory, Moscow, 119992, Russia

    V. K. Beloshapka

Authors
  1. V. K. Beloshapka
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Financially supported by the RBRF under grants no. 05-01-0981 and NSh-2040.2003.1.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Beloshapka, V.K. Representation of the group of holomorphic symmetries of a real germ in the symmetry group of the model surface of the germ. Russ. J. Math. Phys. 14, 213–215 (2007). https://doi.org/10.1134/S1061920807020100

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1061920807020100

Keywords

Search

Navigation