FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2010, VOLUME 16, NUMBER 1, PAGES 135-150

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

A. K. Rybnikov

Abstract

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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.

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Last modified: March 11, 2011