Pseudo-spherical Submanifolds with Degenerate Bianchi Transformation

Abstract

We describe pseudo-spherical submanifolds with degenerate Bianchi transformation in constant curvature spaces.Mathematics Subject Classification (2010). Primary 53A07.

Generalization of the Bianchi–Bäcklund Transformation of Pseudo-Spherical Surfaces

“…No less interesting is the problem of generalization of the considered subject area to the case of many-dimensional pseudo-spherical surfaces with the degenerated Bäcklund transformation. N and H N . Similarly to the case considered in the previous subsection, in [18] pseudo-spherical surfaces with a degenerated into a line Bianchi transformation are discussed in the sphere S N and Lobachevsky space H N .…”

“…A generalization of these results to the case of n-dimensional pseudo-spherical surfaces with the degenerated into a line Bianchi transformation in N -dimensional Euclidean space is obtained in [18].…”

“…In [16] (see also [14,18]), the problem of the description of pseudo-spherical surfaces with the degenerated Bianchi transformation in the Euclidean space E N is considered for an arbitrary N ≥ 3. The following special class of pseudo-spherical surfaces is introduced.…”

“…Moreover, in [16,18], the constructive methods of constructing generalized Beltrami surfaces are discussed. It is shown that any curve in E N −2 can be locally obtained by means of a degenerate Bianchi transformation of some generalized Beltrami surface in E N .…”

Degenerate Bianchi transformations for three-dimensional pseudo-spherical submanifolds in $\mathbb R^5$

“…A complete description of n-dimensional pseudo-spherical submanifolds in R n+p admitting degenerate Bianchi transformations to one-dimensional curves was proved recently in [11], see also [7], [12]. These submanifolds, which are called generalized Beltrami surfaces, are submanifolds of revolution obtained by rotating particular curves, generalized tractrices.…”