Correspondence theorems for hierarchies of equations of pseudo-spherical type

Hierarchies of evolution equations of pseudo-spherical type are introduced, thereby generalizing the notion of a single equation describing pseudo-spherical surfaces due to S.S. Chern and K. Tenenblat, and providing a connection between differential geometry and the study of hierarchies of equations which are the integrability condition of sl(2,R)-valued linear problems. As an application, it is shown that there exist local correspondences between any two (suitably generic) solutions of arbitrary hierarchies of equations of pseudo-spherical type.