RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
General information
Latest issue
Archive

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Mathematical Physics and Computer Simulation:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2014, Issue 4(23), Pages 11–35 (Mi vvgum58)  

This article is cited in 1 scientific paper (total in 1 paper)

Mathematics

On dimensions of affine transformation groups transitively acting on a real hypersurfaces in $\Bbb C^3$

A. V. Loboda

Voronezh State Academy of Building and Architecture

Abstract: The main content of the article constitute the three theorems related to the dimensions of homogeneous manifolds.
Traditionally we mean homogeneity as the existence of a local Lie group, that acts transitively on the manifold under consideration near a selected point. As acting groups, in affine homogeneity case only a subgroups of the group Aff $ (3, \Bbb C)$ are considered. The main instruments of the article are the affine canonical equations for the studied homogeneous surfaces and the Lie algebras of affine vector fields tangent to these manifolds.
The concept of affine homogeneity is closely related, in the case of real hypersurfaces, to holomorphic homogeneity, that is natural for multidimensional complex analysis. But even for the hypersurfaces of $3$-dimensional complex spaces the classification problems in both cases (affine and holomorphic) do not arise until the complete solution. One of the things that could help to obtain such a solution is to understand the situation with possible dimensions of a Lie groups (and Lie algebras) which acts transitively on the homogeneous manifolds under consideration. In this paper the dependence is studied of such dimension from a couple of the Taylor coefficients of the $2$-nd order (specifying the type of surface) of the canonical equation of strictly pseudo convex (SPC) hypersurface.
In this article we obtain (in Theorem 1) a general estimate for the dimension of such groups for an arbitrary strictly pseudo convex affinely homogeneous hypersurfaces in complex space $ \Bbb C^3 $. For one of the several types of homogeneous surfaces, this estimate is sharp: the affine transformation group with the maximal possible dimension $10$ acts transitively on the quadric $ Im   w = | z_1 |^ 2 + | z_2 |^ 2 $.
For affinely homogeneous surfaces, that are not equivalent to this quadric, the dimension of such affine group does not exceed $7$. The proof of this statement is given in the article (in Theorem 2) for the surfaces of the type $(1/2, 0)$. To prove this assertion we use the coefficients structure of affine vector fields tangent to homogeneous surfaces. The description of this structure is also obtained in the paper.
In Theorem 3, the describing problem for the affine homogeneous surfaces of the type $(1/2,0)$ with «rich» symmetry groups in space $ \Bbb C^3 $ is reduced to the study of only $5$-dimensional Lie groups and algebras. Note that in the general context of the homogeneity problem such a reduction is not possible; here the specificity of the studied type of the surfaces plays the crucial role.
Note that a complete description of affinely homogeneous hypersurfaces of the type $(1/2, 0)$ in the space $ \Bbb C^3 $ with $5$-dimensional algebras of tangent vector fields is presented in a joint paper of the author (ArXiv.org, 2014). Because of the sufficiently large dimensions the scheme of the homogeneous varieties description, as well as this article considerations, imply significant use of computer symbolic calculations.

Keywords: affine transformation, homogeneous manifold, vector field, Lie algebra, canonical equation of surface.

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00709


Full text: PDF file (418 kB)
References: PDF file   HTML file
UDC: 517.55+514.74
BBK: 22.161.5+22.151.5

Citation: A. V. Loboda, “On dimensions of affine transformation groups transitively acting on a real hypersurfaces in $\Bbb C^3$”, Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2014, no. 4(23), 11–35

Citation in format AMSBIB
\Bibitem{Lob14}
\by A.~V.~Loboda
\paper On dimensions of affine transformation groups transitively acting
on a real hypersurfaces in $\Bbb C^3$
\jour Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica
\yr 2014
\issue 4(23)
\pages 11--35
\mathnet{http://mi.mathnet.ru/vvgum58}


Linking options:
  • http://mi.mathnet.ru/eng/vvgum58
  • http://mi.mathnet.ru/eng/vvgum/y2014/i4/p11

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. V. Loboda, A. V. Shipovskaya, “Ob affinno-odnorodnykh veschestvennykh giperpoverkhnostyakh obschego polozheniya v $\Bbb C^3$”, Matematicheskaya fizika i kompyuternoe modelirovanie, 20:3 (2017), 111–135  mathnet  crossref
  • Mathematical Physics and Computer Simulation
    Number of views:
    This page:125
    Full text:68
    References:27

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020