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

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 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.