Affine-homogeneous real hypersurfaces of tube type in ℂ3

Abstract

In the present article, the scheme is proposed for constructing a complete description of affine-homogeneous strictly pseudo-convex (SPC) real hypersurfaces of 3-dimensional complex space. The scheme is illustrated by the example of the class of tube-type hypersurfaces that generalizes the known class of tubular manifolds. The estimate is obtained for the dimension of affine transformations group acting on such a surface. It is shown that all homogeneous surfaces of tube type with rich (7-dimensional and 6-dimensional) symmetry groups are affinely equivalent to tubular surfaces (tubes). It is proved that every affine-homogeneous SPC-tube of the space ℂ3 is affinely equivalent to a tube with homogeneous base. Examples are given of affine-homogeneous surfaces of tubular type with 5-dimensional groups that could not be reduced to tubes by means of affine transformations.