GEOMETRIC BÄCKLUND 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 cal-culating from one surface of constant negative curvature a pseudospherical line congruence and matching surface. We construct a generalization of these theo-rems 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. BIOGRAPHICAL SKETCH Matthew Noonan grew up in Kansas City, where he learned to play Go. He attended Hampshire College, where he learned to love mathematics. After some time, he completed graduate work at Cornell University, where he learned to build boats and play the banjo. This strikes him as very high quality of life, for which he is infinitely thankful. ii