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

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{IVM_2013_7_a2,
     author = {A. K. Rybnikov},
     title = {An affine interpretation of {B\"acklund} maps},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {31--44},
     publisher = {mathdoc},
     number = {7},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2013_7_a2/}
}
TY  - JOUR
AU  - A. K. Rybnikov
TI  - An affine interpretation of B\"acklund maps
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2013
SP  - 31
EP  - 44
IS  - 7
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2013_7_a2/
LA  - ru
ID  - IVM_2013_7_a2
ER  - 
%0 Journal Article
%A A. K. Rybnikov
%T An affine interpretation of B\"acklund maps
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2013
%P 31-44
%N 7
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2013_7_a2/
%G ru
%F IVM_2013_7_a2
A. K. Rybnikov. An affine interpretation of B\"acklund maps. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2013), pp. 31-44. http://geodesic.mathdoc.fr/item/IVM_2013_7_a2/

[1] Bianchi L., “Ricerche sulle superficie a curvatura constante e sulle elicoidi”, Ann. Scuola Norm. Sup. Pisa, 2 (1879), 285–341 | MR

[2] Lie S., “Zur Theorie der Flächen konstanter Krümmung, III”, Arch. Math. Naturvidensk., 5:3 (1880), 282–306 | Zbl

[3] Bäcklund A. V., “Zur Theorie der Differentialgleichungen erster Ordnung”, Clebsch Ann., 17 (1880), 285–328 | MR

[4] Darboux G., Lecons sur la Théorie Générale des Surfaces, Part 3, Gauthier-Villars, Paris, 1894 | Zbl

[5] Pirani F. A. E., Robinson D. C., “Sur la définition des transformationa de Bäcklund”, C. R. Acad. Sci. Paris Serie A, 285 (1977), 581–583 | MR | Zbl

[6] Pirani F. A. E., Robinson D. C., Shadwick W. F., Local jet-bundle formulation of Bäcklund transformations, Reidal, Dordrecht, Holland, 1979 | MR | Zbl

[7] Rogers C., Shadwick W. F., Bäcklund transformations and their applications, Academic Press, New York–London, 1982 | MR | Zbl

[8] Rybnikov A. K., “Otobrazheniya Beklunda s tochki zreniya teorii svyaznostei”, Uchen. zap. Kazansk. un-ta. Ser. fiz.-matem. nauki, 151, no. 4, 2009, 93–115 | Zbl

[9] Rybnikov A. K., “Otobrazheniya Beklunda i preobrazovaniya Li–Beklunda kak differentsialno-geometricheskie struktury”, Fundament. i prikl. matem., 16:1 (2010), 135–150 | MR

[10] Vasilev A. M., Teoriya differentsialno-geometricheskikh struktur, Izd-vo MGU, M., 1987 | MR

[11] Rybnikov A. K., “O spetsialnykh svyaznostyakh, opredelyayuschikh predstavleniya nulevoi krivizny dlya evolyutsionnykh uravnenii vtorogo poryadka”, Izv. vuzov. Matem., 1999, no. 9, 32–41 | MR | Zbl

[12] Finikov S. P., Teoriya kongruentsii, GITTL, M.–L., 1950 | MR

[13] Evtushik L. E., Lumiste Yu. G., Ostianu N. M., Shirokov A. P., “Differentsialno-geometricheskie struktury na mnogoobraziyakh”, Itogi nauki i tekhn. Problemy geometrii, 9, VINITI, M., 1979, 5–246 | MR | Zbl

[14] Laptev G. F., “Differentsialnaya geometriya pogruzhennykh mnogoobrazii. Teoretiko-gruppovoi metod differentsialno-geometricheskikh issledovanii”, Tr. Moskovsk. matem. o-va, 2, 1953, 275–382 | MR | Zbl

[15] Laptev G. F., “Teoretiko-gruppovoi metod differentsialno-geometricheskikh issledovanii”, Tr. 3-go Vsesoyuzn. matem. s'ezda (Moskva, 1956), v. 3, AN SSSR, M., 1958, 409–418

[16] Laptev G. F., “Osnovnye infinitezimalnye struktury vysshikh poryadkov na gladkom mnogoobrazii”, Tr. geometrich. semin., 1, VINITI, M., 1966, 139–189 | MR | Zbl

[17] Laptev G. F., “Strukturnye uravneniya glavnogo rassloennogo mnogoobraziya”, Tr. geometrich. semin., 2, VINITI, M., 1969, 161–178 | MR | Zbl

[18] Laptev G. F., “K invariantnoi teorii differentsiruemykh otobrazhenii”, Tr. geometrich. semin., 6, VINITI, M., 1974, 37–42 | MR | Zbl