Degenerate Bianchi transformations for three-dimensional pseudo-spherical submanifolds in \(\mathbb{R}^5\) (Q2046655)

The Bianchi transformation concept provides an effective way to construct new pseudospherical submanifolds (i.e., their sectional curvature is constant, \(K\equiv-1\)) from a given one. Bianchi transformation degenerates if its image is a submanifold, whose dimension is less than the dimension of the original submanifold. The authors study three-dimensional pseudo-spherical submanifolds, \(x(u_1,u_2,u_3)\), in \(\mathbb{R}^5\), whose Bianchi transformations are degenerate of rank 2. In this case, the kernel Bianchi transformation is a 1-dimensional distribution; another 1-dimensional distribution is spanned by \(\partial x/\partial u_1\), and the third 1-dimensional distribution is orthogonal to the two distributions above. The degenerate Bianchi transformation is referred to as holonomically degenerate if the three 2-dimensional distributions determined by the above three 1-dimensional distributions are integrable (i.e., are tangent to 2-dimensional foliations). A complete description of such submanifolds is obtained (Theorem 1.2) in the case where the Bianchi transformations are holonomically degenerate. Two illustrative examples are given.