Geometric Backlund Transformations In Homogeneous Spaces
A classical theorem of Bianchi states that two surfaces in space are the focal surfaces of a pseudospherical line congruence only if each surface has constant negative Gaussian curvature. Lie constructed a partial converse, explicitly calculating from one surface of constant negative curvature a pseudospherical line congruence and matching surface. We construct a generalization of these theorems to submanifolds of arbitrary homogeneous spaces. Applications are given to surfaces in the classical space forms and in a novel geometry related to the group of Lie sphere transformations.
dissertation or thesis