Geodesic
An affine interpretation of Bäcklund maps
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2013), pp. 31-44

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We consider an affine interpretation of Bäcklund maps for second-order differential equations with an unknown function of two arguments. (Note that Bäcklund transformations represent a special case of Bäcklund maps.) Until now, no one has interpreted Bäcklund transformations as transformations of surfaces in a space different from the Euclidean one. In this paper we consider only the so-called Bäcklund maps of class 1. We represent solutions of differential equations as surfaces in an affine space with an induced connection defining a representation of zero curvature. We prove that if a second-order differential equation admits a Bäcklund map of class 1, then for every solution of this equation there exists a congruence of straight lines in an affine space generated by tangents to the affine image of the solution. This congruence is an affine analog of the parabolic congruence in a Euclidean space. One can interpret a Bäcklund map as a transformation of surfaces in the affine space such that the affine image of the solution of the given differential equation is mapped to a certain boundary surface of the congruence.
Keywords: Bäcklund transformations, Bäcklund maps, connection in principal fiber manifold, connection in associated fiber manifold, connections defining representations of zero curvature. 
A. K. Rybnikov. An affine interpretation of Bäcklund maps. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2013), pp. 31-44. http://geodesic.mathdoc.fr/item/IVM_2013_7_a2/
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