Bäcklund maps and Lie–Bäcklund transformations as differential-geometric structures

Abstract

This paper is an exposition of the author’s report prepared for the International Conference devoted to the centennial anniversary of G. F. Laptev (Laptev seminar–2009). In the first section, we consider Bäcklund transformations of second-order partial differential equations. In the present work, the theory of Bäcklund transformations is treated as a special branch of the theory of connections. The second section is devoted to differential-geometric structures generated by the so-called Lie–Bäcklund transformations (or, equivalently, contact transformations of higher order) that are a special case of diffeomorphisms between the manifolds of holonomic jets. Recall that it was G. F. Laptev who pointed out the possibility of considering differentiable mappings as differential-geometric structures.