Brief communications

On complete list of affinely homogeneous surfaces of ($\varepsilon,0$)-types in the space $\mathbb C^3$

A. V. Loboda, A. V. Shipovskaya

Chair of Higher Mathematics, Voronezh State University of Architecture and Civil Engineering, 84 20-letiya Oktyabrya str., Voronezh, 394006 Russia

Abstract: The problem of describing the affine homogeneous real hypersurfaces in complex spaces is an important part of a difficult problem of holomorphic classification for homogeneous manifolds, that have not a complete solution till now, even in the $3$-dimensional case.
The scheme developed by the authors which uses canonical affine equations and techniques of matrix Lie algebras, previously allowed to obtain a full description of two natural classes of affine-homogeneous real hypersurfaces in $3$-dimensional complex space. In this paper we present the complete description of one another class. It comprises the known partial examples, and (obtained with the use of symbolic computations) Lie algebras corresponding to the remaining homogeneous manifolds of discussed types. The main result is obtained by means of the integration of these algebras.

Keywords: complex space, affine transformation, homogeneous manifold, vector field, Lie algebra, canonical equation.

UDC: 517.55

Presented by the member of Editorial Board: V. V. Shurygin
Received: 13.11.2014

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2015, 59:6, 62–67

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